SPDEs with $\alpha$-stable L\'evy noise: a random field approach

This article is dedicated to the study of an SPDE of the form $$Lu(t,x)=\sigma(u(t,x))\dot{Z}(t,x) \quad t>0, x \in \cO$$ with zero initial conditions and Dirichlet boundary conditions, where $\sigma$ is a Lipschitz function, $L$ is a second-order pseudo-differential operator, $\cO$ is a bounded domain in $\bR^d$, and $\dot{Z}$ is an $\alpha$-stable L\'evy noise with $\alpha \in (0,2)$, $\alpha\not=1$ and possibly non-symmetric tails. To give a meaning to the concept of solution, we develop a theory of stochastic integration with respect to $Z$, by generalizing the method of Gin\'e and Marcus (1983) to higher dimensions and non-symmetric tails. The idea is to first solve the equation with "truncated" noise $\dot{Z}_{K}$ (obtained by removing from $Z$ the jumps which exceed a fixed value $K$), yielding a solution $u_{K}$, and then show that the solutions $u_L,L>K$ coincide on the event $t \leq \tau_{K}$, for some stopping times $\tau_K \uparrow \infty$ a.s. A similar idea was used in Peszat and Zabczyk (2007) in the setting of Hilbert-space valued processes. A major step is to show that the stochastic integral with respect to $Z_{K}$ satisfies a $p$-th moment inequality, for $p \in (\alpha,1)$ if $\alpha<1$, and $p \in (\alpha,2)$ if $\alpha>1$. This inequality plays the same role as the Burkholder-Davis-Gundy inequality in the theory of integration with respect to continuous martingales.