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 $\mathrm{\Omega }\subset {\mathbb{R}}^d$, in dimension $d\ge 2$. More precisely, consider the simple random walk on ${\mathbb{Z}}^d$ killed upon exiting the (large) bounded domain ${\mathrm{\Omega }}_N=(N\mathrm{\Omega })\cap {\mathbb{Z}}^d$. 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 $\mathrm{\Omega }$, we give regularity estimates on ${\phi }_N$, namely on its $k$-th order differences (or $k$-th order derivatives), with a uniform control inside ${\mathrm{\Omega }}_N$. 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 ${\phi }_1$ of the corresponding continuous spectral Dirichlet problem, in relation with a Brownian motion killed upon exiting $\mathrm{\Omega }$. 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 ${\phi }_1$ in Lipschitz bounded domains of ${\mathbb{R}}^d$, which can be derived thanks to our estimates.