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

Let D⊂RdD⊂Rd be a bounded domain and letL=12∇⋅a∇+b⋅∇ be a second-order elliptic operator on D. Let ν be a probability measure on D  . Denote by LL the differential operator whose domain is specified by the following nonlocal boundary condition:DL={f∈C2(D¯):∫Dfdν=f|∂D}, and which coincides with L   on its domain. Clearly 0 is an eigenvalue for LL, with the corresponding eigenfunction being constant. It is known that LL 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 LL, indexed by ν, byγ1(ν)≡sup{Reλ:0≠λ is an eigenvalue for L}. In this paper we investigate the eigenvalues of LL in general and the spectral gap γ1(ν)γ1(ν) in particular.The operator LL and its spectral gap γ1(ν)γ1(ν) have probabilistic significance. The operator LL is the generator of a diffusion process with random jumps from the boundary, and γ1(ν)γ1(ν) measures the exponential rate of convergence of this process to its invariant measure.