Schauder estimates for stationary and evolution equations associated to stochastic reaction-diffusion equations driven by colored noise

TL;DR

This paper establishes Schauder estimates for stochastic reaction-diffusion equations driven by colored noise, addressing both stationary and evolution problems and highlighting the influence of noise color on regularity.

Contribution

It provides the first Schauder estimates for these equations considering the effect of colored noise on regularity properties.

Findings

Schauder estimates depend on the noise's color

Results apply to transition semigroups on bounded uniformly continuous functions

Enhances understanding of regularity in stochastic PDEs with colored noise

Abstract

We consider stochastic reaction-diffusion equations with colored noise and prove Schauder type estimates, which will depend on the color of the noise, for the stationary and evolution problems associated with the corresponding transition semigroup, defined on the Banach space of bounded and uniformly continuous functions.