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

Quentin Berger, Nicolas Bouchot

Abstract

We study a discrete and continuous version of the spectral Dirichlet problem in an open bounded connected set $Ω\subset \mathbb{R}^d$, in dimension $d\geq 2$. More precisely, consider the simple random walk on $\mathbb{Z}^d$ killed upon exiting the (large) bounded domain $Ω_N = (NΩ)\cap \mathbb{Z}^d$. 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 $\varphi_1$ 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 $\varphi_1$ in Lipschitz bounded domains of $\mathbb{R}^d$, which can be derived thanks to our estimates.

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

Abstract

We study a discrete and continuous version of the spectral Dirichlet problem in an open bounded connected set , in dimension . More precisely, consider the simple random walk on killed upon exiting the (large) bounded domain . We let its transition matrix and we study the properties of its (-normalized) principal eigenvector , also known as ground state. Under mild assumptions on , we give regularity estimates on , namely on its -th order differences, with a uniform control inside . We provide a completely probabilistic proof of these estimates: our starting point is a Feynman--Kac representation of , 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 of the corresponding continuous spectral Dirichlet problem, in relation with a Brownian motion killed upon exiting . Finally, we take the opportunity to review (and slightly extend) some of the literature on the and uniform convergence of to in Lipschitz bounded domains of , which can be derived thanks to our estimates.
Paper Structure (28 sections, 20 theorems, 143 equations, 1 figure)

This paper contains 28 sections, 20 theorems, 143 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Dirichlet eigenvalue problem and a discrete approximation
  3. Our main results in a nutshell.
  4. Simple random walk and Brownian motion in a (large) bounded domain
  5. Main results: regularity properties of the eigenfunctions
  6. Under the positive reach condition, Assumption \ref{['hyp:D']}
  7. Under the exterior cone condition, Assumption \ref{['hyp:cone']}
  8. Convergence of eigenfunctions and some consequences
  9. Some comments
  10. Some preliminaries and first results
  11. Rough bounds on the first eigenvalue
  12. Random walk and confined random walk, Feynman--Kac representation
  13. Gambler's ruin estimates and a priori bounds on $|\phi_N(x)|$
  14. Regularity of $\phi_N$ and $\varphi_1$ via couplings
  15. Single difference estimates and simple mirror coupling
  16. ...and 13 more sections

Key Result

Theorem 1.1

Suppose that $\Omega$ is a Lipschitz domain (see Assumption hyp:cone below for a precise definition). Then, there exist some (explicit) $p\in (0,1]$ and some constant $C>0$ such that, for all $k\geq 0$: where $D_{i_1,\ldots, i_k}$ is the $k$-th order finite difference operator in direction $(i_1, \ldots, i_k)$, see its definition derivatives below. Note that for $k=0$, the results are simply abou

Figures (1)

  • Figure 1: On the left a successful coupling; the two walks exit $B(z,R)$ through the same point. On the right, the coupling fails, and the two walks reach $\partial B(z,R)$ before $\mathcal{H}$ and the coupling fails; the exit points are different (and symmetric).

Theorems & Definitions (37)

  • Remark 1.1: About the normalization
  • Theorem 1.1
  • Proposition 2.1
  • Remark 2.1
  • Theorem 2.2: Differences of $\phi_N$
  • Remark 2.2
  • Theorem 2.3: $k$-th order differences of $\phi_N$
  • Remark 2.3
  • Theorem 2.4
  • Proposition 2.5
  • ...and 27 more