Initial measures for the stochastic heat equation

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We consider a family of nonlinear stochastic heat equations of the form ∂ₜ u=Lu + σ(u)W, where W denotes space-time white noise, L the generator of a symmetric Lévy process on, 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.