Initial measures for the stochastic heat equation

We consider a family of nonlinear stochastic heat equations of the form @tu = Lu + (u) _ W , where _ W denotes space-time white noise, L the generator of a symmetric L evy process on R, and is Lipschitz continuous and zero at 0. We show that this stochastic PDE has a random-eld solution for every nite initial measure u0. Tight a priori bounds on the moments of the solution are also obtained. In the particular case that Lf = cf 00 for some c > 0, we prove that if u0 is a nite measure of compact support, then the solution is with probability one a bounded function for all times t > 0.