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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.

##### MSC:
 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 35R60 PDEs with randomness, stochastic partial differential equations 60J75 Jump processes (MSC2010)
##### Keywords:
stochastic heat equations; Lévy processes; mild solutions
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##### References:
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