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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:
 [1] Athreya K.B., Ney, P.E., 1972. Branching Processes. Springer Berlin. · Zbl 0259.60002 [2] Dawson D.A., 1993. Measure-valued Markov processes. In P.L. Hennequin, (ed.), Ecole d’Ete de Probabilites de Saint Flour XXI-1991, Lecture Notes in Mathematics Springer, Berlin. · Zbl 0799.60080 [3] Ito K., 1961. Lectures on Stochastic Processes. Tata Institute of Fundamental Research, Bombay, [4] Konno, N.; Shiga, T., Stochastic partial differential equations for some measure-valued diffusions, Probab. theory rel. fields, 79, 201-225, (1988) · Zbl 0631.60058 [5] Lamperti, J., Continuous state branching processes, Bull. amer. math. soc., 73, 3, 382-386, (1967) · Zbl 0173.20103 [6] Mueller, C., Long time existence for the heat equation with a noise term, Probab. theory related fields, 90, 505-518, (1991) · Zbl 0729.60055 [7] Neveu J., A continuous state branching process in relation with the grem model of spin glasses theory. Preliminary version, 1992. [8] Rosinski, J.; Woyczynski, W.A., On ito stochastic integration with respect top-stable motion:inner clock, integrability of sample paths, double and multiple integrals, Ann. probab., 14, 271-286, (1986) · Zbl 0594.60056 [9] Walsh J.B., 1986 An introduction to stochastic partial differential equations. In P. L. Hennequin, (ed.), Ecole d’Ete de Probabilites de Saint Flour XIV-1984, Lecture Notes in Math. 1180 Springer, Berlin. · Zbl 0608.60060 [10] Watanabe, S., A limit theorem of branching processes and continuous state branching processes, J. math. Kyoto U., 3, 141-167, (1968) · Zbl 0159.46201
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