11institutetext: Department of Mathematics, Gonzaga University, Spokane, WA
99258 22institutetext: Sandia National Laboratories, Albuquerque, NM
87185-{1321, 1320}. Sandia is a multi-program laboratory managed and operated
by Sandia Corporation, a wholly subsidiary of Lockheed Martin Corporation, for
the U.S. Department of Energy’s National Nuclear Security Administration under
contract DE-AC04-94AL85000.

# The exit-time problem for a Markov jump process

Nathanial Burch Marta D’Elia 11<EMAIL_ADDRESS>22<EMAIL_ADDRESS>R. B.
Lehoucq 22<EMAIL_ADDRESS>
###### Abstract

The purpose of this paper is to consider the exit-time problem for a finite-
range Markov jump process, i.e, the distance the particle can jump is bounded
independent of its location. Such jump diffusions are expedient models for
anomalous transport exhibiting super-diffusion or nonstandard normal
diffusion. We refer to the associated deterministic equation as a volume-
constrained nonlocal diffusion equation. The volume constraint is the nonlocal
analogue of a boundary condition necessary to demonstrate that the nonlocal
diffusion equation is well-posed and is consistent with the jump process. A
critical aspect of the analysis is a variational formulation and a recently
developed nonlocal vector calculus. This calculus allows us to pose nonlocal
backward and forward Kolmogorov equations, the former equation granting the
various moments of the exit-time distribution.

_Draft as of_ 27 August 2024

##  1 Introduction

The classical Brownian motion model for diffusion is not well-suited for
applications with discontinuous sample paths. For instance, the mean square
displacement of a diffusing particle associated with a jump process often
grows faster than that for the case of Brownian motion, or grows at the same
rate but is of finite variation or activity (terms that we will define in §4).
Such jump diffusions are expedient models for anomalous transport exhibiting
super-diffusion or nonstandard normal diffusion. Examples include various
problems in finance, transport in heterogenous media neta:09 , species
migration or heterogenous heat conduction as suggested by the length and time
scales over which the data is collected. See the edited volume for a
collection of papers klrs  containing further information and an abundance of
references.

The purpose of this manuscript is to discuss the exit-time problem for finite-
range Markov jump processes. A finite-range process restricts the jump-rate to
be zero outside of a bounded neighborhood about the location of the particle.
Because the process may jump outside of the domain, the associated
deterministic equation, in contrast to the classical Fokker-Planck, must be
augmented with a volume-constraint instead of a boundary condition. The
volume-constraint is the nonlocal analogue of a boundary condition necessary
to demonstrate that the equation is well-posed and is consistent with the jump
process. In particular, realizing a Monte Carlo simulation to compute various
exit-time statistics mandates absorbing (or killing) the particle upon
departure out of the domain that rarely, if ever, occurs at the boundary. In
fact, with probability one, a particle originating in a domain does not touch
the boundary; see, e.g., mill:75 . Hence enforcing a volume-constraint for the
deterministic equation is symbiotic with Monte Carlo simulations.

A distinguishing aspect of our treatment for the exit-time problem is that the
deterministic equation is given by a master equation. This allows us to
analyze spatially inhomogeneous problems where the jump-rate depends upon
location and may be asymmetric, i.e., the rate to and from a point may be
distinct. A critical aspect is a recently developed nonlocal vector calculus
that enables striking analogies to be drawn with the classical vector calculus
including Fick’s laws and the backward, forward Kolmogorov equations requiring
the notion of an adjoint operator. The flexibility afforded by the nonlocal
vector calculus enables us to consider exit-time problems over nontrivial
domains in $\mathbb{R}^{n}$ and builds upon work accomplished via the use of
fractional derivative based approaches; see, e.g., delc:06 ; cade:07 ; chmn:12
; kdmm:13  and the references provided. Although we have assumed that the
jump-rate is that associated with a finite-range jump process, the
representation of the master equation in terms of the calculus is valid for a
multitude of classes of jump-rates, including those of infinite range such as
Lévy measures or their tempered, truncated variants; see mast:94 ; cade:07 ;
bame:10  including the recent review samc:14 . Said another way, the nonlocal
vector calculus is jump-rate agnostic and oblivious to a finite domain.

The nonlocal vector calculus also lays the foundation for a variational
formulation of the deterministic exit-time problem. We can then establish that
a broad range of volume-constrained problems are well-posed. This lays the
foundation for stable numerical methods of the volume-constrained problem.
This calculus also enables us to pose the nonlocal analogues of the forward
and backward Kolmogorov equations, useful for determining the various moments
of the exit-time distribution.

The exit-time problem for Lévy motion has received attention in the research
literature, see, e.g., gdls:14  for a recent reference and in particular,
Brownian motion is well-studied. In comparison, the general case of a
spatially inhomogenous Markov process is treated sporadically in the
literature; for instance wesz:83  considers the one-dimensional problem, and
the paper kmst:86  explains that a “boundary layer”, or what amounts to a
volume-constraint, is needed in addition to a boundary condition. To the best
of our knowledge, our paper is the first general treatment for the
deterministic problem associated with the exit-time distribution for a general
class of Markov “pure” jump processes, i.e., no Brownian component or killing
term.

