Variational solutions to nonlinear stochastic differential equations in Hilbert spaces

TL;DR

This paper introduces a new variational approach for solving nonlinear stochastic differential equations in Hilbert spaces, ensuring existence, uniqueness, and continuous dependence of solutions for a broad class of stochastic PDEs with monotone nonlinearities.

Contribution

It develops a novel variational concept of solutions applicable to stochastic PDEs with monotone nonlinearities, extending classical theories to more general and complex cases.

Findings

Existence and uniqueness of solutions for the new variational scheme

Continuous dependence of solutions on initial data

Applicability to stochastic variational inequalities and nonlinear stochastic PDEs

Abstract

One introduces a new variational concept of solution for the stochastic differential equation $dX+A(t)X\,dt+\lambda X\,dt=X\,dW,$ $t\in(0,T)$; $X(0)=x$ in a real Hilbert space where $A(t)=\partial\varphi(t)$, $t\in(0,T)$, is a maximal monotone subpotential operator in $H$ while $W$ is a Wiener process in $H$ on a probability space $\{\Omega,\mathcal{F},\mathbb{P}\}$. In this new context, the solution $X=X(t,x)$ exists for each $x\in H$, is unique, and depends continuously on $x$. This functional scheme applies to a general class of stochastic PDE not covered by the classical variational existence theory ([15], [16], [17]) and, in particular, to stochastic variational inequalities and parabolic stochastic equations with general monotone nonlinearities with low or superfast growth to $+\infty$.