Asymptotic Behavior of the Principal Eigenvalue for a Class of Non-Local Elliptic Operators Related to Brownian Motion with Spatially Dependent Random Jumps

TL;DR

This paper analyzes the asymptotic behavior of the principal eigenvalue of a non-local elliptic operator related to Brownian motion with spatially dependent jumps, revealing how it scales with the jump intensity parameter.

Contribution

It provides explicit asymptotic formulas for the principal eigenvalue as the jump intensity grows large, depending on the boundary behavior of the jump distribution.

Findings

For measures with density near the boundary, eigenvalue scales as gamma^{1/2}.

If the measure's derivatives vanish up to order k-1 at the boundary, eigenvalue scales as gamma^{(1-k)/2}.

Explicit constants are derived for the asymptotic behavior.

Abstract

Let $D\subset R^d$ be a bounded domain and let $\mathcal P(D)$ denote the space of probability measures on $D$. Consider a Brownian motion in $D$ which is killed at the boundary and which, while alive, jumps instantaneously according to a spatially dependent exponential clock with intensity $\gamma V$ to a new point, according to a distribution $\mu\in\mathcal P(D)$. From its new position after the jump, the process repeats the above behavior independently of what has transpired previously. The generator of this process is an extension of the operator $-L_{\gamma,\mu}$, defined by L_{\gamma,\mu}u\equiv -\frac12\Delta u+\gamma V C_\mu(u), with the Dirichlet boundary condition, where $C_\mu$ is the "$\mu$-centering" operator defined by C_\mu(u)=u-\int_Du d\mu. The principal eigenvalue, $\lambda_0(\gamma,\mu)$, of $L_{\gamma,\mu}$ governs the exponential rate of decay of the probability of not exiting $D$ for large time. We study the asymptotic behavior of $\lambda_0(\gamma,\mu)$ as $\gamma\to\infty$. In particular, if $\mu$ possesses a density in a neighborhood of the boundary, which we call $\mu$, then \lim_{\gamma\to\infty}\gamma^{-\frac12}\lambda_0(\gamma,\mu)=\frac{\int_{\partial D}\frac\mu{\sqrt {V}}d\sigma}{\sqrt2\int_D\frac1{V}d\mu}. If $\mu$ and all its derivatives up to order $k-1$ vanish on the boundary, but the $k$-th derivative does not vanish identically on the boundary, then $\lambda_0(\gamma,\mu)$ behaves asymptotically like $c_k\gamma^{\frac{1-k}2}$, for an explicit constant $c_k$.