††thanks: Version of

# Concatenation of dishonest Feller processes, exit laws, and limit theorems
on graphs.

Adam Bobrowski  Lublin University of Technology
Nadbystrzycka 38A
20-618 Lublin, Poland<EMAIL_ADDRESS>
###### Abstract.

We provide a rather explicit formula for the resolvent of a concatenation of
$N$ processes in terms of their exit laws and certain probability measures
characterizing the way the processes are concatenated. As an application, we
prove an averaging principle saying that by concatenating asymptotically
splittable processes one can approximate Markov chains.

###### Key words and phrases:

exit laws, processes on metric graphs, concatenation of processes, non-local
boundary and transmission conditions, Feller processes, snapping out and skew
Brownian motions

###### 1991 Mathematics Subject Classification:

92C42, 35B25, 35F46, 35K57, 35K51, 47D06, 47D07

##  1\. Introduction

###  1.1. Not honest stochastic processes as building blocks of more complex
systems

In modeling natural phenomena one often encounters stochastic processes that
are not honest, that is, are undefined after a random time. Such processes
include e.g. so-called explosive Markov chains (and particular cases of birth
and death processes among them) [fmc, chungboundary, fellerboundaryforMC,
norris], Feller processes killed upon exiting a certain region [chungodkb],
processes generated by fractional Laplace operators perturbed by gradients
(see [bogdanjakub]*Eq. (20)), and killed Lévy processes ([bogdanki]*Section
5.5 and [bogdanros]*Section 4.5).

In biology, they abound for example in the theory of coagulation and
fragmentation processes [banaarlo, bll], and form building blocks of models
involving semi-permeable membranes, such as direct or facilitated diffusion of
ions and molecules across cell/plasma membranes [steinbook, khan]. To wit, a
randomly moving particle that passes through a membrane can be seen as
disappearing from one part of the state-space and reappearing in its other
part, and it is natural to see the entire process as built from simpler, not
honest ones, defined only in parts of the state-space — see below for more on
this subject. Such a structure of the studied process is visible e.g. in the
papers [bobmor] and [zmarkusem] (see also [knigaz]) devoted to reconciliation
of two seemingly different models of so-called fast neurotransmitters, to
describing activity of kinases, and to intracellular calcium dynamics. It is
also apparent, to name a few, (a) in the work of J. E. Tanner [tanner], who
studied diffusion of particles through a sequence of permeable barriers (see
also Powles et al. [powles], for a continuation of the subject), (b) in the
paper by S. S. Andrews [andrews], devoted to describing absorption and
desorption phenomena, and (c) in Fireman et al. [fireman], where a compartment
model with permeable walls (representing e.g., cells, and axons in the white
matter of the brain in particular) is analyzed. See also the references given
by A. Lejay in [lejayn] to articles devoted to modeling of flows between
cells, and the papers with models relevant for astrophysics, ecology,
homogenization, geophysics and finance, provided in [lejayskew]*p. 414.

In fact, it is the abundance of non-honest processes in the natural sciences
that is a primary motivation of this work. As noted e.g. in Hartwell et al.
[hartwell] (see also [alon]), biological systems are built of myriads of
interacting components which can in themselves be decomposed into yet smaller
subsystems with very specific interactions. The idea of _concatenation_ of not
honest processes, that is, of building more complex structures from simpler
ones, provides a unifying language for a wide range of applied models, and we
believe that our main formula (see (5.6)) is an efficient tool for analyzing
them.

###  1.2. Non-honest Feller processes and boundary conditions

A number of examples of not honest processes can be found in the theory of
Feller processes on domains with boundaries — their behavior at these
boundaries is customarily described by means of boundary conditions. For a
simple concrete case, consider a standard Brownian motion on the right half-
axis $\mathbb{R}^{+}\coloneqq[0,\infty)$ which after reaching the boundary
point $x=0$ remains there for an exponential time with parameter, say, $a\geq
0$, and is undefined later, when — figuratively speaking — the Brownian
traveller disappears from the state-space. This process, that will be referred
to as _elementary exit_ (comp. [fellera1]*p. 3 or [knigaz]*p. 19) is related
to the boundary condition

$af^{\prime\prime}(0)+f(0)=0.$ (1.1)

The Robin boundary condition

$bf^{\prime}(0)=f(0),$ (1.2)

where $b>0$ describes a similar process, termed _elastic Brownian motion_ :
initially it behaves precisely as the reflected Brownian motion but the time
it spends ‘at the boundary’ where $x=0$ is measured with the help of the
celebrated Lévy local time; when an exponential time with parameter $b^{-1}$
with respect to the Lévy local time elapses, the process is no longer defined
[itop, ito, karatzas].

