The compact support property for solutions to stochastic heat equations with stable noise

arXiv2212.04520MaRDI QIDQ6420014

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Publication date: 8 December 2022

Abstract: We consider weak non-negative solutions to the stochastic partial differential equation �egin{equation*} partial_t Y_t(x) = Delta Y_t(x) + Y_{t-}(x)^gamma dot{L}(t,x), end{equation*} for $(t,x)inmathbb{R}_{+}imesmathbb{R}^d$, where $gamma>0$ and $dotL$ is a one-sided stable noise of index $alphain(1,2)$. We prove that solutions with compactly supported initial data have compact support for all times if $gammain(2-alpha,1)$ for $d=1$, and if $gammain[1/alpha,1)$ in dimensions $din[2,2/(alpha-1))capmathbbN$. This complements known results on solutions to the equation with Gaussian noise. We also establish a stochastic integral formula for the density of a solution and associated moment bounds which hold in all dimensions for which solutions are defined.

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