Higher-order stochastic partial differential equations with branching noises

In this paper, we propose a class of higher-order stochastic partial differential equations (SPDEs) with branching noises. The existence of weak (mild) solutions is established through weak convergence and tightness arguments.      

Asymptotic analysis of a kernel estimator for parabolic stochastic partial differential equations driven by fractional noises

We study a strongly elliptic partial differential operator with time-varying coeffcient in a parabolic diagonalizable stochastic equation driven by fractional noises. Based on the existence and uniqueness of the solution, we then obtain a kernel estimator of time-varying coeffcient and the convergence rates. An example is given to illustrate the theorem.     

Degenerate parabolic stochastic partial differential equations

We study the Cauchy problem for a scalar semilinear degenerate parabolic partial differential equation with stochastic forcing. In particular, we are concerned with the well-posedness in any space dimension. We adapt the notion of kinetic solution which is well suited for degenerate parabolic problems and supplies a good technical framework to prove the comparison principle. The proof of existence is based on the vanishing viscosity method: the solution is obtained by a compactness argument as the limit of solutions of nondegenerate approximations.     

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.     

A numerical approximation of parabolic stochastic partial differential equations driven by a Poisson random measure

The paper deals with SPDEs driven by Poisson random measures in Banach spaces and its numerical approximation. We investigate the accuracy of space and time approximation. As the space approximation we consider spectral methods and as time approximation the implicit Euler scheme and the explicit Euler scheme. AMS subject classification (2000) 60H15, 35R30     

Impacts of Gaussian noises on the blow-up times of nonlinear stochastic partial differential equations

In this paper, a blow-up problem to nonlinear stochastic partial differential equations driven by Brownian motions is investigated. In particular, the impact of noises on the life span of solutions is studied. It is interesting to know that the noise can be used to postpone the blow-up time of a stochastic nonlinear system.     

Regularities for semilinear stochastic partial differential equations

In this paper, we study the regularities of solutions to semilinear stochastic partial differential equations in general settings, and prove that the solution can be smooth arbitrarily when the data is sufficiently regular. As applications, we also study several classes of semilinear stochastic partial differential equations on abstract Wiener space, complete Riemannian manifold as well as bounded domain in Euclidean space.     

On a jump-type stochastic fractional partial differential equation with fractional noises

We study a class of stochastic fractional partial differential equations of order α>1$\alpha >1$ driven by a (pure jump) Lévy space–time white noise and a fractional noise. We prove the existence and uniqueness of the global mild solution by the fixed point principle under some suitable assumptions.     

Approximate solution for some stochastic differential equations involving both Gaussian and Poissonian white noises

By combining the Kramers-Moyal expansion with fractional Brownian motion of order n, in a formal symbolic calculus, one can obtain an approximation for the solution of some stochastic differential equations involving both Gaussian and Poissonian white noises, in terms of rotating Gaussian white noises on the grid defined by the complex roots of the unity. Illustrative examples are outlined.     

On Runge-Kutta-type methods for two-dimensional stochastic differential equations

In the multidimensional case, second-order weak Runge-Kutta methods for stochastic differential equation (SDE) need simulation of correlated random variables, unless the diffusion matrix of SDE satisfies the commutativity condition. In this paper, we show that this can be avoided for some types of diffusion matrices and test functions important for applications. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 3, pp. 403–412, July–September, 2006.     

Large deviation for stochastic Cahn-Hilliard partial differential equations

In this paper, we prove a large deviation principle for a class of stochastic Cahn-Hilliard partial differential equations driven by space-time white noises.     

Variational solutions for partial differential equations driven by a fractional noise

In this article we develop an existence and uniqueness theory of variational solutions for a class of nonautonomous stochastic partial differential equations of parabolic type defined on a bounded open subset D⊂R^d and driven by an infinite-dimensional multiplicative fractional noise. We introduce two notions of such solutions for them and prove their existence and their indistinguishability by assuming that the noise is derived from an L²(D)-valued fractional Wiener process W^H with Hurst parameter , whose covariance operator satisfies appropriate integrability conditions, and where γ∈(0,1] denotes the Hölder exponent of the derivative of the nonlinearity in the stochastic term of the equations. We also prove the uniqueness of solutions when the stochastic term is an affine function of the unknown random field. Our existence and uniqueness proofs rest upon the construction and the convergence of a suitable sequence of Faedo-Galerkin approximations, while our proof of indistinguishability is based on certain density arguments as well as on new continuity properties of the stochastic integral we define with respect to W^H.     

Shift Harnack inequality and integration by parts formula for semilinear stochastic partial differential equations

Shift Harnack inequality and integration by parts formula are established for semilinear stochastic partial differential equations and stochastic functional partial differential equations by modifying the coupling used by F.-Y. Wang [Ann. Probab., 2012, 42(3): 994–1019]. Log-Harnack inequality is established for a class of stochastic evolution equations with non-Lipschitz coefficients which includes hyperdissipative Navier-Stokes/Burgers equations as examples. The integration by parts formula is extended to the path space of stochastic functional partial differential equations, then a Dirichlet form is defined and the log-Sobolev inequality is established.     

Neutral stochastic functional differential equations with additive perturbations

The paper deals with the solution to the neutral stochastic functional differential equation whose coefficients depend on small perturbations, by comparing it with the solution to the corresponding unperturbed equation of the equal type. We give conditions under which these solutions are close in the (2m)th mean, on finite time-intervals and on intervals whose length tends to infinity as small perturbations tend to zero.     

Implicit Euler method for numerical solution of nonlinear stochastic partial differential equations with multiplicative trace class noise

In this paper, we consider the numerical approximation of stochastic partial differential equations with nonlinear multiplicative trace class noise. Discretization is obtained by spectral collocation method in space, and semi‐implicit Euler method is used for the temporal approximation. Our purpose is to investigate the convergence of the proposed method. The rate of convergence is obtained, and some numerical examples are included to illustrate the estimated convergence rate.     

The interrelation between stochastic differential inclusions and set-valued stochastic differential equations

In this paper we connect the well established theory of stochastic differential inclusions with a new theory of set-valued stochastic differential equations. Solutions to the latter equations are understood as continuous mappings taking on their values in the hyperspace of nonempty, bounded, convex and closed subsets of the space L²$L^2$ consisting of square integrable random vectors. We show that for the solution X$X$ to a set-valued stochastic differential equation corresponding to a stochastic differential inclusion, there exists a solution x$x$ for this inclusion that is a _‖⋅‖L2${\Vert \cdot \Vert }_{L^2}$-continuous selection of X$X$. This result enables us to draw inferences about the reachable sets of solutions for stochastic differential inclusions, as well as to consider the viability problem for stochastic differential inclusions.     

Solutions of stochastic partial differential equations as extremals of convex functionals

Summary Stochastic evolution equations with monotone operators in Banach spaces are considered. The solutions are characterized as minimizers of certain convex functionals. The method of monotonicity is interpreted as a method of constructing minimizers to these functionals, and in this way solutions are constructed via Euler-Galerkin approximations.     

A variational formula for controlled backward stochastic partial differential equations and some application

An optimal control problem for a controlled backward stochastic partial differential equation in the abstract evolution form with a Bolza type performance functional is considered. The control domain is not assumed to be convex, and all coefficients of the system are allowed to be random. A variational formula for the functional in a given control process direction is derived, by the Hamiltonian and associated adjoint system. As an application, a global stochastic maximum principle of Pontraygins type for the optimal controls is established.