Spectral Analysis of a Family of Second-Order Elliptic Operators with Nonlocal Boundary Condition Indexed by a Probabilty Measure

TL;DR

This paper analyzes the spectral properties of a second-order elliptic operator with a nonlocal boundary condition defined by a probability measure, focusing on the spectral gap that indicates the convergence rate of the associated diffusion process.

Contribution

It introduces a detailed spectral analysis of a class of elliptic operators with nonlocal boundary conditions, highlighting the role of the spectral gap in diffusion convergence.

Findings

Eigenvalues are characterized with all but zero having negative real parts.

The spectral gap quantifies the exponential convergence rate of the diffusion process.

The operator models diffusion with boundary jumps, linking spectral properties to probabilistic behavior.

Abstract

Let $D\subset R^d$ be a bounded domain and let \[ L=\frac12\nabla\cdot a\nabla +b\cdot\nabla \] %\[ %L=\frac12\sum_{i,j=1}^da_{i,j}\frac{\partial^2}{\partial x_i\partial x_j}+\sum_{i=1}^db_i\frac{\partial}{\partial x_i}, %\] be a second order elliptic operator on $D$. Let $\nu$ be a probability measure on $D$. Denote by ${\mathcal L}$ the differential operator whose domain is specified by the following non-local boundary condition: $$ {\mathcal D_{{\mathcal L}}}=\{f\in C^2(\ol{D}): \int_D f d\nu = f|_{\partial D}\}, $$ and which coincides with $L$ on its domain. It is known that $\mathcal L$ possesses an infinite sequence of eigenvalues, and that with the exception of the zero eigenvalue, all eigenvalues have negative real part. Define the spectral gap of $\mathcal {L}$, indexed by $\nu$, by \gamma_1(\nu)\equiv\sup\{\re \lambda:0\neq \lambda is an eigenvalue for {\mathcal L}\}. In this paper we investigate the eigenvalues of $\mathcal L$ in general and the spectral gap $\gamma_1(\nu)$ in particular. The operator $\mathcal L$ is the generator of a diffusion process with random jumps from the boundary, and $\gamma_1(\nu)$ measures the exponential rate of convergence of this process to its invariant measure.