A splitting algorithm for stochastic partial differential equations driven by linear multiplicative noise

TL;DR

This paper introduces and analyzes a splitting algorithm based on Douglas-Rachford for solving infinite-dimensional stochastic PDEs with linear multiplicative noise, demonstrating convergence under certain conditions.

Contribution

The paper develops a convergence analysis for a Douglas-Rachford type splitting algorithm applied to stochastic PDEs with nonlinear monotone operators.

Findings

Proves convergence of the splitting algorithm for specific stochastic PDEs.

Extends analysis to maximal monotone operators in Hilbert spaces.

Provides theoretical foundation for numerical methods in stochastic PDEs.

Abstract

We study the convergence of a Douglas-Rachford type splitting algorithm for the infinite dimensional stochastic differential equation $$dX+A(t)(X)dt=X\,dW\mbox{ in }(0,T);\ X(0)=x,$$ where $A(t):V\to V'$ is a nonlinear, monotone, coercive and demicontinuous operator with sublinear growth and $V$ is a real Hilbert space with the dual $V'$. $V$ is densely and continuously embedded in the Hilbert space $H$ and $W$ is an $H$-valued Wiener process. The general case of a maximal monotone operators $A(t):H\to H$ is also investigated.