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
by
Ross G. Pinsky
Vol. 7 No. 4 (2012) P.545~P.564
ABSTRACT
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 $\mu\in\mathcal P(D)$ and
$\delta>0$.
Consider a Markov process $X(t)$ in $D$ which performs diffusion in $D$ generated by the operator $\delta L$ 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 $\mu$. The Markov
process is generated by the operator $L_{\delta,\mu, V}$ defined by
\begin{equation*}
L_{\delta,\mu, V}\phi\equiv \delta L \phi+V(\int_D\phi~d\mu-\phi).
\end{equation*}
Let $\phi_{\delta,\mu,V}$ denote the solution to the Dirichlet
problem
\begin{equation*}\label{Dirprob}
\begin{aligned}
&L_{\delta,\mu,V}\phi=0\ \text{in}\ D;\&\phi=f\ \text{on}\ \partial D,
\end{aligned}
\end{equation*}
where $f$ is continuous. The solution has the stochastic
representation
\begin{equation*}
\phi_{\delta,\mu,V}(x)=E_xf(X(\tau_D)).
\end{equation*}
One has that
$\phi_{0,\mu,V}(f)\equiv\lim_{\delta\to0}\phi_{\delta,\mu,V}(x)$ is
independent of $x\in D$. We evaluate this constant in the case that
$\mu$ has a density in a neighborhood of $\partial D$. We also study
the asymptotic behavior as $\delta\to0$ of the principal eigenvalue
$\lambda_0(\delta,\mu,V)$ for the operator $L_{\delta,\mu, V}$,
which generalizes previously obtained results for the case
$L=\frac12 \Delta$.
KEYWORDS
non-local di erential operator, di usion process, principal eigenvalue, exit problem, random space-dependent jumps
MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 60J60, 35J25, 60J75, 35P15
MILESTONES
Received: 2011-11-18
Revised : 2012-08-29
Accepted: 2012-08-29
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