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

Abstract

We study a discrete and continuous version of the spectral Dirichlet problem in an open bounded connected set $\hat{I}\text{c◯}\subset {\mathbb{R}}^d$, in dimension $d\ge 2$. More precisely, consider the simple random walk on ${\mathbb{Z}}^d$ killed upon exiting the (large) bounded domain $\hat{I}{\text{c◯}}_N=(N\hat{I}\text{c◯})\cap {\mathbb{Z}}^d$. We let $P_N$ its transition matrix and we study the properties of its ($L^2$-normalized) principal eigenvector $\ddot{I}{•}_N$, also known as ground state. Under mild assumptions on $\hat{I}\text{c◯}$, we give regularity estimates on $\ddot{I}{•}_N$, namely on its $k$-th order differences (or \(k\)-th order derivatives), with a uniform control inside $\hat{I}{\text{c◯}}_N$. We provide a completely probabilistic proof of these estimates: our starting point is a Feynman-Kac representation of $\ddot{I}{•}_N$, 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 $\ddot{I}{†}_1$ of the corresponding continuous spectral Dirichlet problem, in relation with a Brownian motion killed upon exiting $\hat{I}\text{c◯}$. Finally, we take the opportunity to review (and slightly extend) some of the literature on the $L^2$ and uniform convergence of $\ddot{I}{•}_N$ to $\ddot{I}{†}_1$ in Lipschitz bounded domains of ${\mathbb{R}}^d$, which can be derived thanks to our estimates.

37 pages, 1 figure. This version contains several improvements compared to the version 1. We include a new coupling to deal with higher-order differences and more complete overview of the $L^2$ and $L^{\infty }$ convergence of the discrete eigenfunction to its continuous counterpart. We include several new discussions with the literature. Comments are welcome !