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The heat equation with Lévy noise. (English) Zbl 0934.60056
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.

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
60J75 Jump processes (MSC2010)
Full Text: DOI
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