# zbMATH — the first resource for mathematics

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
Full Text: