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

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 $γV$ to a new point, according to a distribution $μ\in\mathcal P(D)$. 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 $-L_{γ,μ}$, defined by L_{γ,μ}u\equiv -\frac12Δu+γV C_μ(u), with the Dirichlet boundary condition, where $C_μ$ is the "$μ$-centering" operator defined by C_μ(u)=u-\int_Du dμ. The principal eigenvalue, $λ_0(γ,μ)$, of $L_{γ,μ}$ governs the exponential rate of decay of the probability of not exiting $D$ for large time. We study the asymptotic behavior of $λ_0(γ,μ)$ as $γ\to\infty$. In particular, if $μ$ possesses a density in a neighborhood of the boundary, which we call $μ$, then \lim_{γ\to\infty}γ^{-\frac12}λ_0(γ,μ)=\frac{\int_{\partial D}\fracμ{\sqrt {V}}dσ}{\sqrt2\int_D\frac1{V}dμ}. If $μ$ 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 $λ_0(γ,μ)$ behaves asymptotically like $c_kγ^{\frac{1-k}2}$, for an explicit constant $c_k$.