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

Abstract

Let D ⊂ R d 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: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 γ 1 (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 γ 1 (ν) in particular. 123 The operator L and its spectral gap γ 1 (ν) have probabilistic significance. The operator L is the generator of a diffusion process with random jumps from the boundary, and γ 1 (ν) measures the exponential rate of convergence of this process to its invariant measure.

Spectral analysis of the diffusion operator with random jumps from the boundary

“…Due to the non-self-adjointness of the operator, it is not at all clear in which sense the eigenfunctions can be expected to be a basis of the associated Hilbert space. In these respects we further develop certain strands of research first developed in [8], whose authors calculated among other things the spectrum of the above operator in the case a = 0; see also [3] and [4], where the authors derive results on the spectrum of the above operator including geometric multiplicities of the eigenvalues.…”

Spectral analysis of a class of nonlocal elliptic operators related to Brownian motion with random jumps

“…The principal eigenvalue of the generator of the process is 0, the rest of the spectrum is negative, and the spectral gap, which is the supremum of the real part of the nonzero spectrum, gives the exponential rate of convergence to equilibrium. See [1], [2], [8], [9], [11].…”

Spectral analysis of Brownian motion with jump boundary

“…The equations (1.9) and (1.10) were conjectured in [3] and [2] respectively, and Proposition 2 in [2] shows that (1.10) is a consequence of the main result in Theorem 1.2. In [3], it was also conjectured that…”

“…the jumps are deterministic and concentrated at a single point p ∈ D, then the ergodicity of BMJ was studied in [5], [8] using Laplace transform methods and the theory of analytic semigroups. In the case where ν y = ν for all y ∈ ∂D (ν need not be a point measure), the ergodicity of BMJ (as an important special case of elliptic operators) was systematically studied in [2] using a functional analytic approach. Recently, the most general case, where ν y depends continuously on its exit point y ∈ ∂D, was studied in [3].…”

“…Recently, the most general case, where ν y depends continuously on its exit point y ∈ ∂D, was studied in [3]. The key results in [2], [3] address the construction of the invariant probability measure and relate the rate of convergence of the transition distribution to the invariant measure to the spectral gap of the generator of BMJ.…”