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Comparison principle for stochastic heat equation on \(\mathbb{R}^{d}\). (English) Zbl 1433.60049
Summary: We establish the strong comparison principle and strict positivity of solutions to the following nonlinear stochastic heat equation on \(\mathbb{R}^{d}\)\[\biggl(\frac{\partial}{\partial t}-\frac{1}{2}\Delta \biggr)u(t,x)=\rho\bigl(u(t,x)\bigr)\dot{M}(t,x),\] for measure-valued initial data, where \(\dot{M}\) is a spatially homogeneous Gaussian noise that is white in time and \(\rho\) is Lipschitz continuous. These results are obtained under the condition that \(\int_{\mathbb{R}^{d}}(1+|\xi|^{2})^{\alpha-1}\hat{f}(\mathrm{d}\xi)<\infty\) for some \(\alpha\in(0,1]\), where \(\hat{f}\) is the spectral measure of the noise. The weak comparison principle and nonnegativity of solutions to the same equation are obtained under Dalang’s condition, that is, \(\alpha=0\). As some intermediate results, we obtain handy upper bounds for \(L^{p}(\Omega)$-moments of \(u(t,x)\) for all \(p\ge2\), and also prove that \(u\) is a.s. Hölder continuous with order \(\alpha-\varepsilon\) in space and \(\alpha/2-\varepsilon\) in time for any small \(\varepsilon>0\).

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
60G60 Random fields
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