diff --git "a/-9A0T4oBgHgl3EQfPP_v/content/tmp_files/load_file.txt" "b/-9A0T4oBgHgl3EQfPP_v/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/-9A0T4oBgHgl3EQfPP_v/content/tmp_files/load_file.txt" @@ -0,0 +1,896 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf,len=895 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='02174v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='PR] 5 Jan 2023 Large time behavior of semilinear stochastic partial differential equations perturbed by a mixture of Brownian and fractional Brownian motions Marco Dozzi∗ Ekaterina T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Kolkovska† Jos´e A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' L´opez-Mimbela† Rim Touibi‡ Abstract We study the trajectorywise blowup behavior of a semilinear partial differential equation that is driven by a mixture of multiplicative Brownian and fractional Brownian motion, modeling different types of random perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The linear operator is supposed to have an eigenfunction of constant sign, and we show its influence, as well as the influence of its eigenvalue and of the other parameters of the equation, on the occurrence of a blowup in finite time of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' We give estimates for the probability of finite time blowup and of blowup before a given fixed time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Essential tools are the mild and weak form of an associated random partial differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Keywords Stochastic reaction-diffusion equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' mixed fractional noise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' finite-time blowup of trajectories AMS Mathematics Subject Classification 60H15 60G22 35R60 35B40 35B44 35K58 1 Introduction In this paper we study existence, uniqueness and the blowup behavior of solutions to the fractional stochastic partial differential equation of the form du(x, t) = �1 2k2(t)Lu(x, t) + g(u(x, t)) � dt + u(x, t) dNt, x ∈ D, t > 0, u(x, 0) = ϕ(x) ≥ 0, u(x, t) = 0, x ∈ ∂D, t ≥ 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) where D ⊂ Rd is a bounded Lipschitz domain, L is the infinitesimal generator of a strongly continuous semigroup of contractions which satisfies conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='18), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='19) below, and ϕ ∈ L∞(D), where L∞(D) is the space of real-valued essentially bounded functions on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Additionally, g is a nonnegative locally Lipschitz function and N is a process given by Nt = � t 0 a(s) dB(s) + � t 0 b(s) dBH(s), t ≥ 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='2) ∗corresponding author, marco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='dozzi@univ-lorraine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='fr, UMR-CNRS 7502, Institut Elie Cartan de Lorraine, Nancy, France †Centro de Investigaci´on en Matem´aticas, Guanajuato, Mexico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' ‡UMR-CNRS 7502, Institut Elie Cartan de Lorraine, Nancy, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 1 where B is Brownian motion and BH is fractional Brownian motion with Hurst parameter H > 1/2, a is continuous and b is H¨older continuous of order α > 1 − H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Both, B and BH, are supposed to be defined on a filtered probability space (Ω, F, (Ft, t ≧ 0), P) and adapted to the filtration (Ft, t ≧ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Such models have recently been studied under the name of ‘mixed models’ in the context of stochastic differential equations, see [19] and [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' When N = 0, L = ∆, k = 1, g(u) = u1+β we obtain the classical Fujita equation which was studied in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In [7] and [1] there were considered the cases when N is a Brownian motion, in [5] it was investigated the case when N is a fractional Brownian motion with Hurst parameter H > 1/2 and D ⊂ Rd, and in [6] the case of H ≥ 1/2 and D = Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The fractional Brownian motion (fBm) appears in many stochastic phenomena, where rough exter- nal forces are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The principal difference, compared to Brownian motion, is that fBm is not a semimartingale nor a Markov process, hence classical theory of stochastic integration cannot be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Since H > 1/2, the stochastic integral with respect to BH in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) can be understood as a fractional integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Also the presence of both, Brownian and fractional Brownian motion in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1), due to their different analytic and probabilistic properties, modelize different aspects of the random evolution in time of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The factor k2/2 in front of L affects dissipativity, which in several cases is in favor of retarding or even preventing blowup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' We consider both, weak and mild solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1), which we prove are equivalent and unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Beyond existence and uniqueness of weak and mild solutions we are interested in their qualitative behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In Theorem 3 below we obtain a random time τ ∗ which is an upper bound of the explosion time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In Theorem 8 we obtain a lower bound τ∗ of τ so that a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' τ∗ ≤ τ ≤ τ ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The random times τ∗ and τ ∗ are given by exponential functionals of the mixture of a Brownian and a fractional Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The laws of such kind of functionals presently are not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In order to study the distribution of τ ∗ we use the well-known representation of BH in the form BH t = � t 0 KH(t, s) dWs, where the kernel KH is given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='23) and W is a Brownian motion defined in the same filtered probability space as B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In general, W can be different from the Brownian motion B appearing in the first integral of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' We obtain estimates of the probability P(τ < ∞), and of the tail distribution of τ ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' To achieve this we make use of recent results of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Dung [8, 9] from the Malliavin theory for continuous isonormal Gaussian processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In Theorem 4 we obtain upper bounds for P(τ ∗ ≤ T) in the case when B = W, and in Theorem 5 when B is independent of W, and when B and W are general Brownian motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In Theorem 6 we obtain lower bounds for P(τ < ∞) when B = W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' As a result in the case when W = B we get specific configurations of the coefficients a, b and k under which the weak solution (hence also the mild solution) of equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) exhibits finite time blow-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' To be concrete suppose that g(z) ≥ Cz1+β for 2 some constants C > 0, β > 0, BH t = � t 0 KH(t, s) dBs, and � t 0 a2(r) dr ∼ t2l, � t 0 b2(r) dr ∼ t2m, � t 0 k2(r) dr ∼ t2p as t → ∞ for some nonnegative constants l, m