H\"older estimates for solutions of parabolic SPDEs

TL;DR

This paper proves that solutions to certain second-order parabolic stochastic PDEs with additive noise are almost surely H"older continuous, with finite moments of their H"older norms, under specific integrability conditions on the noise coefficients.

Contribution

It establishes H"older regularity of solutions to parabolic SPDEs with additive noise under $L^p$ conditions on the noise coefficients, extending regularity results in stochastic PDE theory.

Findings

Solutions are almost surely H"older continuous.

H"older norms have finite moments of all orders.

Regularity holds under large $p$-integrability conditions on noise coefficients.

Abstract

This paper considers second-order stochastic partial differential equations with additive noise given in a bounded domain of $\mathbb R^n$. We suppose that the coefficients of the noise are $L^p$-functions with sufficiently large $p$. We prove that the solutions are H\"older-continuous functions almost surely (a.s.) and that the respective H\"older norms have finite momenta of any order.