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Asymptotics for Exit Problem and Principal Eigenvalue for a Class of Non-Local Elliptic Operators Related to Diffusion Processes with Random Jumps and Vanishing Diffusion

Let $D\subset R^d$ be a bounded domain and denote by $\mathcal P(D)$ the space of probability measures on $D$. Let \begin{equation*} L=\frac12\nabla\cdot a\nabla +b\nabla \end{equation*} be a second order elliptic operator. Let $μ\in\mathcal P(D)$ and $δ>0$. Consider a Markov process $X(t)$ in $D$ which performs diffusion in $D$ generated by the operator $δL$ and is stopped at the boundary, and which while running, jumps instantaneously, according to an exponential clock with spatially dependent intensity $V>0$, to a new point, according to the distribution $μ$. The Markov process is generated by the operator $L_{δ,μ, V}$ defined by \begin{equation*} L_{δ,μ, V}ϕ\equiv δL ϕ+V(\int_Dϕdμ-ϕ). \end{equation*} %where $C_μ$ is the % "$μ$-centering" operator defined by %\begin{equation*} %C_μ(ϕ)=ϕ-\int_Dϕdμ. %\end{equation*} Let $ϕ_{δ,μ,V}$ denote the solution to the Dirichlet problem \begin{equation*}\label{Dirprob} \begin{aligned} &L_{δ,μ,V}ϕ=0\ \text{in}\ D;\\ &ϕ=f\ \text{on}\ \partial D, \end{aligned} \end{equation*} where $f$ is continuous. The solution has the stochastic representation \begin{equation*} ϕ_{δ,μ,V}(x)=E_xf(X(τ_D)). \end{equation*} One has that $ϕ_{0,μ,V}(f)\equiv\lim_{δ\to0}ϕ_{δ,μ,V}(x)$ is independent of $x\in D$. We evaluate this constant in the case that $μ$ has a density in a neighborhood of $\partial D$. We also study the asymptotic behavior as $δ\to0$ of the principal eigenvalue $λ_0(δ,μ,V)$ for the operator $L_{δ,μ, V}$, which generalizes previously obtained results for the case $L=\frac12 Δ$.