The role of the _sticky_ , or _slowly reflecting_ boundary

$af^{\prime\prime}(0)-bf^{\prime}(0)+cf(0)=0,$ (1.3)

is similar — see [kostrykin2010brownian], [liggett]*p. 127 or [revuz]*p. 421 —
the only difference lies in the fact that here the process has the tendency to
stick to the boundary for a longer time than in the reflected Brownian motion.

###  1.3. Concatenating processes that are not honest; transmission conditions

In describing such processes it is one of the tricks of the trade to make them
honest by adjoining an additional point (sometimes called the cemetery or
coffin state) to the state-space and agreeing that from the moment when the
process is undefined, it actually stays at this new point forever [kniga,
bass, kallenbergnew, fmc]. There are, however, other, more engaging ways the
process may be continued, that is, extended to a possibly honest process.
Interestingly, such a continuation can often be obtained by appropriate
manipulations on boundary conditions.

For example, in the case of elementary jump from the boundary at $x=0$, we can
require that when the exponential time spent at $x=0$ elapses the process
should start anew at a point $s$ of a locally compact space $S$, and evolve
according to the rules governing there. To this end, one imposes the
transmission condition

$f^{\prime\prime}(0)=a[f(s)-f(0)].$ (1.4)

A similar modification of the elastic Brownian motion results from changing
the boundary condition (1.2) to the transmission condition

$bf^{\prime}(0)=f(0)-f(s),$ (1.5)

see [bobmor, nagrafach, lejayn], consult also [knigaz]*p. 66 and
[zmarkusem]*p. 669, where further references are given.

For a more complex example we refer to [arkuk] (see also [arkuk2]), where a
boundary condition of the form

$f(x)=\int_{\Omega}f\,\mathrm{d}\hskip 0.56917pt\mu_{x}$

commands the process that leaves the state-space $\Omega$ at a point $x$ of
its boundary to start anew inside the state-space at a randomly chosen point
with distribution $\mu_{x}$.

Although processes related to transmission conditions discussed above are
motivating examples for our paper, the idea that the process which is no
longer defined in its original state-space could be continued in an extended
state-space, applies, of course, to more general situations than these
described above, and to processes that may have nothing to do with
transmission or boundary conditions. In fact, such constructions are known
under the name of _concatenation of processes_.

In Section II.14 of Sharpe’s monograph [sharpe] (pp. 77-84) two processes,
say, $X^{1}$ and $X^{2}$, the first of which is not honest, are concatenated
as follows. A particle starting in the state-space of $X^{1}$ moves about
according to the law of evolution of $X^{1}$, but when the lifetime of this
process is over, the particle jumps to the state-space of $X^{2}$ and starts
to move according to the law of $X^{2}$. A particular example of such
concatenation is described in the classical treatise of Ito and McKean
[ito]*p. 105 in intuitive probabilistic terms and by means of transmission
conditions.

In the recent paper of F. Werner [werner] a more general construction is
provided: given a sequence of Markov processes $X^{n},n\geq 1$ their
concatenation is defined as follows: the new process starts at the state-space
of $X^{1}$, behaves like this process for its lifetime, and then starts anew
as $X^{2}$; when the lifetime of $X^{2}$ is over, the process starts as
$X^{3}$, and so on, until it possibly reaches its coffin state.

Figure 1. Processes before (on the left) and after concatenation (on the
right).

