Amplitude equation for the stochastic reaction‐diffusion equations with random Neumann boundary conditions

In this paper, we consider a quite general class of reaction‐diffusion equations with cubic nonlinearities and with random Neumann boundary conditions. We derive rigorously amplitude equations, using the natural separation of time‐scales near a change of stability and investigate whether additive degenerate noise and random boundary conditions can lead to stabilization of the solution of the stochastic partial differential equation or not. The nonlinear heat equation (Ginzburg–Landau equation) is used to illustrate our result. Copyright © 2015 John Wiley & Sons, Ltd.     

On Strongly Petrovskii's Parabolic SPDEs in Arbitrary Dimension and Application to the Stochastic Cahn–Hilliard Equation

In this paper we show that the Cahn–Hilliard stochastic PDE has a function valued solution in dimension 4 and 5 when the perturbation is driven by a space-correlated Gaussian noise. We study the regularity of the trajectories of the solution and the absolute continuity of its law at some given time and position. This is done by showing a priori estimates which heavily depend on the specific equation, and by proving general results on stochastic and deterministic integrals involving general operators on smooth domains of ^d which are parabolic in the sense of Petrovskii, and do not necessarily define a semi-group of operators. These last estimates might be used in a more general framework.     

Discontinuous Galerkin method for elliptic stochastic partial differential equations on two and three dimensional spaces

In this paper,we study a discontinuous Galerkin numerical scheme for a class of elliptic stochastic partial differential equations (abbr.elliptic SPDEs) driven by space white noises with ho- mogeneous Dirichlet boundary conditions for two and three space dimensions.We also establish L~2 error estimates for the scheme.In particular,a numerical test for d=2 is presented at the end of the article.     

Malliavin calculus for regularity structures: The case of gPAM

Malliavin calculus is implemented in the context of Hairer (2014) [16]. This involves some constructions of independent interest, notably an extension of the structure which accommodates a robust, and purely deterministic, translation operator, in $L^2$-directions, between “models”. In the concrete context of the generalized parabolic Anderson model in 2D – one of the singular SPDEs discussed in the afore-mentioned article – we establish existence of a density at positive times.     

On degenerate backward SPDEs in bounded domains under non-local conditions

ABSTRACT

The paper studies backward stochastic partial differential equations (BSPDEs) of parabolic type in bounded domains in the setting where the coercivity condition is not necessary satisfied and under special non-local in time and space boundary conditions replacing the standard Cauchy condition. These conditions connect the terminal value of the solution with a functional over the entire past solution. Uniqueness, solvability, and regularity results are obtained. As an example of applications, it is shown that degenerate BSPDEs with non-local boundary conditions arise naturally in modelling of portfolio selection problems, including models where dividend payoffs and management fees are taken into account.     

L p -Theory of Parabolic SPDE s Degenerating on the Boundary of C 1 Domains

Degenerate stochastic partial differential equations are considered in C ¹ domains. Existence and uniqueness results are given in weighted Sobolev spaces, and H?lder estimates of the solutions are presented. This work was supported by a Korea University grant.     

Averaging principle for multiscale stochastic reaction‐diffusion‐advection equations

Stochastic averaging principle is a powerful tool for studying qualitative analysis of multiscale stochastic dynamical systems. In this paper, we will establish an averaging principle for stochastic reaction‐diffusion‐advection equations with slow and fast time scales. Under suitable conditions, we show that the slow component strongly converges to the solution of the corresponding averaged equation.     

Malliavin calculus for infinite-dimensional systems with additive noise

We consider an infinite-dimensional dynamical system with polynomial nonlinearity and additive noise given by a finite number of Wiener processes. By studying how randomness is spread by the dynamics, we develop in this setting a partial counterpart of Hörmander's classical theory of Hypoelliptic operators. We study the distributions of finite-dimensional projections of the solutions and give conditions that provide existence and smoothness of densities of these distributions with respect to the Lebesgue measure. We also apply our results to concrete SPDEs such as a Stochastic Reaction Diffusion Equation and the Stochastic 2D Navier-Stokes System.     

Hypercontractivity for space–time white noise driven SPDEs with reflection

The hypercontractive property of the Markov semigroup associated with the reflected stochastic partial differential equation driven by the additive space–time white noise is mainly investigated. The main tool for its proof is the general criterion presented recently by F.-Y. Wang [29]. In particular, we obtain the hyperboundedness and the compactness of the Markov semigroup, the exponential convergences of the entropy, the exponential convergences of the Markov semigroup to its unique invariant measure in both $L^2$ and the total variation norm.