Global and Local Properties of the Solution of Stochastic Partial Differential Equations with Measurable Coefficients

This dissertations concerns the {it a priori} global and local properties of solutions of stochastic partial differential equations with measurable coefficients. Under general conditions we show that the solution of a stochastic parabolic partial differential equation of the form ∂tu = div (A∇u) + f( t,x,u;o) + gi(t,x,u;o) w˙it is almost surely Holder continuous in both space and time variables. Under a few slightly refined conditions, we show the solution of the above equation also satisfies a probabilistic Harnack inequality and any solution starts with a non-negative and not identically vanishing initial value will stay strict positive after any small time.