and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' If β ∈ (0, 1/2) and max{p, l} > H + m − 1/2, or if β = 1/2 and p > H +m−1/2, or if β > 1/2 and p > max{l, H +m−1/2}, then all nontrivial positive solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) suffer finite-time blowup with positive probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Our approach here is to transform the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) into a random partial differential equation (RPDE) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5), whose solution blows up at the same random time τ as the solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1), and to work with this equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The blowup behavior of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) is easier to determine because N appears as a coefficient, and not as stochastic integrator as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Such transformations are indeed known for more general SPDEs than (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1), including equations whose stochastic term does not depend linearly on u, see [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' But for the RPDE’s associated to more general SPDE’s it seems difficult to find explicit expressions for upper and lower bounds for the blowup time, and this is an essential point in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Another reason for having chosen the relatively simple form of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) is that we consider the blowup trajectorywise which is a relatively strong notion compared, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=', to blowup of the moments of the solution (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The crucial ingredient in the proofs is the existence of a positive eigenvalue and an eigenfunction with constant sign of the adjoint operator of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Special attention is given to the case H ∈ (3 4, 1) because then the process N is equivalent to a Brownian motion [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' This allows us to apply a result by Dufresne and Yor [27] on the law of exponential functionals of the Brownian motion to get in Theorem 7 an explicit lower bound for the probability of blowup in finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' We finish this section by introducing some notations and definitions we will need in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' A stopping time τ : Ω → (0, ∞) with respect to the filtration (Ft, t ≧ 0) is a blowup time of a solution u of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) if lim sup tրτ sup x∈D |u(x, t)| = +∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Let (P D t , t ≧ 0) and ((P D)∗ t , t ≧ 0) be the strongly continuous semigroups corresponding to the operator L and its adjoint L∗ : � D f(x)P D t g(x)dx = � D g(x)(P D)∗ tf(x)dx, f, g ∈ L2(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='3) As usual, Lf := lim t→0 1 t (P D t f − f) for all f ∈ L2(D) in the domain of L, denoted by Dom(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Due to the Hille-Yosida theorem, Dom(L) and Dom(L∗) are dense in L2(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Let P D t (x, Γ) and (P D)∗ t (x, Γ) denote the associated transition functions, where t > 0, x ∈ D, and Γ ∈ B(D), the Borel sets on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In the sequel we will assume that they admit densities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' there exist families of continuous functions 3 (pD(t, ·, ·), t > 0) and ((pD)∗(t, ·, ·), t > 0) on D × D such that P D t g(x) = � D g(y)P D t (x, dy) = � D g(y)pD(t, x, y)dy, (P D)∗ t f(x) = � D f(y)(P D)∗ t (x, dy) = � D f(y)(pD)∗(t, x, y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Due to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='3), (pD)∗(t, x, y) = pD(t, y, x) for all t > 0 and x, y ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='4) 2 The weak solution of the associated random partial differential equation, equivalence with the mild solution Let us consider the random partial differential equation ∂v ∂t (x, t) = 1 2k2(t)Lv(x, t) − 1 2a2(t)v(x, t) + exp(−Nt)g(exp(Nt)v(x, t)), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) v(x, 0) = ϕ(x), x ∈ D, v(x, t) = 0, t ≥ 0, x ∈ ∂D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In this section we transform the weak form of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) into the weak form of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) using the transformation v(x, t) = exp(−Nt)u(x, t), x ∈ D, t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Hence, if blowup takes place in finite time, it occurs of course at the same time and at the same place x ∈ D for the solutions of both equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In the following we write ⟨·, ·⟩D for the scalar product in L2(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' An (Ft, t ≧ 0)-adapted random field v = (v(x, t), t ∈ [0, T], x ∈ D) with values in L2(D) is a weak solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) if, for all t ∈ [0, T] and all f ∈ Dom(L∗), P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' ⟨v(·, t), f⟩D = ⟨ϕ, f⟩D + � t 0 �1 2k2(s) ⟨v(·, s), L∗f⟩D − 1 2a2(s) ⟨v(·, s), f⟩D � ds + � t 0 exp(−Ns) ⟨g(exp(Ns)v(·, s)), f⟩D ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='6) Since g is supposed to be locally Lipschitz, a blowup in finite time of v may occur, and the blowup time τ depends in general on ω ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' A weak solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) up to τ is defined as an (Ft, t ≧ 0)- adapted random field v that satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='6) for all t ∈ (0, T ∧ τ) P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' If ω is such that v(ω, ·, ·) does not blowup in finite time, we set τ(ω) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' An (Ft, t ≧ 0)-adapted random field u = (u(x, t), t ∈ [0, T], x ∈ D) with values in L2(D) is a weak solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) up to τ if, for all t ∈ (0, T ∧ τ) and all f ∈ Dom(L∗), P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (i) � t 0 a2(s) � 1 + ⟨u(·, s), f⟩2 D � ds < ∞, b(•) ⟨u(·, •), f⟩D ∈ Cβ[0, t] for some β > 1 − H, 4 (ii) � t 0 � k2(s) |⟨u(·, s), L∗f⟩D| + |⟨g(u(·, s)), f⟩D| � ds < ∞, and ⟨u(·, t), f⟩D = ⟨ϕ, f⟩D + � t 0 �1 2k2(s) ⟨u(·, s), L∗f⟩D + ⟨g(u(·, s), f⟩D � ds + � t 0 ⟨u(·, s), f⟩D dNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='7) Conditions (i) and (ii) in the above definition are sufficient for the Itˆo, the fractional and the Lebesgue integrals in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='7) to be well defined P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' We proceed now to the relation between (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' If u is a weak solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) up to a random time τ, then v(x, t) = exp(−Nt)u(x, t) is a weak solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) up to τ, and viceversa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' We notice that ⟨v(·, s), f⟩D is absolutely continuous in s if v is a weak solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' With the choice u(x, t) := exp(Nt)v(x, t) condition (i) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In fact, for t < T ∧ τ(ω), � t 0 ⟨u(·, s), f⟩D a(s) dBs = � t 0 ⟨v(·, s), f⟩D exp(Ns)a(s) dBs is well defined since � t 0( � D v(x, s)f(x) dx)2 exp(2Ns)a2(s) ds < ∞ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Recall that the fractional integral � T 0 f(x)dg(x) is defined (in the sense of Z¨ahle [28]) in [18, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1] for