Martingale solutions and Markov selection of stochastic 3D Navier-Stokes equations with jump

In this paper, we study the existence of martingale solutions of stochastic 3D Navier-Stokes equations with jump, and following Flandoli and Romito (2008) [7] and Goldys et al. (2009) [8], we prove the existence of Markov selections for the martingale solutions.     

Invariant manifolds for stochastic wave equations

In this paper, we consider a class of stochastic wave equations with nonlinear multiplicative noise. We first show that these stochastic wave equations generate random dynamical systems (or stochastic flows) by transforming the stochastic wave equations to random wave equations through a stationary random homeomorphism. Then, we establish the existence of random invariant manifolds for the random wave equations. Due to the temperedness of the nonlinearity, we obtain only local invariant manifolds no matter how large the spectral gap is unlike the deterministic cases. Based on these random dynamical systems, we prove the existence of random invariant manifolds in a tempered neighborhood of an equilibrium. Finally, we show that the images of these invariant manifolds under the inverse stationary transformation give invariant manifolds for the stochastic wave equations.     

Systems of stochastic partial differential equations with reflection: Existence and uniqueness

In this paper, we establish the existence and uniqueness of solutions of systems of stochastic partial differential equations (SPDEs) with reflection in a convex domain. The lack of comparison theorems for systems of SPDEs makes things delicate.     

On signed measure valued solutions of stochastic evolution equations

We study existence, uniqueness and mass conservation of signed measure valued solutions of a class of stochastic evolution equations with respect to the Wiener sheet, including as particular cases the stochastic versions of the regularized two-dimensional Navier–Stokes equations in vorticity form introduced by Kotelenez.     

Lower bound technique in the theory of a stochastic differential equation

We establish a criterion for the existence of an invariant measure for Markov processes acting on measures defined on an arbitrary complete separable metric space. This criterion is applied to time-homogeneous Markov processes associated with a nonlinear heat equation driven by an impulsive noise.     

Backward stochastic differential equations associated to jump Markov processes and applications

In this paper we study backward stochastic differential equations (BSDEs) driven by the compensated random measure associated to a given pure jump Markov process X$X$ on a general state space K$K$. We apply these results to prove well-posedness of a class of nonlinear parabolic differential equations on K$K$, that generalize the Kolmogorov equation of X$X$. Finally we formulate and solve optimal control problems for Markov jump processes, relating the value function and the optimal control law to an appropriate BSDE that also allows to construct probabilistically the unique solution to the Hamilton–Jacobi–Bellman equation and to identify it with the value function.     

Generalized stochastic evolution equations

An approach to generalized stochastic evolution equations is presented which is based on a generalized Ito formula. This allows the consideration of interesting examples which are stochastic generalizations of evolution equations of mixed type or second order in time hyperbolic equations. It includes more standard material involving a Gelfand triple of spaces as a special case. Several examples are given which illustrate the use of the abstract theory presented.     

Strong convergence of split-step backward Euler method for stochastic differential equations with non-smooth drift

In this paper, we are concerned with the numerical approximation of stochastic differential equations with discontinuous/nondifferentiable drifts. We show that under one-sided Lipschitz and general growth conditions on the drift and global Lipschitz condition on the diffusion, a variant of the implicit Euler method known as the split-step backward Euler (SSBE) method converges with strong order of one half to the true solution. Our analysis relies on the framework developed in [D. J. Higham, X. Mao and A. M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM Journal on Numerical Analysis, 40 (2002) 1041-1063] and exploits the relationship which exists between explicit and implicit Euler methods to establish the convergence rate results.     

Stochastic morphological evolution equations

The inadequacy of locally defined set-valued differential equations to describe the evolution of shapes and morphological forms in biology, which are usually neither convex or nondecreasing, was recognised by J.-P. Aubin, who introduced morphological evolution equations, which are essentially nonlocally defined set-valued differential equations with the inclusion vector field also depending on the entire reachable set. This concept is extended here to the stochastic setting of set-valued Itô evolution equations in Hilbert spaces. Due to the nonanticipative nature of Itô calculus, the evolving reachable sets are nonanticipative nonempty closed random sets. The existence of solutions and their dependence on initial data are established. The latter requires the introduction of a time-oriented semi-metric in time-space variables. As a consequence the stochastic morphological evolution equations generate a deterministic nonautonomous dynamical system formulated as a two-parameter semigroup with the complication that the random subsets take values in different spaces at different time instances due to the nonanticipativity requirement. It is also shown how nucleation processes can be handled in this conceptual framework.     

Extremal solutions for stochastic equations indexed by negative integers and taking values in compact groups

Stochastic equations indexed by negative integers and taking values in compact groups are studied. Extremal solutions of the equations are characterized in terms of infinite products of independent random variables. This result is applied to characterize several properties of the set of all solutions in terms of the law of the driving noise.     

Convergence of invariant measures for singular stochastic diffusion equations

It is proved that the solutions to the singular stochastic p$p$-Laplace equation, p∈(1,2)$p\in (1,2)$ and the solutions to the stochastic fast diffusion equation with nonlinearity parameter r∈(0,1)$r\in (0,1)$ on a bounded open domain Λ⊂R^d$\Lambda \subset {\mathbb{R}}^d$ with Dirichlet boundary conditions are continuous in mean, uniformly in time, with respect to the parameters p$p$ and r$r$ respectively (in the Hilbert spaces L²(Λ)$L^2\left(\Lambda \right)$, H⁻¹(Λ)$H^{-1}\left(\Lambda \right)$ respectively). The highly singular limit case p=1$p=1$ is treated with the help of stochastic evolution variational inequalities, where P$\mathbb{P}$-a.s. convergence, uniformly in time, is established.     

Exponential stability for neutral stochastic partial differential equations with delays and Poisson jumps

In this paper, we consider a class of neutral stochastic partial differential equations with delays and Poisson jumps. Sufficient conditions for the existence and exponential stability in mean square as well as almost surely exponential stability of mild solutions are derived by means of the Banach fixed point principle. An example is provided to illustrate the effectiveness of the proposed result.     

Ergodicity of the 3D stochastic Navier-Stokes equations driven by mildly degenerate noise

We prove that any Markov solution to the 3D stochastic Navier-Stokes equations driven by a mildly degenerate noise (i.e. all but finitely many Fourier modes are forced) is uniquely ergodic. This follows by proving strong Feller regularity and irreducibility.     

A stochastic heat equation with the distributions of Lévy processes as its invariant measures

We consider a linear heat equation on a half line with an additive noise chosen properly in such a manner that its invariant measures are a class of distributions of Lévy processes. Our assumption on the corresponding Lévy measure is, in general, mild except that we need its integrability to show that the distributions of Lévy processes are the only invariant measures of the stochastic heat equation.     

Stochastic equations of super-Lévy processes with general branching mechanism

In this work, the process of distribution functions of a one-dimensional super-Lévy process with general branching mechanism is characterized as the pathwise unique solution of a stochastic integral equation driven by time–space Gaussian white noises and Poisson random measures. This generalizes the recent work of Xiong (2013), where the result for a super-Brownian motion with binary branching mechanism was obtained.     

Invariant measure for the stochastic Ginzburg Landau equation

The existence of martingale solutions and stationary solutions of stochastic Ginzburg-Landau equations under general hypothesizes on the dimension, the non linear term and the added noise is investigated. With a few more assumptions, it is established that the transition semi-group is well defined and that the stationary martingale solution yields the existence of an invariant measure. Moreover this invariant measure is shown to be unique.