Initial measures for the stochastic heat equation

We consider a family of nonlinear stochastic heat equations of the form ∂_tu = Lu + σ(u)W, where W denotes space- time white noise, L the generator of a symmetric Lévy process on R, and σ is Lipschitz continuous and zero at 0. We show that this stochastic PDE has a random-field solution for every finite initial measure u₀. Tight a priori bounds on the moments of the solution are also obtained. In the particular case that Lf = cf'' for some c > 0, we prove that if u₀ is a finite measure of compact support, then the solution is with probability one a bounded function for all times t > 0.