Stochastic De Giorgi iteration and regularity of stochastic partial differential equations

Abstract

Under general conditions, we devise a stochastic version of De Giorgi iteration scheme for semilinear stochastic parabolic partial differential equation of the form

\[\partial_{t}u=\operatorname{div}(A\nabla u)+f(t,x,u)+g_{i}(t,x,u)\dot{w}^{i}_{t}\] with progressively measurable diffusion coefficients. We use the scheme to show that the solution of the equation is almost surely Hölder continuous in both space and time variables.