##  2 Markov jump process

The master equation

$\displaystyle u_{t}(x,t)$
$\displaystyle=\int_{\mathbb{R}^{n}}{\big{(}u(y,t)\gamma(y,x)-u(x,t)\gamma(x,y)\big{)}}\,dy\qquad
x\in\mathbb{R}^{n}\,,$ (1)

is useful for describing a spatially inhomogenous Markov process; see, e.g.,
klso:11  for a discussion. The two-point function
$\gamma:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}$ and
$u:\mathbb{R}^{n}\to\mathbb{R}$ are the jump-rate and the density of
particles, respectively. The associated Markov jump process can be simulated
by Monte Carlo realizations of a continuous-time Markov chain over a continuum
state-space, or equivalently, as realizations of an off-lattice continuous-
time random walk (CTRW) with an exponential jump-rate. The realizations are
constructed from the master equation (1). The equation explains that the
temporal rate of change of probability $u(x,t)\,dx$ of locating a particle at
$x$ about the volume $dx$ at time $t$ is given by the difference in
probability gain and loss at the point $x$. The probability gain is given by
the jump-rate $\gamma(y,x)\,dx$ into $dx$ from $y$ given the probability
$u(y,t)\,dy$ and in analogous fashion, the probability loss is given by the
jump-rate $\gamma(x,y)\,dy$ into $dy$ from $x$ given the probability
$u(x,t)\,dx$. As we will demonstrate in §6, the master equation embodies a
nonlocal Fick’s first and second law of diffusion.

We also further suppose that the jump-rate $\gamma(x,y)$ is zero outside a
ball of radius $\lambda$, i.e.,

$\displaystyle\gamma(x,y)$ $\displaystyle=0\,,\text{ for
}|x-y|\geq\lambda>0\,.$ (2)

In words, the particle can jump to $y$ from $x$ when the distance between $x$
and $y$ is no larger than $\lambda$. We say that “nonlocal convection” occurs
when $\gamma(x,y)\neq\gamma(y,x)$. The fluctuations of the particle path can
take a myriad of forms; see §4 for a discussion. The scaling of the mean
square displacement for the jump diffusion ultimately depends upon the
analytic properties of the jump-rate. The scaling is either linear or larger
so that the diffusion is either normal or superdiffusive. Truncating a radial
Lévy measure removes the “heavy tails” rendering a normal jump diffusion that
is not a Brownian motion; see mast:94 .

The master equation (1) is an instance of the more general diffusion equation

$\displaystyle u_{t}(x,t)$
$\displaystyle=\int_{\mathbb{R}^{n}}{\big{(}h(y,x,t)-h(x,y,t)\big{)}}\,dy\,,$

via the relationship $h(x,y,t)=u(x,t)\gamma(x,y)$. Such an equation is a
nonlocal analogue of $u_{t}=-\nabla\cdot\mathbf{q}$. The nonlocal diffusion
arises because points $y\neq x$ determine the rate of diffusion at $x$. An
expression for $h$ (or $\mathbf{q}$) can be derived as an ensemble average in
phase space; see the paper lese:11 . The conclusion is that there is a basis
for nonlocal diffusion in non-equilibrium statistical mechanics; the classical
diffusion arises by postulating Fick’s first law, or assuming that the
particle sample path is continuous. In particular, §6 postulates a nonlocal
analogue of Fick’s first law using the nonlocal vector calculus reviewed in
§5.

##  3 Finite domain

Suppose we have a finite domain, $\Omega$, of interest. By assumption (2), the
jump-rate $\gamma$ is restricted so that the particle can jump at most a
finite distance $\lambda$ from $x$. We define the interaction domain
$\Omega_{\mathcal{I}}$ to be the region to which the particle may jump to when
originating in $\Omega$. Hence, by our assumption (2) on the jump-rate, the
interaction domain is also finite. Figure 1 displays an example of the
interaction domain

$\Omega_{\mathcal{I}}=\bigcup_{x\in\Omega}B_{\lambda}(x)\setminus\Omega\,,$

where $B_{\lambda}(x)$ denotes the ball about $x$ of radius $\lambda$. Such an
interaction domain provides a collar for the domain $\Omega$. Figure 2
provides a more elaborate example of a disconnected domain and its interaction
domain.

Figure 1: Finite domain $\Omega$ and interaction domain $\Omega_{\mathcal{I}}$
given a finite-range process; the particle can jump within the ball about the
point located at center.

Figure 2: A more elaborate example of a domain $\Omega$ and its interaction
domain $\Omega_{\mathcal{I}}$ given a finite-range process; the particle can
jump within the ball about the point located at center.

The master equation (1) describes a spatially inhomogeneous system undergoing
a possibly asymmetric jump-rate (or nonlocal convection) over all of
$\mathbb{R}^{n}$. An example of such a system is given by the jump-rate kernel

$\displaystyle\mathds{1}_{\Omega\cup\Omega_{\mathcal{I}}}(x)\,\mathds{1}_{\Omega\cup\Omega_{\mathcal{I}}}(y)\nu(x-y)$
(3)

where $\nu$ is a Lévy measure and
$\mathds{1}_{\Omega\cup\Omega_{\mathcal{I}}}$ is the indicator function given
by

$\mathds{1}_{\Omega\cup\Omega_{\mathcal{I}}}(x):=\begin{cases}1&x\in\Omega\cup\Omega_{\mathcal{I}}\,,\\\
0&x\notin\Omega\cup\Omega_{\mathcal{I}}\,.\end{cases}$ (4)

Inserting the jump-rate kernel (3) into the free-space master equation (1)
leads to the equation