Our general concatenation theorem, that is, Theorem 5.4, is devoted to the
case where there is a finite number $N\geq 2$ of processes with values in
disjoint compact topological spaces $S_{i},i\in\\{1,\dots,N\\}$, and some of
them are not honest. Those processes that are not honest have a finite,
possibly greater than $1$, number of possible ways of exiting from their
state-spaces (think, in particular, of a diffusion process on a finite
interval which may exit from this interval by either of its end-points, see
Section 6 for more examples). The extended process we construct in the state-
space

$S_{\textnormal{\hskip 0.56917pt\tiny
u}}\coloneqq\dot{\bigcup_{i=1,\dots,N}}S_{i}$

(the dot above the union stresses the fact that $S_{i}$ are disjoint, ‘u’ for
‘union’) intuitively looks as follows (see Figure 1): when started at an $x\in
S_{i}$ the process is initially identical to the original process defined in
$S_{i}$; when the latter is no longer defined, the extended process chooses a
random point in $S$, the distribution of its new position depending both on
the space it came from and the way the process has left it. Starting from this
new position, the process forgets its past and, if its new position happens to
be in $S_{j}$, behaves according to the rules governing the original process
defined in $S_{j}$. Again, after a random time the original process can be
undefined in $S_{j}$; then the extended process chooses a random point in $S$
and the choice once more depends on $j$ and on the way $S_{j}$ was exited, but
not on the previous history, and so on.

There is a number of differences between our approach and that presented in
[werner]; for example, from the technical point of view, F. Werner works with
general strong Markov processes whereas we restrict ourselves to Feller
processes. The most significant of differences, however, is that in [werner]
the order in which the processes are concatenated is deterministic: after
behaving like $X^{n}$ the concatenated process starts anew as $X^{n+1}$. In
our case, the order is stochastic: after exiting from $S_{i}$ the process
chooses the point from which it starts anew randomly; moreover, its new
position depends on the way it exited from $S_{i}$ — this mechanism is
described by Feller exit boundary for the process governing evolution in
$S_{i}$ (see Section 4).

A very similar situation is studied in [kostrykin2012]*Section IV, where two
metric graphs are joined, and a Brownian motion on this larger graph is
constructed by means of Brownian motions constructed beforehand on its
components: a particle can filter from one subgraph to the other via so-called
shadow vertices. It is clear that our approach fits such constructions better
than that in [werner].

###  1.4. The main idea

It should be stressed that, in the cases where there are many exits and many
processes to concatenate, differential operators with domains described by
transmission conditions of the type (1.4)–(1.5) are difficult to handle, and
more often than not, one needs to develop special, frequently involved
techniques to obtain appropriate generation theorems for the related
stochastic processes. This was the case with generation theorems for diffusion
processes on graphs [bobmor, nagrafach, gregosiewicz, binz, gregosiewiczsem,
gregnew, banfal2, marjeta] and in thin layers [zmarkusem, zmarkusem2b,
3Dlayers, sima]. The core of the problem was the fact that in these situations
it is a priori unclear whether the so-called range condition is satisfied —
the other two conditions characterizing generators (and pre-generators) of
Feller semigroups are usually easy to check.

The main idea of the present paper is that in dealing with the problem of
concatenation of processes that are not honest, the limitations of the
previous approaches and technical problems which we face when applying them,
disappear, if instead of working with generators we use _resolvents_ (see our
Section 2 for an explanation of these notions) and Laplace transforms of _exit
laws_. Namely, we show that in the rather general context described above,
whether concatenation involves a change of boundary conditions or the way the
involved generators act, the resolvent of concatenation of processes has the
same, quite explicit form — see (5.5)–(5.6). Remarkably, all terms in the
formula for the resolvent have a clear probabilistic interpretation, whereas
the analytic methods used in the papers cited above seem to be unrelated to
any stochastic intuitions. It should be stressed here that this unification
and simplification of the previous approaches is quite impossible without the
notion of exit law (see Section 5.7).

Put otherwise: our formula for the resolvent of a concatenated process
provides a simple way of checking the range condition for a candidate for a
generator; in fact, it yields the form of the solution to the resolvent
equation. Moreover, it is an interesting feature of formula (5.5)–(5.6) that
it allows inferring properties of concatenation of processes from the
properties of its constituents.

For example, the fact that the resolvent of the process that is obtained as a
concatenation of other processes has a quite explicit and manageable form,
provides an efficient way to deal with convergence problems involving
processes on graphs or similar structures, like those studied in [bobmor,
nagrafach, knigaz, banfal2, zmarkusem, gregosiewicz, gregosiewiczsem,
gregnew].

