Unstable invariant manifolds for stochastic PDEs driven by a fractional Brownian motion

In this paper, we consider a class of stochastic partial differential equations (SPDEs) driven by a fractional Brownian motion (fBm) with the Hurst parameter bigger than 1/2. The existence of local random unstable manifolds is shown if the linear parts of these SPDEs are hyperbolic. For this purpose we introduce a modified Lyapunov-Perron transform, which contains stochastic integrals. By the singularities inside these integrals we obtain a special Lyapunov-Perron's approach by treating a segment of the solution over time interval [0,1] as a starting point and setting up an infinite series equation involving these segments as time evolves. Using this approach, we establish the existence of local random unstable manifolds in a tempered neighborhood of an equilibrium.     

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.     

Macroscopic limits for stochastic partial differential equations of McKean–Vlasov type

A class of quasilinear stochastic partial differential equations (SPDEs), driven by spatially correlated Brownian noise, is shown to become macroscopic (i.e., deterministic), as the length of the correlations tends to 0. The limit is the solution of a quasilinear partial differential equation. The quasilinear SPDEs are obtained as a continuum limit from the empirical distribution of a large number of stochastic ordinary differential equations (SODEs), coupled though a mean-field interaction and driven by correlated Brownian noise. The limit theorems are obtained by application of a general result on the convergence of exchangeable systems of processes. We also compare our approach to SODEs with the one introduced by Kunita.     

Trajectory fitting estimators for SPDEs driven by additive noise

In this paper we study the problem of estimating the drift/viscosity coefficient for a large class of linear, parabolic stochastic partial differential equations (SPDEs) driven by an additive space-time noise. We propose a new class of estimators, called trajectory fitting estimators (TFEs). The estimators are constructed by fitting the observed trajectory with an artificial one, and can be viewed as an analog to the classical least squares estimators from the time-series analysis. As in the existing literature on statistical inference for SPDEs, we take a spectral approach, and assume that we observe the first N Fourier modes of the solution, and we study the consistency and the asymptotic normality of the TFE, as \(N\rightarrow \infty \).     

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.     

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.     

Smooth solutions of non-linear stochastic partial differential equations driven by multiplicative noises

In this paper, we study the regularity of solutions of nonlinear stochastic partial differential equations (SPDEs) with multiplicative noises in the framework of Hilbert scales. Then we apply our abstract result to several typical nonlinear SPDEs such as stochastic Burgers and Ginzburg-Landau equations on the real line, stochastic 2D Navier-Stokes equations (SNSEs) in the whole space and a stochastic tamed 3D Navier-Stokes equation in the whole space, and obtain the existence of their smooth solutions respectively. In particular, we also get the existence of local smooth solutions for 3D SNSEs.     

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     

Strong convergence rates for nonlinearity-truncated Euler-type approximations of stochastic Ginzburg–Landau equations

This article proposes and analyzes explicit and easily implementable temporal numerical approximation schemes for additive noise-driven stochastic partial differential equations (SPDEs) with polynomial nonlinearities such as, e.g., stochastic Ginzburg–Landauequations. We prove essentially sharp strong convergence rates for the considered approximation schemes. Our analysis is carried out for abstract stochastic evolution equations on separable Banach and Hilbert spaces including the above mentioned SPDEs as special cases. We also illustrate our strong convergence rate results by means of a numerical simulation in Matlab.     

Ergodicity for a class of semilinear stochastic partial differential equations

In this paper, we establish the existence and uniqueness of invariant measures for a class of semilinear stochastic partial differential equations driven by multiplicative noise on a bounded domain. The main results can be applied to SPDEs of various types such as the stochastic Burgers equation and the reaction-diffusion equations perturbed by space-time white noise.     

UNIQUENESS PROBLEM FOR SPDES FROM POPULATION MODELS

This is a survey on the strong uniqueness of the solutions to stochastic partial differential equations(SPDEs) related to two measure-valued processes: superprocess and Fleming-Viot process which are given as rescaling limits of population biology models. We summarize recent results for Konno-Shiga-Reimers' and Mytnik's SPDEs, and their related distribution-function-valued SPDEs.     

A Runge–Kutta type scheme for nonlinear stochastic partial differential equations with multiplicative trace class noise

In this paper a new Runge–Kutta type scheme is introduced for nonlinear stochastic partial differential equations (SPDEs) with multiplicative trace class noise. The proposed scheme converges with respect to the computational effort with a higher order than the well-known linear implicit Euler scheme. In comparison to the infinite dimensional analog of Milstein type scheme recently proposed in Jentzen and Röckner (2012), our scheme is easier to implement and needs less computational effort due to avoiding the derivative of the diffusion function. The new scheme can be regarded as an infinite dimensional analog of Runge–Kutta method for finite dimensional stochastic ordinary differential equations (SODEs). Numerical examples are reported to support the theoretical results.     

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.      

A Stochastic Maximum Principle for a Stochastic Differential Game of a Mean-Field Type

We construct a stochastic maximum principle (SMP) which provides necessary conditions for the existence of Nash equilibria in a certain form of N-agent stochastic differential game (SDG) of a mean-field type. The information structure considered for the SDG is of a possible asymmetric and partial type. To prove our SMP we take an approach based on spike-variations and adjoint representation techniques, analogous to that of S.?Peng (SIAM J. Control Optim. 28(4):966?C979, 1990) in the optimal stochastic control context. In our proof we apply adjoint representation procedures at three points. The first-order adjoint processes are defined as solutions to certain mean-field backward stochastic differential equations, and second-order adjoint processes of a first type are defined as solutions to certain backward stochastic differential equations. Second-order adjoint processes of a second type are defined as solutions of certain backward stochastic equations of a type that we introduce in this paper, and which we term conditional mean-field backward stochastic differential equations. From the resulting representations, we show that the terms relating to these second-order adjoint processes of the second type are of an order such that they do not appear in our final SMP equations. A?comparable situation exists in an article by R.?Buckdahn, B.?Djehiche, and J.?Li (Appl. Math. Optim. 64(2):197?C216, 2011) that constructs a SMP for a mean-field type optimal stochastic control problem; however, the approach we take of using these second-order adjoint processes of a second type to deal with the type of terms that we refer to as the second form of quadratic-type terms represents an alternative to a development, to our setting, of the approach used in their article for their analogous type of term.     

带随机初值的随机偏微分方程整体解的存在性

研究了一类带随机初值并且由分数次Brownian运动驱动的随机偏微分方程.借助于Kolmogorov准则,建立了整体Lipschitz条件下此类随机偏微分方程的一个解.同时证明了局部Lipschitz条件下整体解的存在性.     

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 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...     

Forward-backward doubly stochastic differential equations and related stochastic partial differential equations

The notion of bridge is introduced for systems of coupled forward-backward doubly stochastic differential equations (FBDSDEs). It is proved that if two FBDSDEs are linked by a bridge, then they have the same unique solvability. Consequently, by constructing appropriate bridges, we obtain several classes of uniquely solvable FBDSDEs. Finally, the probabilistic interpretation for the solutions to a class of quasilinear stochastic partial differential equations (SPDEs) combined with algebra equations is given. One distinctive character of this result is that the forward component of the FBDSDEs is coupled with the backward variable.     

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.