$\displaystyle u_{t}(x,t)$
$\displaystyle=\int_{\Omega\cup\Omega_{\mathcal{I}}}{\big{(}u(y,t)\nu(y)-u(x,t)\nu(x)\big{)}}\,dy\qquad
x\in\mathbb{R}^{n}\,.$ We remark that the assumption (2) on the jump-rate
kernel satisfies $\displaystyle\gamma(x,y)$ $\displaystyle=0\text{ for
}x\in\Omega\text{ and
}y\in\mathbb{R}^{n}\setminus\big{(}\Omega\cup\Omega_{\mathcal{I}}\big{)}\,,$
(5a) and leads to the confined nonlocal diffusion equation $\displaystyle
u_{t}(x,t)$
$\displaystyle=\int_{\Omega\cup\Omega_{\mathcal{I}}}{\big{(}u(y,t)\gamma(y,x)-u(x,t)\gamma(x,y)\big{)}}\,dy\qquad
x\in\Omega\cup\Omega_{\mathcal{I}}\,.$ (5b)

Realizations of the Markov jump process corresponding to the confined master
equation (5b) are constructed as described in the discussion following (1).

Figure 3: Realizations of a compound Poisson process, a finite-activity,
finite variation process; wait-times are exponentially distributed with
parameter set to $0.2$. The jumps are uniformly distributed in the interval
$(-1,1)$ so that the process is of finite range.

Figure 4: Realizations of a infinite activity, finite variation process with
jump-rate kernel (7a) approximated by (7d) with $\alpha=1/2$, $\lambda=1$,
$m=1$, $\varepsilon=0.001$.

Figure 5: Realizations of a infinite activity, infinite variation process with
jump-rate kernel (7a) approximated by (7d) with $\alpha=3/2$, $\lambda=1$,
$m=1000$, and $\varepsilon=0.001$.

Figure 6: Realizations of a Brownian motion with a diffusion coefficient of
one half so that mean square displacement is equal to $t$.

##  4 Fluctuations

We explained in §2 that the fluctuations of a diffusion where the jump-rate
satisfies (2) can take on a myriad of forms. Since the activity and variation
of the sample path can be finite and infinite, four categories of particle
fluctuations emerge. The activity refers to the number of jumps, or points of
discontinuity, on a finite time interval. An infinite activity process
contains a countable number of jumps and is associated with a jump-rate that
is not integrable, i.e.,

$\displaystyle\int_{-\lambda}^{\lambda}\gamma(x,y)\,dy$
$\displaystyle=\infty\text{ for all }\lambda>0\,.$ (6)

The variation of a one-dimensional function is the vertical distance traversed
along the graph. The definition is somewhat more technical when the function
is $n>1$ dimensional but embodies the same concept.

Figures 3-6 display the four categories of particle fluctuations and provide a
useful visual description. Figures 3 displays a finite activity, finite
variation process induced by a compound Poisson process; the resemblance to a
continuous-time Markov chain is apparent. Figure 6 displays Brownian motion, a
“zero activity” process (since there are a finite number—zero—of jumps) of
infinite variation. The infinite activity processes displayed in Figures 4–5
warrant further discussion since a countable number of jumps over a finite
time interval cannot be computed but can be approximated.

Define the radial jump-rate kernel $\displaystyle\nu(y)$
$\displaystyle:=\frac{1}{m}\frac{1}{|y|^{1+\alpha}}\mathds{1}_{(0,\lambda)}(|y|)\,,\quad
0<\alpha<2\,,$ (7a) and note that this kernel satisfies the condition (6) and
represents a truncated $\alpha$-stable Lévy process where the parameter $m$
controls the rate of diffusion and is used primarily so that the axes of
Figures 3-6 are aligned. In particular, when $0<\alpha<1$, then a finite
variation process occurs because
$\displaystyle\int_{-\lambda}^{\lambda}{|y|\nu(y)}\,dy$
$\displaystyle<\infty\,,$ (7b) whereas when $1<\alpha<2$, an infinite
variation process occurs because $|y|\nu(y)$ is not integrable but
$\displaystyle\int_{-\lambda}^{\lambda}{y^{2}\nu(y)}\,dy$
$\displaystyle<\infty\,.$ (7c)

In order to simulate such a process, we employ the standard procedure to
approximate $\nu$ with the integrable jump-rate

$\displaystyle\nu_{\varepsilon}(y)$
$\displaystyle=\begin{cases}\displaystyle\frac{1}{m}\frac{1}{|y|^{1+\alpha}}&\varepsilon<|y|<\lambda\,,\\\
\displaystyle\frac{1}{m}\frac{1}{\varepsilon^{1+\alpha}}&|y|<\varepsilon\,,\end{cases}$
(7d)

with small $\varepsilon$. Such an approximation enables us to realize a
compound Poisson process that serves to approximate the infinite activity
process.

##  5 Nonlocal vector calculus

We briefly review the nonlocal vector calculus introduced in gule:10  and
extensively developed in dglz:13 . The calculus will enable us to express the
nonlocal diffusion equation in conservative form and demonstrate that the
variational form of the equation is well-posed.

