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

We deal with the following general kind of stochastic partial differential equations: Lu(t,x)= alpha (u(t,x))(F) over dot (t,x)+ beta (u(t,x)), t greater than or equal to0, x is an element of R-d with null initial conditions, L a second-order partial differential operato r and F a Gaussian noise, white in time and correlated in space. Firstly, w e prove that the solution tr(t,x) possesses a smooth density p(t,x) for eve ry t > 0, x is an element of R-d. We the tools of Malliavin Calculus. Secon dly, we apply this general result to two particular cases: the d-dimensiona l spatial heat equation, d greater than or equal to 1, and the wave equatio n, d is an element of {1,2}. (C) 2001 Elsevier Science B.V. All rights rese rved.