About the principal Dirichlet eigenfunction in bounded domains via simple random walk and Brownian motion couplings (preprint)

Abstract

We study a discrete and continuous version of the spectral Dirichlet problem in an open bounded connected set \Omega, in dimension 2 and more. More precisely, consider the simple random walk on \mathbb{Z}^d killed upon exiting the (large) bounded domain \Omega_N = (N \cdot \Omega \cap) \mathbb{Z}^d. We let P_N be its transition matrix and we study the properties of its (L^2-normalized) principal eigenvector ϕ_N, also known as ground state. Under mild assumptions on Omega, we give regularity estimates on ϕ_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 ϕ_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 ϕ_N to \varphi_1 in Lipschitz bounded domains of \mathbb{R}^d, which can be derived thanks to our estimates.