Stochastic viscosity solutions for SPDEs with continuous coefficients

In this note, nonlinear stochastic partial differential equations (SPDEs) with continuous coefficients are studied. Via the solutions of backward doubly stochastic differential equations (BDSDEs) with continuous coefficients, we provide an existence result of stochastic viscosity sub- and super-solutions to this class of SPDEs. Under some stronger conditions, we prove the existence of stochastic viscosity solutions.     

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.     

Stochastic partial differential equations with unbounded and degenerate coefficients

In this article, using DiPerna-Lions theory (DiPerna and Lions, 1989) [1], we investigate linear second order stochastic partial differential equations with unbounded and degenerate non-smooth coefficients, and obtain several conditions for existence and uniqueness. Moreover, we also prove the L¹-integrability and a general maximal principle for generalized solutions of SPDEs. As applications, we study nonlinear filtering problem and also obtain the existence and uniqueness of generalized solutions for a degenerate nonlinear SPDE.     

A stochastic representation for backward incompressible Navier-Stokes equations

By reversing the time variable we derive a stochastic representation for backward incompressible Navier-Stokes equations in terms of stochastic Lagrangian paths, which is similar to Constantin and Iyer’s forward formulations in Constantin and Iyer (Comm Pure Appl Math LXI:330–345, 2008). Using this representation, a self-contained proof of local existence of solutions in Sobolev spaces are provided for incompressible Navier-Stokes equations in the whole space. In two dimensions or large viscosity, an alternative proof to the global existence is also given. Moreover, a large deviation estimate for stochastic particle trajectories is presented when the viscosity tends to zero.     

Stationary solutions of SPDEs and infinite horizon BDSDEs

In this paper we study the existence of stationary solutions for stochastic partial differential equations. We establish a new connection between valued solutions of backward doubly stochastic differential equations (BDSDEs) on infinite horizon and the stationary solutions of the SPDEs. Moreover, we prove the existence and uniqueness of the solutions of BDSDEs on both finite and infinite horizons, so obtain the solutions of initial value problems and the stationary solutions (independent of any initial value) of SPDEs. The connection of the weak solutions of SPDEs and BDSDEs has independent interests in the areas of both SPDEs and BSDEs.     

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.     

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.      

Existence,uniqueness, and numerical approximations for stochastic Burgers equations

Abstract

In this article, we propose an all-in-one statement which includes existence, uniqueness, regularity, and numerical approximations of mild solutions for a class of stochastic partial differential equations (SPDEs) with non-globally monotone nonlinearities. The proof of this result exploits the properties of an existing fully explicit space-time discrete approximation scheme, in particular the fact that it satisfies suitable a priori estimates. We also obtain almost sure and strong convergence of the approximation scheme to the mild solutions of the considered SPDEs. We conclude by applying the main result of the article to the stochastic Burgers equations with additive space-time white noise.     

Random attractors of stochastic partial differential equations: A smooth approximation approach

We use the method of smooth approximation to examine the random attractor for two classes of stochastic partial differential equations (SPDEs). Roughly speaking, we perturb the SPDEs by a Wong-Zakai scheme using smooth colored noise approximation rather than the usual polygonal approximation. After establishing the existence of the random attractor of the perturbed system, we prove that when the colored noise tends to the white noise, the random attractor of the perturbed system with colored noise converges to that of the original SPDEs by invoking some continuity results on attractors in random dynamical systems.     

Backward uniqueness and the existence of the spectral limit for linear parabolic SPDEs

The aim of this article is to study the asymptotic behavior for large times of solutions of linear stochastic partial differential equations of parabolic type. In particular, we will prove the backward uniqueness result and the existence of the spectral limit for abstract SPDEs and then show how these results can be applied to linear SPDEs.     

Stochastic nonhomogeneous incompressible Navier-Stokes equations

We construct solutions for 2- and 3-D stochastic nonhomogeneous incompressible Navier-Stokes equations with general multiplicative noise. These equations model the velocity of a mixture of incompressible fluids of varying density, influenced by random external forces that involve feedback; that is, multiplicative noise. Weak solutions for the corresponding deterministic equations were first found by Kazhikhov [A.V. Kazhikhov, Solvability of the initial and boundary-value problem for the equations of motion of an inhomogeneous viscous incompressible fluid, Soviet Phys. Dokl. 19 (6) (1974) 331-332; English translation of the paper in: Dokl. Akad. Nauk SSSR 216 (6) (1974) 1240-1243]. A stochastic version with additive noise was solved by Yashima [H.F. Yashima, Equations de Navier-Stokes stochastiques non homogènes et applications, Thesis, Scuola Normale Superiore, Pisa, 1992].The methods here extend the Loeb space techniques used to obtain the first general solutions of the stochastic Navier-Stokes equations with multiplicative noise in the homogeneous case [M. Capiński, N.J. Cutland, Stochastic Navier-Stokes equations, Applicandae Math. 25 (1991) 59-85]. The solutions display more regularity in the 2D case. The methods also give a simpler proof of the basic existence result of Kazhikhov.     

