The heat equation with Lévy noise (Q1805744)

Let \(D\subset \mathbb R^{d}\) be a bounded open domain, \(\gamma >0\), and \(\dot L( ds dy)\) a nonnegative Lévy noise of index \(p\in \left ]0,1\right [\). Denote by \(\Delta_\alpha = -(-\Delta) ^{\alpha /2}\) the \(\frac {\alpha}2\)-th power of the Laplacian, \(\alpha \in \left ]0,2\right ]\), let \(u_{0}\geq 0\) be a continuous function on \(D\). A stochastic heat equation driven by the Lévy noise \(\dot L\) is considered. It is proven that if \(d< ((\gamma +1)p - 1)^{-1}(1-p)\alpha\), then there exists a random time \(\tau >0\) such that with probability 1 the equation \[ \frac {\partial u}{\partial t} = \Delta_\alpha u + u^\gamma \dot L, \quad t>0, \;x\in D,\qquad u(t,\cdot) = 0 \;\text{on \(\mathbb R^{d}\setminus D\)}, \quad u(0,\cdot) = u_{0} \] has a mild solution \(u\) valid for \(t<\tau \). Moreover, it is shown that the constructed solution is minimal among all solutions.