To explain, we recall that the classical Trotter–Kato–Neveu theorem [kniga,
knigaz, ethier, goldstein, pazy] states that a sequence of semigroups
converges to a limit semigroup iff their resolvents converge to the resolvent
of the limit semigroup — and this is equivalent to convergence in distribution
of the corresponding processes (see [kallenbergnew]*p. 385). Since generation
theorems obtained in [bobmor, nagrafach, gregosiewicz, gregosiewiczsem,
gregnew, banfal2, marjeta, binz] almost never provide explicit formulae for
the resolvent, direct use the Trotter–Kato–Neveu theorem was so far
impossible. Instead, quite involved manipulations on generators, based for
example on the Sova–Kurtz version of the approximation theorem [sova, ethier,
kniga, knigaz, kurtz] or asymptotic analysis (see e.g. [banfal1]) had to be
performed. As exemplified in our Section 9, formula (5.5) allows us to use the
original version of the Trotter–Kato–Theorem, to prove a class of known
convergence theorems with greater ease and to obtain new theorems of this type
(see further down for more details).

###  1.5. Structure of the paper

In Section 2 we recall basic facts from the theory of Feller semigroups. Next,
in Section 3, we study the notion of exit laws. In the following Section 4, a
link between exit laws and excessive functions is recalled, and a definition
of regular Feller boundary is given. The main Section 5 contains our master
theorem (Theorem 5.4) saying that any finite number of Feller processes with
regular Feller boundary can be concatenated by means of probability measures
describing distributions of positions of points at which the process starts
anew after exiting from one of the components of the state-space. Formula
(5.6) (or its more formal, but equivalent version (5.11)) provides a quite
explicit form of the resolvent of such a concatenated process.

Section 6 contains examples of Feller processes with regular Feller boundary,
that is, of processes that can be concatenated within the theory presented in
this paper. Notably, we provide exact formulae for their excessive functions
and exit laws. These processes can be in particular used as building blocks in
the theorem on fast processes on graphs we discuss in Section 9.

In Section 7, by presenting a sample generation theorem, we provide a link
between the approach via resolvents developed in this paper, and the approach
via generators, used before.

The concluding Section 9 is devoted to a limit theorem, an averaging
principle, describing in particular fast processes on graphs (see Section 1.6
for more information); it is preceded by Section 8 where the theory of
convergence of semigroups is briefly recalled. In the Appendix (Section 10)
supplementary material is collected.

###  1.6. An averaging principle

As already mentioned, as a sample application of our general concatenation
theorem (Theorem 5.4), in Section 9, we revisit the question of fast
diffusions on graphs. To recall, in [nagrafach], following the previous work
devoted to reconciling two models of evolution of neurotransmitters [bobmor],
the following theorem was proved.

Consider a diffusion process on a graph $G$ without loops, and assume that
each of its vertices is a semi-permeable membrane through which particles
diffusing on the graph’s edges can filter. Permeability of the membrane
depends on the edge from which the particles filter and on the edge to which
they filter; permeability differs also in two possible vertices by which the
edges are connected. Now, assume that diffusion on each edge is accelerated
and at the same time permeability of the membranes situated at vertices is
lowered so that the fluxes through each of them remain constant. In the limit,
as diffusion is faster and faster, by averaging property of diffusion, points
in each edge become indistinguishable and so each edge collapses to a single
point, and the entire graph becomes a collection of disjoint points. Since the
fluxes remain constant, however, these points communicate: the limit process
is a Markov chain on the state-space composed of points formed from former
edges. Intensities of jumps in the limit Markov chain are functions of fluxes
of the approximating diffusions and thus functions of permeability
coefficients.

There is a couple of other versions, extensions and generalizations of this
result — see [banfal1, banfal2, zmarkusem, gregosiewicz, gregosiewiczsem,
gregnew] where also a spectrum of applications to models of applied
mathematics is discussed. In particular, Banasiak et al. [banfal1] (see also
[bobmor]) expound the fact that in biological systems often some interactions
occur at a much faster time scale than other, and thus justify the need for
theorems of this type.

