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

Overview & Synopsis

We study a discrete and continuous version of the spectral Dirichlet problem in an open bounded connected set Ω ⊂ R d , in dimension d ≥ 2. More precisely, consider the simple random walk on Z d killed upon exiting the (large) bounded domain Ω N = (N Ω) ∩ Z d . We let P N 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 Ω, we give regularity estimates on ϕ N , namely on its k-th order differences, with a uniform control inside Ω 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 φ 1 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 L 2 and uniform convergence of ϕ N to φ 1 in Lipschitz bounded domains of R d , which can be derived thanks to our estimates.