Some properties of the principal Dirichlet eigenfunction in Lipschitz domains, via probabilistic couplings

TL;DR

This paper provides probabilistic regularity estimates for the principal eigenfunction of Dirichlet problems in Lipschitz domains, using couplings and Feynman-Kac representations for both discrete and continuous cases.

Contribution

It introduces a novel probabilistic coupling method to derive regularity estimates for eigenfunctions in Lipschitz domains, applicable to both discrete and continuous spectral problems.

Findings

Regularity estimates for eigenfunctions' differences and derivatives

Probabilistic proof via Feynman-Kac and gambler's ruin techniques

Extension of convergence results for eigenfunctions in Lipschitz domains

Abstract

We study a discrete and continuous version of the spectral Dirichlet problem in an open bounded connected set $\Omega\subset \mathbb{R}^d$, in dimension $d\geq 2$. More precisely, consider the simple random walk on $\mathbb{Z}^d$ killed upon exiting the (large) bounded domain $\Omega_N = (N\Omega)\cap \mathbb{Z}^d$. We let $P_N$ its transition matrix and we study the properties of its ($L^2$-normalized) principal eigenvector $\phi_N$, also known as ground state. Under mild assumptions on $\Omega$, we give regularity estimates on $\phi_N$, namely on its $k$-th order differences (or \(k\)-th order derivatives), with a uniform control inside $\Omega_N$. We provide a completely probabilistic proof of these estimates: our starting point is a Feynman-Kac representation of $\phi_N$, combined with gambler's ruin estimates and a new ``multi-mirror'' coupling, which may be of independent interest. We also obtain the same type of estimates for the first eigenfunction $\varphi_1$ of the corresponding continuous spectral Dirichlet problem, in relation with a Brownian motion killed upon exiting $\Omega$. Finally, we take the opportunity to review (and slightly extend) some of the literature on the $L^2$ and uniform convergence of $\phi_N$ to $\varphi_1$ in Lipschitz bounded domains of $\mathbb{R}^d$, which can be derived thanks to our estimates.