To explain the idea of the generalization we present in Section 9, we consider
the space of continuous functions on the closed interval $[0,r]$ and the
operator $Af=\frac{1}{2}f^{\prime\prime}$ with Feller–Wentzel boundary
conditions (see (6.3)) — processes generated by such operators, when
concatenated, describe diffusions on graphs. If $c$ and $d$ in (6.3) are
replaced by $\varepsilon c$ and $\varepsilon d$, respectively, and $A$ is
replaced by $\varepsilon^{-1}A$, in the limit as $\varepsilon\to 0$ we obtain
a process with the state-space composed of a single point (all points of the
interval $[0,r]$ are lumped together). This limit process is described by two
independent exponential random variables, say, $T_{1}$ and $T_{2}$. After
$T\coloneqq\min(T_{1},T_{2})$ the process leaves its state-space and is no
longer defined, if $T=T_{1}$ the process leaves through the trace of the left
end of the interval, if $T=T_{2}$ it leaves through its right end — see
Section 6.2 and Example 5 for details.

Such a limit property is shared by a large class of Feller processes, except
that the number of exponential variables can be different that $2$ — we single
out such processes in Definition 9.1. For instance, in the Walsh-type Skew
Brownian motion on a star-like graph discussed in Section 6.3 the number of
variables coincides with the number of edges, and in a variety of other
examples this number is $1$. Our averaging principle of Section 9 (Theorem
9.2) says that such processes, when concatenated, in the limit as
$\varepsilon\to 0$ converge to Markov chains.

##  2\. Preliminaries on Feller semigroups and resolvents

To recall (see e.g. [bass, kniga, kallenbergnew]): if $S$ is a compact,
metrizable and separable space, and $C(S)$ is the space of continuous
functions on $S$, then a Feller semigroup is a strongly continuous family of
positive contraction operators $\left(T(t)\right)_{t\geq 0}$ in $C(S)$ such
that $T(0)=I_{C(S)}$ (the identity operator) and

$T(t)T(s)=T(t+s)\qquad s,t\geq 0.$

Because of the Riesz representation theorem, for each $x\in S$ and $t\geq 0$
there is a Borel measure $\textsf{m}_{x,t}$ on $S$ such that
$\textsf{m}_{x,t}(S)\leq 1$ and

$T(t)f(x)=\int_{S}f\,\mathrm{d}\hskip 0.56917pt\textsf{m}_{x,t},\qquad
f\in\mbox{$C(S)$};$ (2.1)

if $\textsf{m}_{x,t}(S)=1$ for all $x$ and $t$, the process is said to be
conservative or honest.

It is well-known that with each Feller semigroup one can associate a Markov
process $\left(X(t)\right)_{t\geq 0}$ with càdlàg paths [bass, kallenbergnew]
(i.e., paths that are right-continuous and possess left limits) in such a way
that

$T(t)f(x)=E_{x}f(X(t))1_{\\{t\leq\tau\\}},\qquad f\in\mbox{$C(S)$},t\geq
0,x\in S$ (2.2)

where $E_{x}$ denotes expectation conditional on $X(0)=x$ and $\tau$ is the
lifetime of the process, that is, $\tau=\tau(\omega)$ is the random time up to
which the path $X(t,\omega)$ is defined. (Comparing Eqs. (2.1) and (2.2) we
see that $\textsf{m}_{x,t}(\Gamma),\Gamma\subset S$ should be interpreted as
the probability that the related process starting at $x$ at time $0$ will be
in $\Gamma$ at a time $t\geq 0$.)

Feller semigroups are conveniently described by their generators. The
generator $A$ of a Feller semigroup is defined by

$Af=\lim_{t\to 0}t^{-1}(T(t)f-f)$

on the domain $D(A)$ composed of $f$ such that the above strong limit exists
(that is, the right-hand side converges to $Af$ uniformly on $S$). It is well-
known that the generator $A$ characterizes the semigroup uniquely (in
particular, different semigroups have different generators) and that an
operator $A$ is a generator of a Feller semigroup (shortly: a Feller
generator) in $C(S)$ iff the following three conditions are met

  * 1\.

$A$ is densely defined,

  * 2\.

$A$ satisfies the positive maximum principle,

  * 3\.

$A$ satisfies the range condition: for any $g\in C(S)$ and $\lambda>0$ there
is an $f\in D(A)$ such that $\lambda f-Af=g.$

The semigroup generated by $A$ will in what follows be denoted
$\mbox{$\left(\mathrm{e}^{t{A}}\right)_{t\geq 0}$}.$

In defining Feller semigroups it is customary to require also that the related
process is honest or conservative (which comes down to the requirement that
$1_{S}\in D(A)$ and $A1_{S}=0$) but in this paper, for obvious reasons, we
will allow the process to be dishonest. It is easy to see that the process is
honest iff $T(t)1_{S}=1_{S}$ for all $t\geq 0$ and this holds iff $\tau$ of
equation (2.1), that is, the lifetime of the process, is a.s. equal to
$\infty$.