Let
${\boldsymbol{\alpha}},\mathbf{f}:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$
where ${\boldsymbol{\alpha}}$ is antisymmetric in $x$ and $y$, i.e.,
${\boldsymbol{\alpha}}(x,y)=-{\boldsymbol{\alpha}}(y,x)$. Define the nonlocal
divergence

$\displaystyle\mathcal{D}(\mathbf{f})(x):=\int_{\mathbb{R}^{n}}\big{(}\mathbf{f}(x,y)+\mathbf{f}(y,x)\big{)}\cdot{\boldsymbol{\alpha}}(x,y)\,dy\,.$
(8)

The nonlocal divergence of a vector is a scalar and the definition implies
that the antisymmetric part of $\mathbf{f}$, i.e.,
$1/2\big{(}\mathbf{f}(x,y)-\mathbf{f}(y,x)\big{)}$ is annihilated. If we
suppose that $\mathbf{f}$ is differentiable, then the choice

$\displaystyle{\boldsymbol{\alpha}}(x,y)$
$\displaystyle=-\nabla_{y}\,\delta(y-x)$ (9a) and an integration by parts
implies that $\displaystyle\mathcal{D}(\mathbf{f})(x)$
$\displaystyle\equiv\nabla\cdot\mathbf{f}(x,x)\,.$ (9b)

The antisymmetry of the integrand of (8) grants the nonlocal divergence
theorem

$\displaystyle\int_{\Omega}\mathcal{D}(\mathbf{f})\,dx$
$\displaystyle=-\int_{\mathbb{R}^{n}\setminus\Omega}\mathcal{D}(\mathbf{f})\,dx\,.$
(10a) If we restrict ${\boldsymbol{\alpha}}$ so that
${\boldsymbol{\alpha}}(x,y)=\mathbf{0}$ for $x\in\Omega$ and
$y\in\mathbb{R}^{n}\setminus\big{(}\Omega\cup\Omega_{\mathcal{I}}\big{)}$
where the interaction domain $\Omega_{\mathcal{I}}$ was defined in §3, then
$\displaystyle\int_{\mathbb{R}^{n}\setminus\Omega}\mathcal{D}(\mathbf{f})\,dx$
$\displaystyle=\int_{\Omega_{\mathcal{I}}}\mathcal{D}(\mathbf{f})\,dx\,.$
(10b) Both integrals represent the flux of the vector field into the region
external to $\Omega$. The nonlocal divergence theorem is the nonlocal analogue
of the classical divergence theorem
$\displaystyle\int_{\Omega}\nabla\cdot\mathbf{f}\,dx$
$\displaystyle=\int_{\partial\Omega}\mathbf{f}\cdot\mathbf{n}\,d^{n-1}x\,,$

where we abuse notation to suppose that $\mathbf{f}$ is a vector field of only
one variable. The relationship is clear when the relations (9) are invoked.
The analogy between divergence theorems suggests that the orientation given by
the unit normal $\mathbf{n}$, or the sense of direction, is embodied by the
antisymmetry of the kernel ${\boldsymbol{\alpha}}$.

We now define the operator

$\big{(}\mathcal{D}^{\ast}u\big{)}(x,y):=-\big{(}u(y)-u(x)\big{)}\,{\boldsymbol{\alpha}}(x,y)$

so that given the density $u$, the function $\mathcal{D}^{\ast}u$ is a vector
field of the points $x$ and $y$. In a similar fashion to (9b), the choice (9a)
for the kernel ${\boldsymbol{\alpha}}$ implies

$\int_{\mathbb{R}^{n}}\mathcal{D}^{\ast}u\,dy=-\nabla u\,,$

so that we may conclude that the operator $\mathcal{D}^{\ast}$ can be seen as
a nonlocal gradient. Because $\mathcal{D}^{\ast}u$ is an example of a vector
field $\mathbf{f}$, its nonlocal divergence can be determined. The identity

$\displaystyle\int_{\Omega\cup\Omega_{\mathcal{I}}}v\,\mathcal{D}\,\mathbf{f}\,dx=\int_{\Omega\cup\Omega_{\mathcal{I}}}\int_{\Omega\cup\Omega_{\mathcal{I}}}\mathcal{D}^{\ast}v\cdot\mathbf{f}\,dy\,dx\,,$
(11)

is established by direct substitution and invoking the antisymmetry of
${\boldsymbol{\alpha}}$. In words, the identity states that
$\mathcal{D}^{\ast}$ is the adjoint operator for $\mathcal{D}$. This is
analogous to the classical identity that the divergence $\nabla\cdot$ is the
negative adjoint operator for the gradient $\nabla$. Inserting
$\mathbf{f}=\mathcal{D}^{\ast}u$ into the previous identity and using
linearity of the integral on the left-hand side grants a nonlocal Green’s
identity

$\displaystyle\int_{\Omega}v\mathcal{D}\,\mathcal{D}^{\ast}u\,dx$
$\displaystyle=\int_{\Omega\cup\Omega_{\mathcal{I}}}\int_{\Omega\cup\Omega_{\mathcal{I}}}\mathcal{D}^{\ast}v\cdot\mathcal{D}^{\ast}u\,dy\,dx-\int_{\Omega_{\mathcal{I}}}v\,\mathcal{D}\,\mathcal{D}^{\ast}u\,dx\,,$
(12) which is seen to be the nonlocal analogue of the conventional Green’s
identity $\displaystyle\int_{\Omega}v\Delta u\,dx$
$\displaystyle=-\int_{\Omega}\nabla v\cdot\nabla
u\,dx+\int_{\partial\Omega}v\big{(}\nabla
u\cdot\mathbf{n}\big{)}\,d^{n-1}x\,.$ (13)

Let $v$ be a smooth, compactly supported function. Then, we can show that

$\int_{\Omega_{\mathcal{I}}}v\,\mathcal{D}\,\mathcal{D}^{\ast}u\,dx=\int_{\partial\Omega}v\,\big{(}\mathcal{D}^{\ast}u\cdot\mathbf{n}\big{)}\,d^{n-1}x\,;$

see (dglz:13, , §5.1) for details. This demonstrates in what sense the
nonlocal identity is a generalization of the conventional identity by avoiding
spatial derivatives and replacing surfaces by volumes for boundary and volume
data, respectively. This more general Green’s identity will allow us to
demonstrate that the deterministic exit-time problem is associated with a
general class of Markov jump processes is well-posed.

##  6 Nonlocal convection-diffusion equation

We define the nonlocal convection-diffusion equation

$\left\\{\begin{aligned} u_{t}&=\mathcal{D}\,\mathbf{f}\\\
\mathbf{f}&={\boldsymbol{\mu}}\,u-{\boldsymbol{\Theta}}\,\mathcal{D}^{\ast}u\,,\end{aligned}\right.$

where $\boldsymbol{\mu}:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$
and
${\boldsymbol{\Theta}}:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}^{n}\times\mathbb{R}^{n}$.
The first and second lines represent a nonlocal Fick’s first and second law of
diffusion and are the analogues of the classical laws

$\left\\{\begin{aligned} u_{t}&=-\nabla\cdot\mathbf{q}\\\
\mathbf{q}&=\mathbf{b}\,u+\nabla\mathbf{A}\,u\,.\end{aligned}\right.$

Combining Fick’s first and second laws leads to the nonlocal convection-
diffusion equation

$\displaystyle u_{t}$
$\displaystyle=\mathcal{D}\,\big{(}\boldsymbol{\mu}\,u-{\boldsymbol{\Theta}}\,\mathcal{D}^{\ast}u\big{)}\qquad\text{on
}\Omega\cup\Omega_{\mathcal{I}}\,.$ (14a) This equation is simply a rewrite of
the confined master equation (5b) with the identification
$\displaystyle\gamma$
$\displaystyle={\boldsymbol{\alpha}}\cdot{\boldsymbol{\Theta}}{\boldsymbol{\alpha}}-{\boldsymbol{\mu}}\cdot{\boldsymbol{\alpha}}\,.$
(14b) An immediate consequence for equation (14a) is
$\displaystyle\begin{aligned}
\frac{d}{dt}\int_{\Omega\cup\Omega_{\mathcal{I}}}u\,dx&=\int_{\Omega\cup\Omega_{\mathcal{I}}}\mathcal{D}\,\big{(}\boldsymbol{\mu}\,u-{\boldsymbol{\Theta}}\,\mathcal{D}^{\ast}u\big{)}\,dx\\\
&=-\int_{\Omega_{\mathcal{I}}}\mathcal{D}\,\big{(}\boldsymbol{\mu}\,u-{\boldsymbol{\Theta}}\,\mathcal{D}^{\ast}u\big{)}\,dx+\int_{\Omega_{\mathcal{I}}}\mathcal{D}\,\big{(}\boldsymbol{\mu}\,u-{\boldsymbol{\Theta}}\,\mathcal{D}^{\ast}u\big{)}\,dx\\\
&=0\,,\end{aligned}$ (14c)

where we invoked the nonlocal divergence theorem (10) for the second equality.
In words, the probability flux out of $\Omega$ into $\Omega_{\mathcal{I}}$ is
equal and opposite to the probability flux out of $\Omega_{\mathcal{I}}$ into
$\Omega$. This is an instance of the more general principle of action-
reaction, where $\Omega_{\mathcal{I}}$ and $\Omega$ can be replaced by regions
that have no overlap and are separated by a finite distance. Hence the
interaction is not restricted to regions that are in direct contact so leading
to nonlocal diffusion.

The paper (dglz:13, , Theorem 2.1) demonstrates that a well-formulated
nonlocal balance law is given by the following four equivalent conditions:

  1. 1.

antisymmetry of $\psi(x,y):=u(y,t)\gamma(y,x)-u(x,t)\gamma(x,y)$, the
integrand of
$\mathcal{D}\,\big{(}\boldsymbol{\mu}\,u-{\boldsymbol{\Theta}}\,\mathcal{D}^{\ast}u\big{)}$;

  2. 2.

no self interaction, i.e.,
$\displaystyle\int_{\Omega}\mathcal{D}\,\big{(}\boldsymbol{\mu}\,u-{\boldsymbol{\Theta}}\,\mathcal{D}^{\ast}u\big{)}\,dx=0$
for all $\Omega$;

  3. 3.

action-reaction, i.e.,
$\displaystyle\int_{\Omega}\int_{\Omega^{\prime}}\psi(x,y)\,dy\,dx+\int_{\Omega^{\prime}}\int_{\Omega}\psi(x,y)\,dy\,dx=0$
for all $\Omega,\Omega^{\prime}$ that have no overlap;

  4. 4.

additivity, i.e.,
$\displaystyle\int_{\Omega\cup\Omega^{\prime}}\mathcal{D}\,\big{(}\boldsymbol{\mu}\,u-{\boldsymbol{\Theta}}\,\mathcal{D}^{\ast}u\big{)}\,dx=\int_{\Omega}\mathcal{D}\,\big{(}\boldsymbol{\mu}\,u-{\boldsymbol{\Theta}}\,\mathcal{D}^{\ast}u\big{)}\,dx+\int_{\Omega^{\prime}}\mathcal{D}\,\big{(}\boldsymbol{\mu}\,u-{\boldsymbol{\Theta}}\,\mathcal{D}^{\ast}u\big{)}\,dx$
for all $\Omega,\Omega^{\prime}$ that have no overlap.

