Reaction-diffusion models for a class of infinite-dimensional non-linear stochastic differential equations

Noise and Stability in Reaction-diffusion Equations

We study the stability of reaction-diffusion equations in presence of noise. The relationship of stability of solutions between the stochastic ordinary different equations and the corresponding stochastic reaction-diffusion equation is firstly established. Then, by using the Lyapunov method, sufficient conditions for mean square and stochastic stability are given. The results show that the multiplicative noise can make the solution stable, but the additive noise will be not.

A Khasminskii type averaging principle for stochastic reaction-diffusion equations

We prove that an averaging principle holds for a general class of stochastic reaction-diffusion systems, having unbounded multiplicative noise, in any space dimension. We show that the classical Khasminskii approach for systems with a finite number of degrees of freedom can be extended to infinite dimensional systems.

An integral inequality for the invariant measure of a stochastic reaction--diffusion equation

We consider a reaction--diffusion equation perturbed by noise (not necessarily white). We prove an integral inequality for the invariant measure $ν$ of a stochastic reaction--diffusion equation. Then we discuss some consequences as an integration by parts formula which extends to $ν$ a basic identity of the Malliavin Calculus. Finally, we prove the existence of a surface measure for a ball and a half-space of $H$.

具有满足三角质量控制的超线性反应的随机反应扩散系统的全局时间解

Global in time solutions to stochastic reaction-diffusion systems with superlinear reactions satisfying a triangular control of mass

研究受乘性噪声扰动的反应扩散系统，在反应项满足拟正性、三角质量控制和多项式增长条件下，证明全局解的存在性。

随机与确定性的非自治反应扩散方程

Stochastic and deterministic non-autonomous reaction-diffusion equations

本文研究了具有多项式反应项的非自治确定性与随机反应扩散方程，证明了其适定性，并得出了非自治随机卷积时空正则性的新结果。

A phase diagram for a stochastic reaction diffusion system

In this paper a stochastic reaction diffusion system is considered, which models the spread of a finite population reacting with a non-renewable resource in the presence of individual based noise. A two-parameter phase diagram is established to describe the large time evolution, distinguishing between certain death or possible life of the population.

Time--space white noise eliminates global solutions in reaction diffusion equations

We prove that perturbing the reaction--diffusion equation $u_t=u_{xx} + (u_+)^p$ ($p>1$), with time--space white noise produces that solutions explodes with probability one for every initial datum, opposite to the deterministic model where a positive stationary solution exists.

Homogenization of a stochastic nonlinear reaction-diffusion equation with a large reaction term: the almost periodic framework

Homogenization of a stochastic nonlinear reaction-diffusion equation with a large non- linear term is considered. Under a general Besicovitch almost periodicity assumption on the coefficients of the equation we prove that the sequence of solutions of the said problem converges in probability towards the solution of a rather different type of equation, namely, the stochastic non- linear convection-diffusion equation which we explicitly derive in terms of appropriated functionals. We study some particular cases such as the periodic framework, and many others. This is achieved under a suitable ge…

EDP-convergence for a linear reaction-diffusion system with fast reversible reaction

We perform a fast-reaction limit for a linear reaction-diffusion system consisting of two diffusion equations coupled by a linear reaction. We understand the linear reaction-diffusion system as a gradient flow of the free energy in the space of probability measure equipped with a geometric structure, which contains the Wasserstein metric for the diffusion part and cosh-type functions for the reaction part. The fast-reaction limit is done on the level of the gradient structure by proving EDP-convergence with tilting. The limit gradient system induces a diffusion system with Lagrange multipliers…