APPROXIMATION AND SUPPORT THEOREM IN HOLDER NORM FOR PARABOLIC STOCHASTIC PARTIAL-DIFFERENTIAL EQUATIONS

The solution u(t, x) of a parabolic stochastic partial differential equation is a random element of the space C-alpha,C-beta of Holder continuous functions on [0, T] x [0, 1] of order (alpha = 1/4 - epsilon in the time variable and beta = 1/2 - epsilon in the space variable, for any epsilon > 0. We prove a support theorem in C-alpha,C-beta for the law of u. The proof is based on an approximation procedure in Holder norm (which should have its own interest) using a space-time polygonal interpolation for the Brownian sheet driving the SPDE, and a sequence of absolutely continuous transformations of the Wiener space.