f, g belonging to fractional Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' If 0 < ε < H, f and g are H¨older continuous of exponents α and H − ε respectively, and α + H − ε > 1, this fractional integral coincides with the corresponding generalized Riemann-Stieltjes integral;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' see [18, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Hence, the fractional integral � t 0 ⟨u(·, s), f⟩D b(s) dBH s = � t 0 ⟨v(·, s), f⟩D exp(Ns)b(s) dBH s (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='8) is well defined for t < T ∧ τ(ω) because, on the one hand, N· = � · 0(a(s) dBs + b(s) dBH s ) is P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' H¨older continous of order 1/2 − ǫ for all ǫ > 0 by the theorem of Kolmogorov and [22, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' On the other hand b(·) is α-H¨older continuous (with α > 1 − H) and BH is H¨older continuous with exponent H − ε for any ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Hence, choosing ε < min{H/2 − 1/4, α + H − 1} we get that the integrand on the right side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='8) is H¨older continuous of order min{α, 1/2 − ε}, and therefore H −ε+min{α, 1/2−ε} > 1 and the integral is well defined as a generalized Riemann-Stieltjes integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Let T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' It suffices to prove the assertion for t ∈ (0, T ∧ τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' We apply (a slight generalisation 5 of) the Itˆo formula in [18, page 184].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Let Y 1 t = � t 0 a(s) dBs , Y 2 t = � t 0 ⟨u(·, s), f⟩D a(s) dBs, Y 3 t = � t 0 b(s) dBH s , Y 4 t = � t 0 ⟨u(·, s), f⟩D b(s) dBH s , Y 5 t = ⟨ϕ, f⟩D + � t 0 �1 2k2(s) ⟨u(·, s), L∗f⟩D + ⟨g(u(·, s)), f⟩D � ds, and let F(y1, y2, y3, y4, y5) = exp(−y1 − y3)(y5 + y2 + y4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Then F(Y 1 t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' , Y 5 t ) = exp(−Nt) ⟨u(·, t), f⟩D = ⟨v(·, t), f⟩D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The above mentioned Itˆo formula then reads F(Y 1 t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' , Y 5 t ) = F(Y 1 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' , Y 5 0 ) + 5 � i=1 � t 0 ∂F ∂yi (Y 1 s , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' , Y 5 s ) dY i s + 1 2 2 � i,j=1 � t 0 ∂2F ∂yi∂yj (Y 1 s , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' , Y 5 s ) d � Y i s , Y j s � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Since u is a weak solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1), ⟨v(·, t), f⟩D = ⟨ϕ, f⟩D − � t 0 exp(−Ns) ⟨u(·, s), f⟩D � a(s) dBs + b(s) dBH s � + � t 0 exp(−Ns) ⟨u(·, s), f⟩D � a(s)dBs + b(s)dBH s � + � t 0 exp(−Ns) �1 2k2(s) ⟨u(·, s), L∗f⟩D + ⟨g(u(·, s)), f⟩D � ds −1 2 � t 0 exp(−Ns) ⟨u(·, s), f⟩D a2(s)ds = ⟨ϕ, f⟩D + � t 0 �1 2k2(s) ⟨v(·, s), L∗f⟩D − 1 2a2(s) ⟨v(·, s), f⟩D � ds + � t 0 exp(−Ns) ⟨g(exp(Ns)v(·, s)), f⟩D ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Therefore v is a weak solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Similarly we obtain the viceversa result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In order to define the mild solutions of equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) we define first the evolution families of contractions corresponding to the generator 1 2k2(t)L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' For 0 ≤ s < t let K(t, s) = 1 2 � t s k2(r) dr, A(t, s) = 1 2 � t s a2(r) dr, K(t) = K(t, 0), A(t) = A(t, 0), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='9) 6 and set pD(s, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, y) = pD(K(t, s), x, y), x, y ∈ D × D, 0 ≦ s < t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' For f ∈ L2(D) the corresponding evolution families of contractions on L2(D) are given by U D(t, s)f(x) = � D pD(s, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, y)f(y)dy = P D K(t,s)f(x), (U D)∗(t, s)f(x) = � D pD(s, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, x)f(y)dy = (P D)∗ K(t,s)f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' An (Ft, t ≧ 0)-adapted random field v = (v(x, t), t ≧ 0, x ∈ D) with values in L2(D) is a mild solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) on [0, T] if, for all t ∈ [0, T], P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' v(x, t) = U D(t, 0)ϕ(x) − 1 2 � t 0 a2(s)U D(t, s)v(x, s) ds + � t 0 exp(−Ns)U D(t, s) � g((exp Ns)v(x, s)) � ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The mild form of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) can be written as v(x, t) = exp(−A(t))U D(t, 0)ϕ(x) + � t 0 exp(−Ns − A(t, s))U D(t, s)g(exp(Ns)v(·, s))(x)ds, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='10) where A(t, s) and A(t) are given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Since g and ϕ are supposed to be nonnegative, v(x, t) ≥ exp(−A(t))U D(t, 0)ϕ(x) ≥ 0 for all x ∈ D and t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Let w(x, t) = exp(A(t))v(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' For f ∈ L2(D),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' we get from the definition of the mild solution d dt⟨w(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' f⟩D = 1 2a2(t) exp(A(t))⟨v(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' f⟩D + exp(A(t)) d dt⟨v(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' f⟩D = 1 2a2(t) exp(A(t))⟨v(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' f⟩D + exp(A(t)) �1 2k2(t) ⟨v(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' L∗f⟩D − 1 2a2(t) ⟨v(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' f⟩D � + exp(A(t)) exp(−Nt) ⟨g(exp(Nt)v(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' f⟩D = 1 2 exp(A(t))k2(t) ⟨v(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' L∗f⟩D + exp(A(t) − Nt) ⟨g(exp(Nt)v(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' f⟩D = 1 2k2(t) ⟨w(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' L∗f⟩D + exp(A(t) − Nt) ⟨g(exp(Nt − A(t))w(·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' f⟩D ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' with boundary conditions w(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 0) = ϕ(x) for x ∈ D and w(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t) = 0 for x ∈ ∂D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Therefore w is a weak solution of the RPDE formally given by d dtw(x, t) = 1 2k2(t)Lw(x, t) + exp(A(t) − Nt)g(exp(Nt − A(t))w(x, t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 7 By the definition of the mild solution w(x, t) = U D(t, 0)ϕ(x) + � t 0 exp(A(s) − Ns)U D(t, s)g(exp(Ns − A(s))w(·, s))(x) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Consequently, v(x, t) = exp(−A(t))w(x, t) = exp(−A(t))U D(t, 0)ϕ(x) + � t 0 ds exp(−A(t, s) − Ns)U D(t, s)g(exp(Ns)v(·, s))(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='10) has a unique non-negative local mild solution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' there exists t > 0 such that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='10) has a mild solution in L∞([0, t[×D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Let T > 0 and denote ET = {v : [0, T] × D → L∞(D) : []v[] < ∞} , where []v[] := sup 0≤t≤T ∥v(t, ·)∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Let PT = {v ∈ ET : v ≥ 0} and for R > 0 let CR = {v ∈ ET : []v[] ≤ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Then ET is a Banach space and PT and CR are closed subsets of ET .