A Schauder estimate for stochastic PDEs

TL;DR

This paper establishes a sharp Schauder estimate for parabolic stochastic PDEs with random coefficients, providing a key tool for analyzing their solutions in vector-valued Hölder spaces.

Contribution

It introduces a novel Schauder estimate for stochastic PDEs with random coefficients, advancing the regularity theory in this area.

Findings

Proves a sharp Schauder estimate for stochastic PDEs.

Establishes existence and uniqueness of solutions to the Cauchy problem.

Enhances understanding of regularity properties in stochastic PDEs.

Abstract

Considering stochastic partial differential equations of parabolic type with random coefficients in vector-valued H\"older spaces, we obtain a sharp Schauder estimate. As an application, the existence and uniqueness of solution to the Cauchy problem is also proved.