Stochastic De Giorgi Iteration and Regularity of Stochastic Partial Differential Equation

TL;DR

This paper proves that solutions to a broad class of stochastic parabolic PDEs are almost surely Hölder continuous in space and time, under general conditions, extending regularity results to stochastic settings.

Contribution

It establishes almost sure Hölder continuity for solutions of stochastic parabolic PDEs with general conditions, broadening the understanding of regularity in stochastic PDEs.

Findings

Solutions are almost surely Hölder continuous in space and time.

Regularity results hold under broad, general conditions.

Extends deterministic PDE regularity theory to stochastic PDEs.

Abstract

Under general conditions we show that the solution of a stochastic parabolic partial differential equation of the form \[ \partial_t u = \mathrm{div} (A \nabla u) + f(t,x, u) + g_i (t,x,u) \dot{w}^i_t \] is almost surely H\"older continuous in both space and time variables.