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

Abstract

Let $u(t,x)=(u_1(t,x),\dots ,u_d(t,x))$ be the solution to the systems of nonlinear stochastic heat equations \[ \begin{split} \frac{\partial}{\partial t} u(t, x) &= \frac{\partial^2}{\partial x^2} u(t, x) + σ(u(t, x)) \dot{W}(t, x),\\ u(0, x) &= u_0(x), \end{split} \] where $t\ge 0$, $x\in \mathbb{R}$, $\dot{W}(t,x)=({\dot{W}}_1(t,x),\dots ,{\dot{W}}_d(t,x))$ is a vector of $d$ independent space-time white noises, and $\sigma :{\mathbb{R}}^d\to {\mathbb{R}}^{d\times d}$ is a matrix-valued function. We say that a subset $S$ of ${\mathbb{R}}^d$ is polar for $\{u(t,x),t\ge 0,x\in \mathbb{R}\}$ if \[ \mathbb{P}\{u(t,x) \in S \text{ for some } t>0 \text{ and } x\in\mathbb{R} \}=0. \] The main result of this paper shows that, in the critical dimension $d=6$, all points in ${\mathbb{R}}^d$ are polar for $\{u(t,x),t\ge 0,x\in \mathbb{R}\}$. This solves an open problem of Dalang, Khoshnevisan and Nualart (2009, 2013) and Dalang, Mueller and Xiao (2021). We also provide a sufficient condition for a subset $S$ of ${\mathbb{R}}^d$ to be polar.

arXiv commentThere is a crucial error in the paper because the formula (4.2) is not true in general. As a result, the decomposition (4.3) is not valid, which leads to a gap in the proof