Polarity of almost all points for systems of non-linear stochastic heat equations in the critical dimension

Abstract

We study vector-valued solutions $u(t,x)\in {\mathbb{R}}^d$ to systems of nonlinear stochastic heat equations with multiplicative noise: \begin{equation*} \frac{\partial}{\partial t} u(t,x)=\frac{\partial^2}{\partial x^2} u(t,x)+σ(u(t,x))\dot{W}(t,x). \end{equation*} Here $t\ge 0$, $x\in \mathbb{R}$ and $\dot{W}(t,x)$ is an ${\mathbb{R}}^d$-valued space-time white noise. We say that a point $z\in {\mathbb{R}}^d$ is polar if \begin{equation*} P\{u(t,x)=z\text{ for some $t>0$ and $x\in \mathbb{R}$}\}=0. \end{equation*} We show that in the critical dimension $d=6$, almost all points in ${\mathbb{R}}^d$ are polar.

30 pages

References in corpus (3)

• Hitting Probabilities for Systems of Non-Linear Stochastic Heat Equations with Additive Noise
• Potential theory for hyperbolic SPDEs
• Multiple Points of Gaussian Random Fields