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 $\Omega\subset \mathbb{R}^d$, in dimension $d\geq 2$. More precisely, consider the simple random walk on $\mathbb{Z}^d$ killed upon exiting the (large) bounded domain $\Omega_N = (N\Omega)\cap \mathbb{Z}^d$. 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 $\Omega$, we give regularity estimates on $\phi_N$, namely on its $k$-th order differences (or $k$-th order derivatives), with a uniform control inside $\Omega_N$. 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 $\varphi_1$ 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 $\phi_N$ to $\varphi_1$ in Lipschitz bounded domains of $\mathbb{R}^d$, which can be derived thanks to our estimates.