##  7 Backward and forward Kolmogorov equations

A consequence of the identity (14c) implies that the equation (14a) is
formally understood as a forward Kolmogorov, or Fokker-Planck, equation
because the transition measure for the jump diffusion is evolved. Convention
denotes the corresponding operator via its action on the density

$\displaystyle\mathcal{A}^{\ast}u$
$\displaystyle:=\mathcal{D}\,\big{(}\boldsymbol{\mu}\,u\big{)}-\mathcal{D}\,{\boldsymbol{\Theta}}\,\mathcal{D}^{\ast}u\,,$
(15a) where $\mathcal{A}^{\ast}$ denotes the adjoint operator for
$\mathcal{A}$. In order to consider the mean exit-time problem, the nonlocal
backward Kolmogorov equation $\displaystyle u_{t}$
$\displaystyle=\mathcal{A}\,u$ (15b) is also helpful. An explicit expression
for $\mathcal{A}$, the generator for the Markov process, is possible by
exploiting the nonlocal vector calculus. To simplify matters, we assume,
without loss of generality that
${\boldsymbol{\Theta}}(x,y)={\boldsymbol{\Theta}}^{T}(x,y)={\boldsymbol{\Theta}}(y,x)\,,$
so that by the adjoint identity (11) with
$\mathbf{f}(x,y)={\boldsymbol{\mu}}(x,y)\,u(x)$, the requisite operator is
$\displaystyle\big{(}\mathcal{A}\,u\big{)}(x)$
$\displaystyle:=\int_{\Omega\cup\Omega_{\mathcal{I}}}{\big{(}{\boldsymbol{\mu}}\cdot\mathcal{D}^{\ast}u\big{)}(x,y)}\,dy-\big{(}\mathcal{D}\,{\boldsymbol{\Theta}}\,\mathcal{D}^{\ast}\big{)}u(x)\,.$
(15c)

When the jump-rate satisfies $\gamma(x,y)=\gamma(y,x)$, or equivalently by
(14b), when ${\boldsymbol{\mu}}(x,y)=-{\boldsymbol{\mu}}(y,x)$, then
$\mathcal{A}=\mathcal{A}^{\ast}$.

##  8 Exit-time problem

Let $X_{t}$ be a finite-range Markov jump process for the confined master
equation (5b) conditioned on $X_{0}=x\in\Omega$ and let a generalized exit-
time random variable be given by

$\displaystyle T_{x}$
$\displaystyle:=\inf\\{t:X_{t}\in\Omega_{d}\subseteq\Omega_{\mathcal{I}}\\}\,,$
(16)

where $\emptyset\subseteq\Omega_{d}\subseteq\Omega_{\mathcal{I}}$. We denote
the exit-time generalized since the particle cannot exit to the region of
$\Omega_{\mathcal{I}}$ not in $\Omega_{d}$ (except when
$\Omega_{d}\equiv\Omega_{\mathcal{I}}$).

In contrast to the classical exit-time problem for Brownian motion, we cannot
expect the finite-range jump process $X_{t}$ to hit the boundary
$\partial\Omega$ upon departing $\Omega$. In fact, with probability almost
surely one, the process never encounters the boundary. Instead, $X_{t}$
departs to a location in the interaction domain $\Omega_{\mathcal{I}}$. The
density of particles that have not yet exited $\Omega$ to $\Omega_{d}$ evolves
according to the nonlocal convection-diffusion, or Fokker-Planck, equation:

$\left\\{\begin{aligned} \displaystyle
u_{t}(x,t)&=\big{(}\mathcal{A}^{\ast}u\,\mathds{1}_{\Omega\cup\Omega_{d}}\big{)}(x,t)\\\
&=\int_{\Omega\cup\Omega_{d}}{\big{(}u(y,t)\gamma(y,x)-u(x,t)\gamma(x,y)\big{)}}\,dy,&x&\in{\Omega}\\\
\displaystyle u(x,t)&=0,&x&\in\Omega_{d}\\\ \displaystyle
u(x,0)&=u_{0}(x),&x&\in\Omega\,,\end{aligned}\right.$ (17)

where we used (4), (14b) and (15a) for the second equality, and without loss
of generality we suppose that $u_{0}$ is a probability density over $\Omega$.
We define the constraint on the density over $\Omega_{d}$ to be a volume-
constraint, the generalization of a boundary condition, or committing a
semantic abuse of terminology, a nonlocal boundary condition. The volume
constraint is necessary because the particle may jump out of $\Omega$ into
$\Omega_{\mathcal{I}}$. As we will review in §9, the volume constraint is
crucial in showing that the nonlocal convection-diffusion equations is well-
posed. Moreover, the volume-constraint enables us to link the finite-range
jump process with a deterministic equation and corresponds to how Monte Carlo
realizations are implemented.

Figure 7: Absorbed/censored jump process depicted by the possible steps taken
by the random walker stickman.

