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

Let $u(t,x)=(u_1(t,x),\dots ,u_d(t,x))$ be the solution to the systems of nonlinear stochastic heat equations $$\begin{array}{rl}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}u(t,x)}& {\displaystyle =\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}u(t,x)+\sigma (u(t,x))\dot{W}(t,x),}\\ {\displaystyle u(0,x)}& {\displaystyle =u_0(x),}\end{array}$$ 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.