Initial measures for the stochastic heat equation

We consider a family of nonlinear stochastic heat equations of the form ${\mathrm{\partial }}_{t}u=\mathcal{L}u+\sigma (u)\dot{W}$, where $\dot{W}$ denotes space-time white noise, $\mathcal{L}$ the generator of a symmetric L'evy process on $\mathbb{R}$, 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 $\mathcal{L}f=cf\mathrm{"}$ for some $c>0$, we prove that if $u_0$ is a finite measure of compact support, then the solution is with probability one a bounded function for all times $t>0$.