Initial measures for the stochastic heat equation

We consider a family of nonlinear stochastic heat equations of the form partial derivative(t)u = Lu + sigma(u)(W)over dot, where (W)over dot denotes space time white noise, L the generator of a symmetric Levy process on R, and sigma is Lipschitz continuous and zero at 0. We show that this stochastic PDE has a random-field solution for every finite initial measure u(0). Tight a priori bounds on the moments of the solution are also obtained. In the particular case that L f = c '' for some c> 0, we prove that if up is a finite measure of compact support, then the solution is with probability one a bounded function for all times t > 0.