On a Class of Martingale Problems on Banach Spaces

TL;DR

This paper develops a framework linking martingale problems to weak solutions for semilinear stochastic evolution equations in Banach spaces, and explores their Markov properties and applications to complex stochastic PDEs.

Contribution

It introduces a novel local martingale problem approach for semilinear stochastic evolution equations in Banach spaces, establishing solution equivalences and Markov properties.

Findings

One-to-one correspondence between martingale problem solutions and weak solutions.

Solutions to well-posed equations are strong Markov processes.

Application to stochastic reaction-diffusion equations with H"older continuous noise.

Abstract

We introduce the local martingale problem associated to semilinear stochastic evolution equations driven by a cylindrical Wiener process and establish a one-to-one correspondence between solutions of the martingale problem and (analytically) weak solutions of the stochastic equation. We also prove that the solutions of well-posed equations are strong Markov processes. We apply our results to semilinear stochastic equations with additive noise where the semilinear term is merely measurable and to stochastic reaction-diffusion equations with H\"older continuous multiplicative noise.