On a Class of Martingale Problems on Banach Spaces

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 Hölder continuous multiplicative noise.

Incorporated referee's comments; final version

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• Stochastic integration in UMD Banach spaces
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Cited by in corpus (4)

• An optimal regularity result for Kolmogorov equations and weak uniqueness for some critical SPDEs
• Weak convergence rates for numerical approximations of stochastic partial differential equations with nonlinear diffusion coefficients in UMD Banach spaces
• Stochastic reaction-diffusion systems with Hölder continuous multiplicative noise
• On the equivalence of solutions for a class of stochastic evolution equations in a Banach space