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Let us now define ψ(v)(t, x) = e−A(t)U D(t, 0)ϕ(x) + � t 0 e−A(t,s)−NsU D(t, s)g � eNsv(·, s) � (x) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' We will prove that for sufficiently big R and sufficiently small T, ψ is contraction on PT ∩ CR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Let v1, v2 ∈ PT ∩ CR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Then []ψ(v1) − ψ(v2)[] ≤ sup 0≤t≤T � t 0 ��e−Ns � g � eNsv1 � − g � eNsv2 ���� ∞ ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Let AT = sup0≤s≤T e|Ns| and GR = sup|x| 0 centered at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Then, sup 0≤s≤T ��e−Nsvi(s, ·) �� ∞ ≤ AT R, i = 1, 2, and ��e−Nsg � eNsv1(s, ·) � − e−Nsg � eNsv2(s, ·) ��� ∞ ≤ A2 T KAT R ∥v1(s) − v2(s)∥∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Therefore, []ψ(v1) − ψ(v2)[] ≤ sup 0≤t≤T � t 0 A2 T KAT R []v1 − v2[] ds = TA2 T KAtR []v1 − v2[].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 8 We need TA2 T KAT R < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='11) In addition, we require that CR ∩ PT be mapped by ψ into itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Let v ∈ CR ∩ PT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Using that for 0 ≤ s ≤ T the operator U D(t, s) is a contraction, and that ∥eNsv(·, s)∥∞ ≤ AT R, we get ∥g(eNsv(·, s))∥∞ ≤ GAT R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' It follows that []ψ(v)[] ≤ ∥ϕ∥∞ + sup 0≤t≤T � t 0 ���e−N(s)��� ∞ ds GAT R ≤ ∥ϕ∥∞ + TAT GAT R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Hence, we need that ∥ϕ∥∞ + TAT GAT R < R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='12) Let R be such that R ≥ 2∥ϕ∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Since limT→0 AT = 1, we choose ε1 > 0 so that AT < 2 if T < ε1, and ε < R 4G2R ∧ 1 4K2R ∧ ε1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Using that GA ≤ GB and KA ≤ KB if A ≤ B, we get for R > 2∥ϕ∥∞ and T < ε, ∥ϕ∥∞ + TAT GAT R ≤ ∥ϕ∥∞ + 2εG2R < R 2 + R 2 = R and TA2 T KAT R < 4εK2R < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' We proceed to prove equivalence of weak and mild solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The proof of this theorem follows the method in [24, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='15], where this equivalence is shown for SPDE’s with autonomous differential operators and driven by L´evy noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' For a comparison of weak and mild solutions of SPDEs driven by fractional Brownian motion we refer to [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' We state first the Kolmogorov backward and forward equations for U D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' By the Kolmogorov back- ward equation for P D, the transition density pD(u, x, y) satisfies, for any y fixed, ∂ ∂upD(u, x, y) = LpD(u, x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Then (s, x) → pD(s, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, y) satisfies, for (t, y) fixed, the equation − ∂ ∂spD(s, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, y) = − ∂ ∂spD(K(t, s), x, y) = − ∂ ∂upD(u, x, y) |u=K(t,s) ∂ ∂sK(t, s) = 1 2k2(s)LpD(K(t, s), x, y) = 1 2k2(s)LpD(s, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='13) Similarly, by the Kolmogorov forward equation for P D, for any x fixed, pD(u, x, y) satisfies ∂ ∂upD(u, x, y) = L∗pD(u, x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 9 Then (t, y) → pD(s, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, y) satisfies, for (s, x) fixed, the equation ∂ ∂tpD(s, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, y) = ∂ ∂tpD(K(t, s), x, y) = ∂ ∂upD(u, x, y) |u=K(t,s) ∂ ∂tK(t, s) = 1 2k2(t)L∗pD(K(t, s), x, y) = 1 2k2(t)L∗pD(s, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='14) Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Consider the random partial differential equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Then v is a weak solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) on [0, T] if and only if v is a mild solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) on [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Assume that v is a weak solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Let h ∈ C1([0, ∞), R), f ∈ Dom(L∗), and G(x, t) := − 1 2a2(t)v(x, t) + exp(−Nt)g(exp(Nt)v(x, t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The integration by parts formula is applicable since h ∈ C1([0, ∞), R) (see [24] Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='16) and yields ⟨v(·, t), h(t)f(·)⟩D = ⟨v(·, 0), h(0)f(·)⟩D + � t 0 ⟨v(·, s), h′(s)f(·)⟩D ds + � t 0 ⟨v(·, s), 1 2h(s)k2(s)L∗f(·)⟩D ds + � t 0 ⟨G(·, s), h(s)f(·)⟩D ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Since the functions h · f are dense in C1([0, ∞), Dom(L∗)), for each z ∈ C1([0, ∞), Dom(L∗)) we have ⟨v(·, t), z(·, t)⟩D = ⟨v(·, 0), z(·, 0)⟩D + � t 0 ⟨v(·, s), ∂ ∂sz(·, s)⟩D ds (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='15) + � t 0 ⟨v(·, s), 1 2k2(s)L∗z(·, s)⟩D ds + � t 0 ⟨G(·, s), z(·, s)⟩D ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' For each f ∈ Dom(L∗) we define ψ(x, s) := (U D)∗(t, s)f(x) = \uf8f1 \uf8f2 \uf8f3 ⟨pD∗(s, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, ·), f(·)⟩D if s < t, f(x) if s = t, hence ψ ∈ C1([0, ∞), Dom(L∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Taking z = ψ(x, s) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='15) we get, for any t ∈ [0, T] fixed, ⟨v(·, t), ψ(·, t)⟩D = ⟨v(·, 0), ψ(·, 0)⟩D + � t 0 � v(·, s), d dsψ(·, s) + 1 2k2(s)L∗ψ(·, s) � D ds + � t 0 ⟨G(·, s), ψ(·, s)⟩D ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='16) Now we evaluate the terms above: ⟨v(·, 0), ψ(·, 0)⟩D = � D v(x, 0) � D pD∗(0, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, y)f(y) dy dx = � D f(y) � D pD∗(0, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, y)v(x, 0) dx dy = � U D(t, 0)v(·, 0), f(·) � D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 10 By applying the Kolmogorov backward equation to (x, s) → (U D)∗(t, s)f(x) we get − d dsψ(x, s) = − ∂ ∂s � (pD)∗(s, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, ·), f(·) � D = 1 2k2(s)L∗ � (pD)∗(s, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, ·), f(·) � D = 1 2k2(s)L∗ψ(x, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Moreover, from Fubini’s theorem and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='4) ⟨G(·, s), ψ(·, s)⟩D = � D G(x, s) � D pD∗(s, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, y)f(y) dy dx = � D f(y) � D pD(s, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, x)G(x, s) dx dy = � U D(t, s)G(·, s), f(·) � D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Therefore, from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='16), ⟨v(·, t), f(·)⟩D = � U D(t, 0)v(·, 0), f(·) � D + � t 0⟨U D(t, s)G(·, s), f(·)⟩D ds for all f ∈ Dom(L∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Since Dom(L∗) is dense in L2(D) we obtain that v is a mild solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) on [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' To prove the converse let v be a mild solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) on [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' For f ∈ Dom(L∗), � t 0 � v(·, s), 1 2k2(s)L∗f(·) � D ds = � t 0 � U D(s, 0)v(·, 0), 1 2k2(s)L∗f(·) � D ds + � t 0 �� s 0 χ[0,s](r)U D(s, r)G(·, r) dr, 1 2k2(s)L∗f(·) � D ds = � t 0 � v(·, 0), (U D)∗(s, 0)1 2k2(s)L∗f(·) � D ds + � t 0 � t r � U D(s, r)G(·, r), 1 2k2(s)L∗f(·) � D ds dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='17) By applying the Kolmogorov forward equation to (U D)∗ we get for the first integral on the right side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='17): (U D)∗(s, 0)(1 2k2(s)L∗f)(x) = � D pD∗(0, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' s, y)1 2k2(s)L∗f(y) dy = � D (1 2k2(s)L)pD∗(0, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' s, y)f(y) dy = � D ∂ ∂spD∗(0, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' s, y)f(y) dy, and therefore � t 0 � v(·, 0), (U D)∗(s, 0)(1 2k2(s)L∗)f(·) � D ds = � t 0 � v(·, 0), � D ∂ ∂spD∗(0, ·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' s, y)f(y) dy � D ds = � v(·, 0), � D pD∗(0, ·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, y)f(y)dy − f(·) � D = � v(·, 0), (U D)∗(t, 0)f(·) � D − ⟨v(·, 0), f(·)⟩D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 11 In the same way we get for the second integral on the right side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='17) � U D(s, r)G(·, r), 1 2k2(s)L∗f(·)) � D = � G(·, r), (U D)∗(s, r)(1 2k2(s)L∗f)(·) � D = � G(·, r), � D ∂ ∂spD∗(r, ·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' s, y)f(y)dy � D , and therefore � t r � U D(s, r)G(·, r), 1 2k2(s)L∗f(·) � D ds = � t r � G(·, r), � D ∂ ∂spD∗(r, ·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' s, y)f(y)dy � D ds = � G(·, r), � D pD∗(r, ·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' t, y)f(y)dy − f(·) � D = � G(·, r), (U D)∗(t, r)f(·) − f(·) � D = � U D(t, r)G(·, r), f(·) � D − ⟨G(·, r), f(·)⟩D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In this way we obtain � t 0 ⟨v(·, s), 1 2k2(s)L∗f(·)⟩D ds = � U D(t, 0)v(·, 0) + � t 0 U D(t, r)G(·, r)dr, f(·)⟩D − ⟨v(·, 0), f(·) � D − � t 0 ⟨G(·, r), f(·)⟩D dr = ⟨v(·, t), f(·)⟩D − ⟨v(·, 0), f(·)⟩ D − � t 0 ⟨G(·, r), f(·)⟩D dr, since v is a mild solution on [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' It follows that v is a weak solution on [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) possess unique weak solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Theorem 2 and Proposition 1 show the existence and uniqueness of a local weak and mild solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5), and Proposition 1 shows the uniqueness of a weak solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' We refer to [23] for an existence and uniqueness theorem of the variational solution of an SPDE with a nonautonomous second order differential operator and driven by fractional Brownian motion, and to [26] for the existence and uniqueness of the mild solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In [20] the existence and uniqueness of the mild solution is shown for equations with the same differential operator and driven by mixed noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 3 An upper bound for the blowup time and probability estimates 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1 An upper bound for the blowup time In the remaining part of the paper we will assume that L and L∗ admit strictly positive eigenfunctions: there exists a positive eigenvalue λ0 and strictly positive eigenfunctions ψ0 ∈ Dom(L) for P D t and 12 ϕ0 ∈ Dom(L∗) for (P D)∗ t with � D ψ0(x)dx = � D ϕ0(x)dx = 1 such that (P D t − e−λ0t)ψ0 = ((P D)∗ t − e−λ0t)ϕ0 = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='18) hence (L + λ0)ψ0 = (L∗ + λ0)φ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='19) For generators of a general class of L´evy processes, properties (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='18) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='19) follow from [14, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Another example are the diffusion processes: for f ∈ C2 0(D), the set of twice continously differentiable functions with compact support in D, let us define the differential operator Lf = d � j,k=1 ∂ ∂xj � ajk ∂ ∂xk f � + d � j=1 bj ∂ ∂xj f − cf, where aj,k, bj, j, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=', d are bounded smooth functions on D and c is bounded and continous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' We assume that the matrix (aj,k, j, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=', d) is symmetric and uniformly elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In this case properties (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='18) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='19) follow from [12, Theorem 11, Chapter 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Assume (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='19) and let g(z) ≥ Cz1+β for all z > 0, where C > 0, β > 0, are given constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Let us define τ ∗ = inf � t > 0 : � t 0 exp [−β(λ0K(r) + A(r)) + βNr] dr ≥ 1 Cβ ⟨ϕ, φ0⟩−β D � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='20) where the functions K and A are defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Then, on the event {τ ∗ < ∞} the solution v of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) and the solution u of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) blow up in finite time τ, and τ ≤ τ ∗ P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Using the hypothesis on g and Jensen’s inequality we get for the terms in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='6): ⟨v(·, s), L∗φ0⟩D = −λ0⟨v(·, s), φ0⟩D, exp(−Ns) ⟨g(exp(Ns)v(·, s)), φ0⟩D ≧ C exp(βNs) � v1+β(·, s), φ0 � D , ≧ C exp(βNs)⟨v(·, s), φ0⟩1+β D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Applying these lower bounds to (⟨v(·, t + ε), φ0⟩D − ⟨v(·, t), φ0⟩D)/ε and letting ε → 0 we get d dt⟨v(·, t), φ0⟩D ≧ −1 2(λ0k2(t) + a2(t))⟨v(·, t), φ0⟩D + C exp(βNt)⟨v(·, t), φ0⟩1+β D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='21) The corresponding differential equality reads d dtI(t) = −1 2(λ0k2(t) + a2(t))I(t) + C exp(βNt)I(t)1+β, and I(t) is a subsolution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='21), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' ⟨v(·, t), φ0⟩D ≧ I(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Then I(t) = exp[−(λ0K(t) + A(t))] � ⟨ϕ, φ0⟩−β D − βC � t 0 exp [−β(λ0K(s) + A(s)) + βNs] ds �−1/β 13 for all t ∈ [0, τ ∗), where τ ∗ is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Therefore τ ∗ is an upper bound for the blowup time of ⟨v(·, t), φ0⟩D, and the function t �→ ∥v(·, t)∥∞ = exp(−Nt)∥u(·, t)∥∞ can not stay finite on [0, τ ∗] if τ ∗ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Therefore u and v blow up before τ ∗ if τ ∗ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Notice that τ ∗ depends on L only by the positive eigenvalue λ0 and the associated eigen- function φ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Moreover, τ ∗ is a decreasing function of ϕ, φ0 and C, and an increasing function of λ0K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Therefore small functions ϕ, φ0 and a small constant C, as well as high values of λ0K postpone the blowup of I and have, in this sense, the tendency to postpone the blowup of v and u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='2 A tail probability estimate for the upper bound of the blowup time In the following theorem we apply a tail probability estimate for exponential functionals of fBm studied by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Dung [8] to estimate the probability that τ ∗ occurs before a fixed time T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Here we assume that the process BH is given by the formula BH t = � t 0 KH(t, s) dBs, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='22) where the kernel KH is given for H > 1/2 by KH(t, s) = \uf8f1 \uf8f2 \uf8f3 CHs1/2−H � t s (σ − s)H−3/2σH−1/2dσ if t > s, 0 if t ≦ s, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='23) where CH = [ H(2H−1) B(2−2H,H−1/2)] 1 2 and B is the usual beta function (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='3 in [21] for a general representation formula of fBm with H > 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Notice that BH and B are dependent in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Under assumptions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='19) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='22), let g(z) ≥ Cz1+β for all z > 0, where C > 0, β > 0, are given constants, and let µ(T) = � T 0 exp[−β(λ0K(t) + A(t))]E [exp(βNt)] dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Then, for any T > 0 such that 1 Cβ⟨ϕ, φ0⟩−β D > µ(T), P {τ ∗ ≤ T} ≤ 2 exp \uf8eb \uf8ed− ln2 � Cβ⟨ϕ, φ0⟩β D µ(T) � 2M(T) \uf8f6 \uf8f8 , where M(T) = 2β2 � T 0 a2(r) dr + 4β2HT 2H−1 � T 0 b2(u) du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' For t ≥ 0, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='22), we have the following representation: Xt := −β(λ0K(t) + A(t)) + βNt (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='24) = −β(λ0K(t) + A(t)) + β �� t 0 a(s) dBs + � t 0 � t s b(r) ∂ ∂rKH(r, s) dr dBs � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 14 From [8, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1] it follows that for any T ≥ 0 and any x > µ(T), there holds P �� T 0 eXtdt ≥ x � ≤ 2 exp � −(ln x − ln µ(T))2 2M(T) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='25) where µ(T) = � T 0 E � eXt� dt and M(T) is such that sup t∈[0,T] � T 0 |DrXt|2 dr ≤ M(T) P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='26) Here DrXt denotes the Malliavin derivative of Xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In the following we will find an upper bound M(T) such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='26) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' For r < t we have, using the representation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='25), DrXt = β � a(r) + � t r b(s) ∂ ∂sK(s, r) ds � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Hence � t 0 |DrXt|2 dr ≤ 2β2 � t 0 a2(r) dr + 2β2 � t 0( � t r b(s) ∂ ∂sK(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' r) ds)2 dr and � t 0 �� t r b(s) ∂ ∂sK(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' r) ds �2 dr = � t 0 �� t r b(s) ∂ ∂sK(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' r) ds � �� t r b(s′) ∂ ∂s′ K(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' r) ds′ � dr = � t 0 b(s) ds � s 0 ∂ ∂sK(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' r) dr � t r b(s′) ∂ ∂s′ K(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' r) ds′ = � t 0 ds b(s) � t 0 dr1[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='s](r) ∂ ∂sK(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' r) � t r b(s′) ∂ ∂s′ K(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' r) ds′ = � t 0 ds b(s) � t 0 ds′b(s′) � s′ 0 1[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='s](r) ∂ ∂sK(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' r) ∂ ∂s′ K(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' r) dr = � t 0 ds � t 0 ds′ b(s)b(s′) � s∧s′ 0 ∂ ∂sK(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' r) ∂ ∂s′ K(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' r) dr = � t 0 ds � t 0 ds′ b(s)b(s′)Φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' s′) = � t 0 ds � s 0 ds′ b(s)b(s′)Φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' s′) + � t 0 ds � t s ds′ b(s)b(s′)Φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' s′) = 2 � t 0 ds � s 0 ds′ b(s)b(s′)Φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' s′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' where Φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' s′) = � s∧s′ 0 ∂ ∂sK(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' r) ∂ ∂s′ K(s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' r) dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Since ∂ ∂sK(s, r) = CHr1/2−H(s − r)H−3/2sH−1/2, using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='7) in [21] we obtain Φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' s′) = C2 H(ss′)H−1/2 � s∧s′ 0 r1−2H(s − r)H−3/2(s′ − r)H−3/2 dr = H(2H − 1)(s − s′)2H−2 15 for s′ < s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' hence � t 0 �� t r b(s) ∂ ∂sK(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' r) ds �2 dr ≤ 2H(2H − 1) � t 0 ds � s 0 |b(s)b(s′)|(s − s′)2H−2 ds′ ≤ H(2H − 1) �� t 0 b(s)2 � s 0 (s − s′)2H−2 ds′ ds + � t 0 � s 0 b(s′)2(s − s′)2H−2 ds′ ds � = H � t 0 b(s)2s2H−1 ds + H(2H − 1) � t 0 b(s′)2 � t s′ (s − s′)2H−2 ds ds′ = H � t 0 b(s)2(s2H−1 + (t − s)2H−1) ds ≤ 2Ht2H−1 � t 0 b(s)2 ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='27) From the above inequalities we obtain sup t∈[0,T] � T 0 |DrXt|2dr ≤ 2β2 � T 0 a2(r)dr + 4β2HT 2H−1 � T 0 b2(u)du := M(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='28) Now, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='20) P(τ ∗ ≦ T) = P �� T 0 exp[−β(λ0K(t) + A(t)) + βNt] dt ≧ 1 Cβ ⟨ϕ, φ0⟩−β D � = P �� T 0 exp[X(t)] dt ≥ x � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='29) where x = 1 Cβ⟨ϕ, φ0⟩−β D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The result follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='25) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In the following theorem we obtain upper bounds for the tail of τ ∗ in the case when the Brownian motion B and the fractional Brownian motion BH have general dependence structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Assume (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='19) and let g(z) ≥ Cz1+β for all z > 0, where C > 0, β > 0, are given constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Assume that BH t = � t 0 KH(t, s) dWs, where W is a Brownian motion defined in the same proba- bility space, and adapted to the same filtration as the Brownian motion B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Then P(τ ∗ ≤ T) ≤ Cβ⟨ϕ, φ0⟩β D � T 0 � e−βλ0 � t 0 k2(s) ds+2β2 � t 0 a2(s) ds + e−β � t 0 a2(s) ds+4β2Ht2H−1 � t 0 b2(s) ds� dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' If B and BH are independent, then P(τ ∗ ≤ T) ≤ Cβ⟨ϕ, φ0⟩β D � T 0 e−βλ0K(t)+ β2−β 2 � t 0 a2(s) ds+β2Ht2H−1 � t 0 b2(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Using H¨older’s and Chebishev’s inequalities we obtain P(τ ∗ ≤ T) = P �� T 0 e−βλ0K(t)+β � t 0 a(s) dBs−βA(t)+β � t 0 b(s) dBH s dt ≥ 1 Cβ⟨ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' φ0⟩−β D � ≤ P \uf8ee \uf8f0 �� T 0 e−2βλ0K(t)+2β � t 0 a(s) dBs dt � 1 2 × �� T 0 e−2βA(t)+2β � t 0 b(s) dBH s dt � 1 2 ≥ 1 Cβ ⟨ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' φ0⟩−β D \uf8f9 \uf8fb ≤ P �� T 0 e−2βλ0K(t)+2β � t 0 a(s) dBs dt ≥ 1 Cβ ⟨ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' φ0⟩−β D � +P �� T 0 e−2βA(t)+2β � t 0 b(s) dBH s dt ≥ 1 Cβ⟨ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' φ0⟩−β D � ≤ E �� T 0 e−2βλ0K(t)+2β � t 0 a(s) dBs dt � + E �� T 0 e−2βA(t)+2β � t 0 b(s) dBH s dt � 1 Cβ⟨ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' φ0⟩−β D ≤ � T 0 � e−2βλ0K(t)+2β2 � t 0 a2(s) ds� dt + � T 0 e−2βA(t)E � e2β � t 0 b(s) dBH s � dt 1 Cβ⟨ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' φ0⟩−β D ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='30) where we have used the fact that E � exp �� t 0 f(s) dB(s) �� = exp � 1 2 � t 0 f 2(s) ds � to obtain the last inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In addition, E � e2β � t 0 b(s) dBH s � = E � e2β � t 0 � t s b(r) ∂ ∂r KH(r,s) dr dWs� = e2β2 � t 0[ � t s b(r) ∂ ∂r KH(r,s) dr] 2 ds, where the last equality follows from [13, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Therefore, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='27) we get E � e2β � t 0 b(s) dBH s � ≤ exp � 4β2Ht2H−1 � t 0 b2(s) ds � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='31) Substituting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='31) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='30) we obtain the desired bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Using Chebishev’s inequality, the independence of B and BH and the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='31), P(τ ∗ ≤ T) = P �� T 0 e−βλ0K(t)+β � t 0 a(s) dBs−βA(t)+β � t 0 b(s) dBH s ) dt ≥ 1 Cβ ⟨ϕ, φ0⟩−β D � ≤ Cβ⟨ϕ, φ0⟩β D � T 0 E � e−βλ0K(t)+β � t 0 a(s) dBs� E � e−βA(t)+β � t 0 b(s) dBH s � dt ≤ Cβ⟨ϕ, φ0⟩β D � T 0 exp � −βλ0K(t) + β2 − β 2 � t 0 a2(s) ds + β2Ht2H−1 � t 0 b2(s) ds � dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 4 Lower bounds for the blowup time and for the probability of finite time blowup 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1 A lower