Optimal Regularity of Stochastic Evolution Equations in M-type 2 Banach Spaces

TL;DR

This paper establishes the well-posedness and optimal regularity of solutions to stochastic evolution equations in M-type 2 Banach spaces, extending existing methods and solving an open problem for the stochastic heat equation.

Contribution

It introduces a novel approach combining Hilbert space techniques with Burkholder inequalities in M-type 2 Banach spaces, advancing the understanding of stochastic evolution equations.

Findings

Proves well-posedness of stochastic evolution equations in M-type 2 Banach spaces.

Establishes optimal trajectory regularity for solutions.

Provides a solution to an open problem for the stochastic heat equation.

Abstract

In this paper, we prove the well-posedness and op- timal trajectory regularity for the solution of stochastic evolution equations driven by general multiplicative noises in martingale type 2 Banach spaces. The main idea of our method is to combine the approach in [HL] dealing with Hilbert setting and a version of Burkholder inequality in M-type 2 Banach space. Applying our main results to the stochastic heat equation gives a positive an- swer to an open problem proposed in [JR12].