Stability of solutions of parabolic PDEs with random drift and viscosity limit

Let u(alpha) be the solution of the Ito stochastic parabolic Cauchy problem partial derivative u/partial derivative t - L(alpha)u = xi . del u, u\(t=0 ) = f, where xi is a space-time noise. We prove that u(alpha) depends conti nuously on alpha, when the coefficients in L-alpha converge to those in L-0 . This result is used to study the diffusion limit for the Cauchy problem i n the Stratonovich sense: when the coefficients of L-alpha tend to 0 the co rresponding solutions u(alpha) converge to the solution u(0) of the degener ate Cauchy problem partial derivative u(0)/partial derivative t = xi circle del u(0), u(0)\(t=0) = f. These results are based on a criterion for the e xistence of strong limits in the space of Hida distributions (S)*. As a by- product it is proved that weak solutions of the above Cauchy problem are in fact strong solutions.