Jump-diffusions in Hilbert spaces: existence,stability and numerics

By means of an original approach, called ‘method of the moving frame’, we establish existence, uniqueness and stability results for mild and weak solutions of stochastic partial differential equations (SPDEs) with path-dependent coefficients driven by an infinite-dimensional Wiener process and a compensated Poisson random measure. Our approach is based on a time-dependent coordinate transform, which reduces a wide class of SPDEs to a class of simpler SDE (stochastic differential equation) problems. We try to present the most general results, which we can obtain in our setting, within a self-contained framework to demonstrate our approach in all details. Also, several numerical approaches to SPDEs in the spirit of this setting are presented.     

Wong–Zakai approximation and support theorem for SPDEs with locally monotone coefficients

In this paper we present the Wong–Zakai approximation results for a class of nonlinear SPDEs with locally monotone coefficients and driven by multiplicative Wiener noise. This model extends the classical monotone one and includes examples like stochastic 2d Navier–Stokes equations, stochastic porous medium equations, stochastic p-Laplace equations and stochastic reaction–diffusion equations. As a corollary, our approximation results also describe the support of the distribution of solutions.     

Regular solutions for multiplicative stochastic Landau-Lifshitz-Gilbert equation and blow-up phenomena

The two-dimensional Landau-Lifshitz-Gilbert equation of motion for a classical magnetic moment perturbed by a multiplicative noise is considered. This equation is highly nonlinear in nature and, for this reason, many mathematical results in stochastic partial differential equations (SPDEs) cannot be applied. The aim of this work is to introduce the difference method to handle SPDEs and prove the existence of regular martingale solutions in dimension two. Some blow-up phenomena are presented, which are drastically different from the deterministic case. Finally, to yield correct thermal-equilibrium properties, Stratonovitch integral is used instead of Ito integral.     

Stochastic partial differential equations with unbounded coefficients and applications i

The present paper is the first instalment of a three-part study of stochastic partial differentia! equations (SPDEs) having unbounded coefficients. In this paper we prove existence and uniqueness theorems for a large class of parabolic SPDEs (having unbounded data), including a class of systems of SPDEs     

Harnack and Shift Harnack Inequalities for Degenerate (Functional) Stochastic Partial Differential Equations with Singular Drifts

Journal of Theoretical Probability - The existence and uniqueness of the mild solutions for a class of degenerate functional stochastic partial differential equations (SPDEs) are obtained, where...     

The stability of stochastic partial differential equations II

We prove general results on stability (in finite time intervals) of SPDEs (stochastic partial differential equations) with unbounded coefficients, with respect to the simultaneous perturbations of the driving semimartingales, of all data, and of the underlying probability space. Hence we derive support theorems for SPDEs (with unbounded coefficients). In particular, we get theorems on supports and theorems on robustness for the nonlinear filter of diffusion processes with unbounded drift and diffusion coefficients. (The above results were proved in the case of bounded coefficients in our earlier papers [4] and [5].) Finally we treat an application in a problem of kinematic dynamo     

Anticipating stochastic differential systems with memory

This article establishes existence and uniqueness of solutions to two classes of stochastic systems with finite memory subject to anticipating initial conditions which are sufficiently smooth in the Malliavin sense. The two classes are semilinear stochastic functional differential equations (sfdes) and fully nonlinear sfdes with a sublinear drift term. For the semilinear case, we use Malliavin calculus techniques, existence of the stochastic semiflow and an infinite-dimensional substitution theorem. For the fully nonlinear case, we employ an anticipating version of the Itô–Ventzell formula due to Ocone and Pardoux [D. Ocone, E. Pardoux, A generalized Itô–Ventzell formula. Application to a class of anticipating stochastic differential equations, Annales de l’Institut Henri Poincaré. Probabilité s et Statistiques 25 (1) (1989) 39–71]. In both cases, the use of Malliavin calculus techniques is necessitated by the infinite dimensionality of the initial condition.     

Stochastic Volterra equations in Banach spaces and stochastic partial differential equation

In this paper, we study the existence-uniqueness and large deviation estimate for stochastic Volterra integral equations with singular kernels in 2-smooth Banach spaces. Then we apply them to a large class of semilinear stochastic partial differential equations (SPDE), and obtain the existence of unique maximal strong solutions (in the sense of SDE and PDE) under local Lipschitz conditions. Moreover, stochastic Navier-Stokes equations are also investigated.     

Nonlinear instability for nonhomogeneous incompressible viscous fluids

We investigate the nonlinear instability of a smooth steady density profile solution to the three-dimensional nonhomogeneous incompressible Navier-Stokes equations in the presence of a uniform gravitational field, including a Rayleigh-Taylor steady-state solution with heavier density with increasing height (referred to the Rayleigh-Taylor instability). We first analyze the equations obtained from linearization around the steady density profile solution. Then we construct solutions to the linearized problem that grow in time in the Sobolev space H ^k , thus leading to a global instability result for the linearized problem. With the help of the constructed unstable solutions and an existence theorem of classical solutions to the original nonlinear equations, we can then demonstrate the instability of the nonlinear problem in some sense. Our analysis shows that the third component of the velocity already induces the instability, which is different from the previous known results.