Two cases are of special interest.

$\Omega_{d}\equiv\emptyset$:

The system (17) is then a nonlocal analogue of the classical Fokker-Planck
with a homogenous Neumann boundary condition. The corresponding jump process
models a particle confined to the region $\Omega$ and is an instance of a
censored process; see the paper bobc:03  for a mathematical discussion.

$\Omega_{d}\equiv\Omega_{\mathcal{I}}$:

The system (17) is then a nonlocal analogue of the classical Fokker-Planck
with a homogenous Dirichlet boundary condition. The corresponding jump process
models the exit-time of a particle conditioned on $X_{0}\in\Omega$.

The general case corresponds to a system for a mixed absorbed/censored process
and is the analogue of the classical Fokker-Planck with a mixed
Dirichlet/Neumann boundary condition. Figure 7 depicts this general case and
this section provides the infrastructure needed to extend the analyses of the
narrow escape problem hosc:14  beyond classical diffusion to Markov jump
processes.

The homogenous Dirichlet volume-constraint is explicit in the system (17);
where is the homogenous Neumann volume-constraint? Suppose that instead of a
homogenous condition, we consider a given source density $g$. Then we may
augment (17) with the condition

$\int_{\Omega_{\mathcal{I}}\setminus\Omega_{d}}{\big{(}u(y,t)\gamma(y,x)-u(x,t)\gamma(x,y)\big{)}}\,dy=g(x)\qquad
x\in\Omega\,,$

to obtain a modified system. If we specify $g\equiv 0$ over $\Omega$ then we
see that the homogenous Neumann volume-constraint is also explicitly given by
the formulation of the system (17).

How does the deterministic system (17) conserve the probability of the
location of the particle? The derivation provides the insight needed to
properly understand the distinction between the nonlocal and classical systems
evolving the density. By invoking the volume-constraint, we have

$\displaystyle\frac{d}{dt}\int_{\Omega}u(x,t)\,dx$
$\displaystyle=\int_{\Omega}\int_{\Omega\cup\Omega_{d}}{\big{(}u(y,t)\gamma(y,x)-u(x,t)\gamma(x,y)\big{)}}\,dy\,dx$
$\displaystyle=\int_{\Omega}\int_{\Omega_{d}}{\big{(}u(y,t)\gamma(y,x)-u(x,t)\gamma(x,y)\big{)}}\,dy\,dx$
$\displaystyle=-\int_{\Omega}\int_{\Omega_{d}}{u(x,t)\gamma(x,y)}\,dy\,dx<0\,,$

where the second and third equalities follow from the antisymmetry of the
integrand and applying the volume constraint a second time, respectively. By
the nonlocal divergence theorem, we may also conclude that the last iterated
integral is the nonlocal flux of probability out of $\Omega$ into
$\Omega_{d}$. Therefore the probability is conserved over
$\Omega\cup\Omega_{d}$, i.e., the particle is located either in $\Omega$ or
has exited to $\Omega_{d}$.

We now provide a probabilistic interpretation. Because the probability that
the particle remains in $\Omega$ is given by

$\displaystyle\text{Prob}(T_{x}>t)$ $\displaystyle=\int_{\Omega}u(y,t)\,dy\,,$
then the exit-time distribution is given by
$\displaystyle\text{Prob}(T_{x}\leq t)$
$\displaystyle=1-\int_{\Omega}u(y,t)\,dy$
$\displaystyle=\int_{0}^{t}\int_{\Omega}\int_{\Omega_{d}}{u(y,s)\gamma(y,z)}\,dy\,dz\,ds\,,$

where the last equality is the time integrated probability flux. In words, the
rate of change of the probability of the particle exiting to $\Omega_{d}$
increases to one so that as time increases the density of particles located in
$\Omega$ decreases to zero. We also see the effect of decreasing the size of
$\Omega_{d}$ is to delay the time required for the probability of a particle
to exit to $\Omega_{d}$. And when $\Omega_{d}\equiv\emptyset$, then the
probability of a particle to exiting $\Omega$ is zero for all time.

The mean exit-time

$\langle T_{x}\rangle:=\int_{0}^{\infty}\int_{\Omega}u(x,t)\,dx\,dt\,,$

then, in direct analogy with the classical exit time problem, see, e.g., the
paper nkms:90 , is given by the solution of the steady-state volume-
constrained problem

$\left\\{\begin{aligned} \mathcal{A}\,\langle
T_{x}\rangle&=-1&x&\in{\Omega}\\\ \langle
T_{x}\rangle&=0,&x&\in\Omega_{d}\,\,(\neq\emptyset)\,,\end{aligned}\right.$
(18)

where the operator $\mathcal{A}$ is given by (15c). An equation for the
remaining moments of the exit-time random variable $T_{x}$ may also be
determined via repeated integration, in an analogous fashion to the classical
case; see, e.g., gard:04 .

An important question is whether the exit-time distribution and moments are
finite. Given sufficient conditions on the kernel $\gamma$, the distribution
and moments are indeed finite; see §9 for a discussion.

The above analysis allows us to consider far more interesting problems than
that depicted by Figure 7; for instance, consider the Figure 2. Such an exit-
time analysis is simply not possible with Brownian motion since $\Omega$ is
the union of a disconnected set of regions—a particle undergoing Brownian
motion cannot jump outside of $\Omega$. The report bule:13  provides an
example of such an analysis.

