Polarity of almost all points for systems of nonlinear stochastic heat equations in the critical dimension

Abstract

We study vector-valued solutions $\mathit{u}(\mathit{t},\mathit{x})\in {\mathbb{R}}^{\mathit{d}}$ to systems of nonlinear stochastic heat equations with multiplicative noise,

$$\frac{\partial }{\partial \mathit{t}}\mathit{u}(\mathit{t},\mathit{x})=\frac{{\partial }^2}{\partial {\mathit{x}}^2}\mathit{u}(\mathit{t},\mathit{x})+\mathit{\sigma }(\mathit{u}(\mathit{t},\mathit{x}))\dot{\mathit{W}}(\mathit{t},\mathit{x}).$$

Here, $\mathit{t}\ge 0$, $\mathit{x}\in \mathbb{R}$ and $\dot{\mathit{W}}(\mathit{t},\mathit{x})$ is an ${\mathbb{R}}^{\mathit{d}}$-valued space–time white noise. We say that a point $\mathit{z}\in {\mathbb{R}}^{\mathit{d}}$ is polar if

$$\mathit{P}\{\mathit{u}(\mathit{t},\mathit{x})=\mathit{z}\text{for some}\mathit{t}>0\text{and}\mathit{x}\in \mathbb{R}\}=0.$$

We show that, in the critical dimension $\mathit{d}=6$, almost all points in ${\mathbb{R}}^{\mathit{d}}$ are polar.