Spectral analysis of a family of second-order elliptic operators with nonlocal boundary condition indexed by a probability measure

Let D ⊂ Rd be a bounded domain and let L = 1 2∇ · a∇ +b · ∇ be a second-order elliptic operator on D. Let ν be a probability measure on D. Denote by L the differential operator whose domain is specified by the following nonlocal boundary condition: DL = f ∈ C2(D): D f dν = f |∂D , and which coincides with L on its domain. Clearly 0 is an eigenvalue for L, with the corresponding eigenfunction being constant. It is known that 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 L, indexed by ν, by γ1(ν) ≡ sup{Re λ: 0 = λ is an eigenvalue for L}. In this paper we investigate the eigenvalues of L in general and the spectral gap γ1(ν) in particularThe operator L and its spectral gap γ1(ν) have probabilistic significance. The operator L is the generator of a diffusion process with random jumps from the boundary, and γ1(ν) measures the exponential rate of convergence of this process to its invariant measure. © 2007 Elsevier Inc. All rights reserved