A linear stochastic biharmonic heat equation: hitting probabilities

TL;DR

This paper analyzes the hitting probabilities of solutions to a linear stochastic biharmonic heat equation on a torus, providing precise anisotropic descriptions and bounds on the likelihood of the process hitting sets.

Contribution

It introduces a detailed pseudo-distance for the solution, characterizes anisotropies, and establishes bounds on hitting probabilities for the process.

Findings

Derived the canonical pseudo-distance for the solution

Described anisotropies including a logarithmic term in 2D

Established bounds on hitting probabilities and Hausdorff dimensions

Abstract

Consider the linear stochastic biharmonic heat equation on a $d$-dimensional torus ($d=1,2,3$), driven by a space-time white noise and with periodic boundary conditions: \begin{equation} \label{0} \left(\frac{\partial}{\partial t}+(-\Delta)^2\right) v(t,x)= \sigma \dot W(t,x),\ (t,x)\in(0,T]\times \mathbb{T}^d,\ v(0,x)=v_0(x). \end{equation} We find the canonical pseudo-distance corresponding to the random field solution, therefore the precise description of the anisotropies of the process. We see that for $d=2$, they include a $z(\log \tfrac{c}{z})^{1/2}$ term. Consider $D$ independent copies of the random field solution to the SPDEs. Applying the criteria proved in [4], we establish upper and lower bounds for the probabilities that the path process hits bounded Borel sets. This yields results on the polarity of sets and on the Hausdorff dimension of the path process.