Stability and genericity for spde's driven by spatially correlated noise

Abstract

We consider stochastic partial differential equations, on $\mathbb{R}^{d}$ $(d\geq 1)$, driven by a Gaussian noise white in time and colored in space. Assuming pathwise uniqueness holds, we establish various strong stability results. As consequence, we give an application to the convergence of the Picard successive approximation. Finally, we show that in the sense of Baire category, almost all stochastic partial differential equations with continuous and bounded coefficients have the properties of existence and pathwise uniqueness of solutions as well as the continuous dependence on the coefficients.