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Let $D\subset R^d$ be a bounded domain and let \[ L=\frac12\nabla\cdota\nabla +b\cdot\nabla \] %\[%L=\frac12\sum_{i,j=1}^da_{i,j}\frac{\partial^2}{\partial x_i\partialx_j}+\sum_{i=1}^db_i\frac{\partial}{\partial x_i}, %\] be a second orderelliptic operator on $D$. Let $\nu$ be a probability measure on $D$.  Denote by ${\mathcal L}$ the differential operator whose domain is specifiedby the following non-local boundary condition: $$ {\mathcal D_{{\mathcalL}}}=\{f\in C^2(\ol{D}): \int_D f d\nu = f|_{\partial D}\}, $$ and whichcoincides with $L$ on its domain. It is known that $\mathcal L$ possesses aninfinite sequence of eigenvalues, and that with the exception of the zeroeigenvalue, 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 andthe spectral gap $\gamma_1(\nu)$ in particular. The operator $\mathcal L$ isthe generator of a diffusion process with random jumps from the boundary, and$\gamma_1(\nu)$ measures the exponential rate of convergence of this process toits invariant measure.

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