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

• Published in: arXiv.org 2023-08
• Main Authors: Cheuk Yin Lee, Xiao, Yimin
• Format: Article
• Language: English
• Subjects: Mathematical analysis; Nonlinear systems; Thermodynamics
• ISSN: 2331-8422

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