bound for the probability of finite time blowup In the following theorem we give a lower bound for the probability of finite time blow up of the weak solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' If f, g are nonnegative functions and c is a constant, we write f(t) ∼ cg(t) as t → ∞ if limt→∞ f(t)/g(t) = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Assume (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='19) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Let g(z) ≥ Cz1+β and � t 0 a2(r) dr ∼ C1t2l, � t 0 b2(r) dr ∼ C2t2m, � t 0 k2(r) dr ∼ C3t2p as t → ∞ for some nonnegative constants l, m, p and positive constants C, β, C1, C2 and C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Suppose additionally that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' if β ∈ (0, 1/2), then max{p, l} > H + m − 1 2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' if β = 1/2, then H+m − 1 2 < p, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' if β > 1/2, then p > max{l, H + m − 1 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Under these assumptions the solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) blows up in finite time with positive probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Moreover, P(τ < ∞) ≧ P(τ ∗ < ∞) ≧ 1 − exp � −(mξ − 1)2 2Lξ � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='32) where ξ = 1 Cβ ⟨ϕ, φ0⟩−β D , Lξ = sup t≧0 M(t) (ln(ξ + 1) + f(t))2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='33) 18 with f(t) = tmax{H+m−1/2, l} and mξ = E \uf8ee \uf8f0sup t≧0 ln �� t 0 exp (−β(λ0K(s) + A(s)) + βNs) ds + 1 � + f(t) ln(ξ + 1) + f(t) \uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='34) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='29) it follows that P(τ ∗ < ∞) = P( � ∞ 0 eXt dt ≥ ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In order to estimate P( � ∞ 0 eXt dt ≥ ξ) we use [9, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1], with a = 0 and σ = 1 : Proposition 3 ([9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Assume that the stochastic process X is adapted and satisfies a) � ∞ 0 EeXs ds < ∞, b) For each t ≥ 0, Xt ∈ D1,2, c) There exists a function f : R+ → R+ such that limt→∞ f(t) = ∞ and for each x > 0, sup t≧0 sups∈[0,t] � t 0 |DrXs|2dr (ln(x + 1) + f(t))2 ≤ Lx < ∞ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='35) Then P �� ∞ 0 eXt dt < x � ≤ exp � −(mx − 1)2 2Lx � , where mx = E � sup t≥0 ln( � t 0 eXs ds + 1) + f(t) ln(x + 1) + f(t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' We now verify that conditions a) - c) of the above proposition hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' For condition a) we have from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='25),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' � ∞ 0 E exp[Xt] dt = � ∞ 0 E exp � −βλ0 2 � t 0 k2(s) ds − β 2 � t 0 a2(s) ds + β �� t 0 a(s) dBs + � t 0 � t s b(r) ∂ ∂rKH(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' s) dr dBs �� dt = � ∞ 0 E exp � −βλ0 2 � t 0 k2(s) ds − β 2 � t 0 a2(s) ds + β � t 0 � a(s) + � t s b(r) ∂ ∂rKH(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' s) dr � dBs � dt = � ∞ 0 exp � −βλ0 2 � t 0 k2(s) ds − β 2 � t 0 a2(s) ds + β2 2 � t 0 � a(s) + � t s b(r) ∂ ∂rKH(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' s) dr �2 ds � dt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' again,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' we have used [13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='12] to obtain the last equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Therefore, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='27), � ∞ 0 E exp[Xt] dt ≤ � ∞ 0 exp � −βλ0 2 � t 0 k2(s) ds − β 2 � t 0 a2(s) ds + β2 2 � t 0 2a2(s) ds + 2β2Ht2H−1 � t 0 b2(s) ds � dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='36) 19 The integral (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='36) is finite if and only if the leading power of t in the term −βλ0 2 � t 0 k2(s) ds + 2β2 − β 2 � t 0 a2(s) ds + 2β2Ht2H−1 � t 0 b2(s) ds has negative coefficient, which follows from our assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Condition b) is a consequence of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' For condition c) we use the inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='28), which implies that for any x > 0 and any fixed function f, sup t≧0 sups∈[0,t] � t 0 |DrXs|2dr (ln(x + 1) + f(t))2 ≤ sup t≥0 M(t) (ln(x + 1) + f(t))2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='37) Due to our assumptions, for big t, the leading power of t in the numerator is max{2l, 2H + 2m − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' It follows that lim t→∞ M(t) � ln(x + 1) + tmax{l,H+m−1/2}�2 < ∞, and therefore the supremum in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='37) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The result follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The cases when a = 0 (presence only of fractional Brownian motion) or b = 0 (presence only of Brownian motion), are simpler: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Under the assumptions in Theorem 6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' When a(t) ≡ 0 and p > H + m − 1/2 the solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) explodes in finite time with positive probability for all β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' If a(t) ≡ 0 and p = H + m − 1/2, the solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) explodes in finite time with positive probability for all β > 0 satisfying β < C3λ0 4C2H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' When b(t) ≡ 0 and 0 < β ≤ 1 2 the solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) exhibits explosion in finite time with positive probability for all values of p and l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' If b(t) �� 0 and β > 1/2, the solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) exhibits explosion in finite time with positive probability if p > l or if p = l and C3λ0 > C1(2β − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Notice that mξ given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='34) satisfies mξ > 1 due to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1 in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' The formula for mξ shows interactions between ϕ and K that have an influence on the lower bound in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Increasing values of K decrease the lower bound in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In this sense high values of K are in favour of absence of finite time blowup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='2 The case H > 3/4 and independent B and BH In order to find more explicit lower bounds for P(τ < +∞), we consider in this subsection the case H ∈ (3/4, 1) and suppose that B and BH are independent and b(s) = ca(s) for all s ≧ 0, where c is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Then Nt = � t 0 a(s)dMs with Ms = Bs + cBH s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' By [3] M is equivalent to a Brownian motion �B, and therefore Nt is equivalent to ˜Nt := � t 0 a(s) d �Bs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Here equivalence means equality of the laws of the processes on (C[0, T], B), the space of continous functions defined on [0, T] endowed with the σ−algebra generated by the cylinder sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Furthermore, ( ˜Nt)t≧0 is a continous martingale and therefore a time-changed Brownian motion: ˜Nt = �B2A(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Assume (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Let H ∈ (3/4, 1), B and BH be independent and b(s) = ca(s) for all s ≧ 0, where c is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='We assume also that g(z) ≥ Cz1+β, that the functions k and a are