##  9 Analysis and approximation of nonlocal diffusion equation

The goal of this section is to briefly review analyses demonstrating that the
volume-constrained nonlocal diffusion equation is well-posed. For simplicity,
we suppose that the interaction domain $\Omega_{\mathcal{I}}\equiv\Omega_{d}$.
A crucial aspect of the analysis is that a nonlocal variational formulation
based upon the nonlocal vector calculus introduced in §5 is exploited.

The analysis proceeds by first demonstrating that the steady-state problem

$\displaystyle\left\\{\begin{aligned} \mathcal{A}^{\ast}u&=b&\text{on
}&\Omega\\\ u&=0&\text{on }&\Omega_{\mathcal{I}}\,,\end{aligned}\right.$ (19)

where $\mathcal{A}^{\ast}$ was defined in (15a), is well-posed. Then, standard
results are invoked to demonstrate that the time dependent nonlocal diffusion
equation is well-posed. This sets the stage for the development of stable
numerical methods for the discretization of (17) that offer an alternative to
averaging the results of realizations of the jump process; see bule:11 ;
bule:13 ; duhl:14  for numerical results demonstrating the consistency of the
numerical solutions of the volume constrained diffusion equation and Monte
Carlo simulations.

We now review the analysis of the steady-state problem. Given a Hilbert space
$V$ with the inner product between elements $u,v$

$\displaystyle(u,v)_{V}$
$\displaystyle:=\int_{\Omega\cup\Omega_{\mathcal{I}}}\int_{\Omega\cup\Omega_{\mathcal{I}}}\big{(}u\,v+\mathcal{D}^{\ast}u\cdot\mathcal{D}^{\ast}v\big{)}\,dy\,dx\,,$
the variational problem is: Find $u\in V$ such that $\displaystyle a(u,v)$
$\displaystyle=\int_{\Omega}u\,b\,dx\qquad\forall\,v\in V$ (20) where the
bilinear form $a(\cdot,\cdot)$ is given by $\displaystyle a(u,v)$
$\displaystyle=\int_{\Omega\cup\Omega_{\mathcal{I}}}\int_{\Omega\cup\Omega_{\mathcal{I}}}\mathcal{D}^{\ast}u\cdot{\boldsymbol{\Theta}}\mathcal{D}^{\ast}v\,dy\,dx-\int_{\Omega}\mathcal{D}\big{(}\boldsymbol{\mu}u\big{)}\,v\,dx$
$\displaystyle=\int_{\Omega\cup\Omega_{\mathcal{I}}}\int_{\Omega\cup\Omega_{\mathcal{I}}}\big{(}u(y)-u(x)\big{)}\big{(}v(y)-v(x)\big{)}\,\big{(}{\boldsymbol{\alpha}}\cdot{\boldsymbol{\Theta}}{\boldsymbol{\alpha}}\big{)}(x,y)\,dy\,dx$
$\displaystyle\,\,\,\,-\int_{\Omega}\int_{\Omega\cup\Omega_{\mathcal{I}}}\big{(}{\boldsymbol{\mu}}(x,y)u(x)+{\boldsymbol{\mu}}(y,x)u(y)\big{)}\,v(x)\,dy\,dx\,,$

and an expression for $\gamma$ in terms of
${\boldsymbol{\alpha}},{\boldsymbol{\Theta}},{\boldsymbol{\mu}}$ was given in
(14). The Lax-Milgram theorem then provides sufficient conditions that when
satisfied, demonstrate that (20) has a unique density $u$, i.e., the system
(19) has a weak solution. The Hilbert space $V$ is identified with a volume-
constrained subspace of square integrable functions or a fractional Sobolev
space given conditions on the integrability of the jump-rate $\gamma$. The
latter space contains the densities corresponding to infinite activity
processes. Continuous dependence upon the data implies the energy is bounded
by the data, i.e.,

$\displaystyle(u,u)_{V}\leq C\int_{\Omega}b^{2}\,dx\,;$

see duhl:14 ; ddgl:14  for details and further discussion. If
$\gamma(x,y)=\gamma(y,x)$ then the bilinear form $a(\cdot,\cdot)$ is symmetric
and the variational problem (20) is the Euler-Lagrange equation for the
minimization problem

$\min_{V}\frac{1}{2}a(u,v)-\int_{\Omega}u\,b\,dx\,,$

this is the case considered in dglz:12 . To the best of our knowledge, this
latter paper was the first to demonstrate that the deterministic exit-time
problem for an infinite activity and finite variation jump diffusion is well-
posed. The paper degu:13  exploits the analysis in dglz:12  to demonstrate how
the truncated fractional Laplacian converges to the fractional Laplacian as
the length $\lambda$ of the finite-range increases.

One of the hypothesis associated with the Lax-Milgram theorem is to show that
the bilinear form $a(\cdot,\cdot)$ is coercive. Equivalently, coercivity means
that

$\displaystyle\sigma=\inf_{v\in V}\frac{a(v,v)}{(u,v)_{V}}\,.$

is a positive number. Coercivity can then be used to demonstrate that the
moments of the exit-time are finite; see bule:13 ; ddgl:14  for a
demonstration.

## Acknowledgements

We thank Professors Scott McKinley (University of Florida) and Renming Song
(University Illinois) for helpful discussions, and Professor Michael Mascagni
of Florida State University for a careful reading.

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