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

We study a discrete and continuous version of the spectral Dirichlet problem in an open bounded connected set Ω⊂Rd, in dimension d≥2. More precisely, consider the simple random walk on Zd killed upon exiting the (large) bounded domain ΩN=(NΩ)∩Zd. We let PN its transition matrix and we study the properties of its (L2-normalized) principal eigenvector ϕN, also known as ground state. Under mild assumptions on Ω, we give regularity estimates on ϕN, namely on its k-th order differences (or k-th order derivatives), with a uniform control inside ΩN. We provide a completely probabilistic proof of these estimates: our starting point is a Feynman–Kac representation of ϕN, 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 φ1 of the corresponding continuous spectral Dirichlet problem, in relation with a Brownian motion killed upon exiting Ω. Finally, we take the opportunity to review (and slightly extend) some of the literature on the L2 and uniform convergence of ϕN to φ1 in Lipschitz bounded domains of Rd, which can be derived thanks to our estimates.