Stochastic evolution equations driven by nuclear-space-valued martingales

This paper presents a theory of stochastic evolution equations for nuclear-space-valued processes and provides a unified treatment of several examples from the field of applications. (C ₀ , 1) reversed evolution systems on countably Hilbertian nuclear spaces are also investigated.Research supported by the Air Force Office of Scientific Research Grant No. F49620 85 C 0144. The research of V. Perez-Abreu was also supported by CONACYT Grants PCEXCNA-040651 and PCMTCNA-750220 (Mexico).     

Stochastic Volterra equations driven by cylindrical Wiener process

In this paper, stochastic Volterra equations driven by cylindrical Wiener process in Hilbert space are investigated. Sufficient conditions for existence of strong solutions are given. The key role is played by convergence of α-times resolvent families. Both authors are supported partially by project “Proyecto Anillo: Laboratorio de Analisis Estocastico; ANESTOC”.     

Stochastic generalized Burgers equations driven by fractional noises

The existence and uniqueness are proved for the solutions to a class of stochastic generalized Burgers equations driven by multi-parameter fractional noises. In addition, the existence and moment estimate are also obtained for the density of such a solution.     

Stochastic Cahn-Hilliard equations driven by Poisson random measures

We study a stochastic Cahn-Hilliard equation driven by a Poisson random measure with Neumann boundary conditions. The global weak solution is established for the equation. Moreover, the existence of a Lyapunov function for the equation and an invariant measure associated with the transition semigroup are proved.     

Stochastic Volterra equations driven by fractional Brownian motion

This paper is devoted to study a class of stochastic Volterra equations driven by fractional Brownian motion. We first prove the Driver type integration by parts formula and the shift Harnack type inequalities. As a direct application, we provide an alternative method to describe the regularities of the law of the solution. Secondly, by using the Malliavin calculus, the Bismut type derivative formula is established, which is then applied to the study of the gradient estimate and the strong Feller property. Finally, we establish the Talagrand type transportation cost inequalities for the law of the solution on the path space with respect to both the uniform metric and the L₂-metric.     

Stochastic evolution equations driven by Liouville fractional Brownian motion

Let H be a Hilbert space and E a Banach space. We set up a theory of stochastic integration of ℒ(H,E)-valued functions with respect to H-cylindrical Liouville fractional Brownian motion with arbitrary Hurst parameter 0 < β < 1. For 0 < β < ? we show that a function Φ: (0, T) → ℒ(H,E) is stochastically integrable with respect to an H-cylindrical Liouville fractional Brownian motion if and only if it is stochastically integrable with respect to an H-cylindrical fractional Brownian motion.     

Local times of stochastic differential equations driven by fractional Brownian motions

In this paper, we study the existence and (Hölder) regularity of local times of stochastic differential equations driven by fractional Brownian motions. In particular, we show that in one dimension and in the rough case $H<1/2$, the Hölder exponent (in $t$) of the local time is $1?H$, where $H$ is the Hurst parameter of the driving fractional Brownian motion.     

Stochastic delay fractional evolution equations driven by fractional Brownian motion

In this paper, we consider a class of stochastic delay fractional evolution equations driven by fractional Brownian motion in a Hilbert space. Sufficient conditions for the existence and uniqueness of mild solutions are obtained. An application to the stochastic fractional heat equation is presented to illustrate the theory. Copyright © 2014 John Wiley & Sons, Ltd.     

Stochastic wave equations with dissipative damping

We prove existence and (in some special case) uniqueness of an invariant measure for the transition semigroup associated with the stochastic wave equations with nonlinear dissipative damping.     

Stochastic partial differential equations with gradient driven by space-time fractional noises

We establish a class of stochastic partial differential equations (SPDEs) driven by space-time fractional noises, where we suppose that the drfit term contains a gradient and satisfies certain non-Lipschitz condition. We prove the strong existence and uniqueness and joint Hölder continuity of the solution to the SPDEs.     

Stochastic differential equations with non-negativity constraints driven by fractional Brownian motion

In this paper, we consider stochastic differential equations with non-negativity constraints, driven by a fractional Brownian motion with Hurst parameter H > 1/2. We first study an ordinary integral equation, where the integral is defined in the Young sense, and we prove an existence result and the boundedness of the solutions. Then, we apply this result pathwise to solve the stochastic problem.     

The stochastic wave equations driven by fractional and colored noises

We investigate a wave equation in the plane with an additive noise which is fractional in time and has a non-degenerate spatial covariance. The equation is shown to admit a process-valued solution. Also we give a continuity modulus of the solution, and the HSlder continuity is presented.     

On a type of stochastic differential equations driven by countably many Brownian motions

Consider the one-dimensional SDE , where Wⁱ is an infinite sequence of independent standard Brownian motions, i=1,2,3,…. We prove two theorems on the existence and uniqueness of solutions with non-Lipschitz coefficients, and give a non-contact property and a strong comparison theorem for solutions.     

Stochastic averaging principles for multi-valued stochastic differential equations driven by poisson point Processes

The purpose of this article is to investigate an averaging principle for multi-valued stochastic differential equations (MSDEs) driven by Poisson point processes. The solutions to MSDEs driven by Poisson point processes can be approximated by solutions to averaged MSDEs in the sense of both convergence in mean square and convergence in probability. Finally, an example is presented to illustrate the averaging principle.     

Stochastic PDIEs and backward doubly stochastic differential equations driven by Lévy processes

In this paper, a new class of backward doubly stochastic differential equations driven by Teugels martingales associated with a Lévy process satisfying some moment condition and an independent Brownian motion is investigated. We obtain the existence and uniqueness of solutions to these equations. A probabilistic interpretation for solutions to a class of stochastic partial differential integral equations is given.     

Stochastic inertial manifold of dissipative wave equations with time delays

利用修正的Lyapunov-Perron方法研究随机耗散时滞波方程不变流形的存在性,证明了当谱间隙条件成立和时滞适当小时,随机耗散时滞波方程存在随机惯性流形,并且谱间隙条件与确定型时滞耗散波方程的一致.