Stochastic De Giorgi iteration and regularity of stochastic partial differential equations

Under general conditions, we devise a stochastic version of De Giorgi iteration scheme for semilinear stochastic parabolic partial differential equation of the form .tu=div(A.u)+f(t,x,u)+gi(t,x,u).wit 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.