positive continuous on R+ and that there exist constants η ∈ (0, +∞] and c1 > 0 such that 1 a2(t) exp(−βλ0K(t)) ≥ c1 exp � −2β A(t) η � , t ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='38) Then P(τ < +∞) ≥ P(Zµ ≤ θ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='39) where τ is the blowup time of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1), Zµ is a gamma-distributed random variable with parameter µ := 2 β( 1 η + 1 2), θ := 2c1 β2ξ and ξ := 1 Cβ⟨ϕ, φ0⟩−β D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' From Theorem 3, P(τ ∗ = +∞) = P �� t 0 dr exp � −β(λ0K(r) + A(r)) + β ˜Nr � < ξ for all t > 0 � = P �� ∞ 0 dr exp � −β(λ0K(r) + A(r)) + β ˜Nr � ≤ ξ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' By the change of variable q = 2A(r) we get P(τ ∗ = +∞) = P �� ∞ 0 dr exp � −β(λ0K(r) + A(r)) + β ˜B2A(r) � ≤ ξ � = P �� ∞ 0 dq a2(A−1(q/2)) exp � −β(λ0K(A−1(q/2)) + 1 2q) + β ˜Bq � ≤ ξ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Applying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='38) to t = A−1(q/2) yields 1 a2(A−1(q/2)) exp � −β(λ0K(A−1(q/2)) � ≥ c1 exp � −β η q � , q ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 21 Therefore P(τ ∗ = +∞) ≤ P � c1 � ∞ 0 dq exp � −βq �1 η + 1 2 � + β ˜Bq � ≤ ξ � = P �� ∞ 0 dq exp � β( ˜Bq − ˜µq) � ≤ ξ c1 � , where ˜µ := 1 η + 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' A second change of variable q = 4s β2 yields P(τ ∗ = +∞) ≤ P �� ∞ 0 ds exp � 2( ˜Bs − µs) � ≤ β2ξ 4c1 � , where µ := ˜µ 2 β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Due to [27, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='2, page 95], � ∞ 0 e2( ˜ Bs−µs) ds L= 1 2Zµ , where Zµ is a gamma-distributed random variable with parameter µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Therefore P(τ = +∞) ≤ P(τ ∗ = +∞) ≤ P � 1 2Zµ ≤ β2ξ 4c1 � = P � Zµ ≥ 2c1 β2ξ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' This implies the statement of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' If k, a and b are constants, a more explicit lower bound for P(τ < +∞) is available without the assumption (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Indeed, starting with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='20), a straightforward calculation gives a lower bound in terms of a gamma-distributed random variable Z again, but this time with parameter �µ := (λ0k2 + a2)/(a2β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' More precisely, P(τ < ∞) ≧ P(τ ∗ < ∞) = P � Z�µ ≦ 2C a2β ⟨ϕ, φ0⟩β D � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='3 A lower bound for the blowup time Our next goal is to obtain a lower bound for the blowup time τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Since the proofs of the following results are close to those in [1] (where b = 0), we omit them here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Let the function g be such that g(0) = 0, z → g(z)/z is increasing, and g(z) ≤ Λz1+β for some positive constant Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Then τ ≥ τ∗, where τ∗ = inf � t > 0 : � t 0 exp(β(Nr − A(r))) ��U D(r, 0)ϕ ��β ∞ dr ≧ 1 Λβ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='40) Let us define for 0 ≦ t < τ∗, J(t) = � 1 − Λβ � t 0 exp(β(Nr − A(r))) ��U D(r, 0)ϕ ��β ∞ dr �−1/β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 22 Then the solution u of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) satisfies, for x ∈ D, 0 ≦ t < τ∗, P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 0 ≦ u(x, t) ≦ J(t) exp(Nt − A(t))U D(t, 0)ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='41) Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' More precisely, the proof of this theorem shows that the mild solution v of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='41) without the factor exp(Nt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' By Theorem 2, v is also the weak solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5), hence the weak solution u(·, t) = exp(Nt)v(·, t) of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Assume that Λβ � ∞ 0 exp[β(Nr − A(r))] ��U D(r, 0)ϕ ��β ∞ dr < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Then the solution u of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='41) P-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' For the special choice of ϕ = pψ0, p > 0, the integrals appearing in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='20) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='40) are the same exponential functionals of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In fact, U D(r, 0)ψ0 = exp(−λ0K(r))ψ0, and τ∗ becomes τ∗ = inf � t > 0 : � t 0 exp � β(Nr − λ0K(r) − A(r)) � dr ≧ p−β Λβ ∥ψ0∥−β ∞ � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='42) whereas τ ∗ = inf � t > 0 : � t 0 exp � β(Nr − λ0K(r) − A(r)) � dr ≥ p−β Cβ ⟨ψ0, φ0⟩−β D � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='43) In fact τ∗ ≦ τ ∗ if C ≦ Λ, since ⟨ψ0, φ0⟩D ≦ ∥ψ0∥∞ � D φ0(x)dx = ∥ψ0∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' In order to apply both bounds simultaneously, we have to suppose Cz1+β ≦ g(z) ≦ Λz1+β, z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' It is therefore of interest to know the law of the integral appearing in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='42) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' This seems possible only for bH = 0, since, to our best knowledge, the law of exponential functionals of fractional Brownian motion is still unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' For the moment it seems that only estimates of the type of those in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='2 are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' See also Theorem 7 for H > 3/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 5 A sufficient condition for finite time blowup We consider now the mild form of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5) obtained in Proposition 2, and obtain a sufficient condition for finite time blowup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Suppose that g(z) ≥ Cz1+β and that there exists w∗ > 0 such that exp(βA(w∗)) ∥ U D(w∗, 0)ϕ ∥−β ∞ < βC � w∗ 0 exp(βNs) ds .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='44) Then for the explosion time τ of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='1) there holds τ ≤ w∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' 23 Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Inequality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='44) is understood trajectorywise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Therefore w∗ is random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='44) is harder to satisfy with a small initial condition ϕ and with a small value of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Due to the different interpretations of the integrals in N, the effects on blowup of B and BH are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' If N = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='44) reads ∥ U D(w∗, 0)ϕ ∥−β ∞ < βCw∗ and in this case w∗ is deterministic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' if in addition ϕ = ψ0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='44) reads exp(λ0βK(w∗)) ∥ ψ0 ∥−β ∞ < βCw∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' We use the approach in [25, Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='6];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' see also [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Suppose that v(x, t), x ∈ D, t ≥ 0, is a global solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content='5), and let 0 < t < t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' Using the semigroup property of the evolution system (U D(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9A0T4oBgHgl3EQfPP_v/content/2301.02174v1.pdf'} +page_content=' r))0≦r