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

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.