On stochastic partial differential equations with spatially correlated noise: smoothness of the law

AbstractWe deal with the following general kind of stochastic partial differential equations:Lu(t,x)=α(u(t,x))Ḟ(t,x)+β(u(t,x)),t⩾0,x∈Rdwith null initial conditions, L a second-order partial differential operator and F a Gaussian noise, white in time and correlated in space. Firstly, we prove that the solution u(t,x) possesses a smooth density pt,x for every t>0,x∈Rd. We use the tools of Malliavin Calculus. Secondly, we apply this general result to two particular cases: the d-dimensional spatial heat equation, d⩾1, and the wave equation, d∈{1,2}