Stochastic De Giorgi Iteration and Regularity of Stochastic Partial Differential Equation

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.