An alternative description of a Feller semigroup is provided via Feller
resolvents. A family $R_{\lambda},\lambda>0$ of non-negative operators in
$C(S)$ is said to be a Feller resolvent if the following conditions are met:

  * 1\.

the Hilbert equation holds:

$(\lambda-\mu)R_{\mu}R_{\lambda}=R_{\mu}-R_{\lambda},\qquad\lambda,\mu>0,$
(2.3)

  * 2\.

for each $f\in\mbox{$C(S)$}$, $\lim_{\lambda\to\infty}\lambda R_{\lambda}f=f,$

  * 3\.

$\lambda R_{\lambda}1_{S}\leq 1_{S}$ for all $\lambda>0$.

It may be argued (see the already cited references) that for each Feller
resolvent there is a Feller generator $A$ such that
$R_{\lambda}=\left(\lambda-A\right)^{-1}$, and thus also the related Feller
semigroup. Moreover, the related process is conservative iff $\lambda
R_{\lambda}1_{S}=1_{S}$ for all $\lambda>0$.

A word about terminology: a family $R_{\lambda},\lambda>0$ of bounded
operators is said to be a pseudoresolvent, if it satisfies the Hilbert
equation. This family is said to be regular, if additionally condition 2.
given above holds.

##  3\. Exit laws

Let $S$, as above, be a compact, metrizable and separable space, and let $A$
be the generator of a Feller semigroup in $C(S)$ with resolvent
$R_{\lambda},\lambda>0$. By definition, $\ell_{\lambda},\lambda>0$ is the
_Laplace transform of an exit law_ for $A$ iff (compare [neveu62]*p. 324,
[fellerboundaryforMC]*p. 538, or [fmc]*Section 3.6.10, cf. also the section on
_exit systems_ in [blum]; condition (c), below, bears remarkable resemblance
to the characteristic equation in _empathy theory_ of N. Sauer – see e.g.
[sauer]*eq. (7))

  * (a)

$(0,\infty)\ni\lambda\mapsto\ell_{\lambda}\in\mbox{$C(S)$}$ is nontrivial
(i.e., $\ell_{\lambda}\not=0$ for at least one $\lambda$), locally bounded and
non-negative,

  * (b)

$\lim_{\lambda\to 0+}\ell_{\lambda}(x)\leq 1$ for each $x\in S$, and

  * (c)

we have

$(\lambda-\mu)R_{\lambda}\ell_{\mu}=\ell_{\mu}-\ell_{\lambda},\qquad\lambda,\mu>0.$
(3.1)

In this section we collect basic properties of such objects. In what follows,
they will, for simplicity, often be called ‘ _exit laws_ ’.

######  Proposition 3.1.

Each locally bounded function
$(0,\infty)\ni\lambda\mapsto\ell_{\lambda}\in\mbox{$C(S)$}$ satisfying (3.1)
is infinitely differentiable with

$\ell_{\lambda}^{(n)}=(-1)^{n}n!(R_{\lambda})^{n}\ell_{\lambda},\qquad n\geq
1$ (3.2)

###### Proof.

Fix a $\lambda\in(0,\infty)$, and let $M$ be such that $\|\ell_{\mu}\|\leq M$
for $\mu$ in a neighborhood of $\lambda$. Since $\|\lambda R_{\lambda}\|\leq
1,$ the left-hand side of (3.1) does not exceed
$|\lambda-\mu|\frac{M}{\lambda}$ in this neighborhood. It follows that, as
$\mu\to\lambda$, $\ell_{\mu}$ converges to $\ell_{\lambda}$, establishing
continuity of $\lambda\mapsto\ell_{\lambda}$.

To prove (3.2) we proceed by induction. For $n=1$ the formula is a direct
consequence of (3.1) and the already established continuity. Moreover,
assuming (3.2) is true for some $n$, we have

$\displaystyle\frac{\ell_{\mu}^{(n)}-\ell_{\lambda}^{(n)}}{\mu-\lambda}$
$\displaystyle=\frac{(-1)^{n}n!}{\mu-\lambda}\left[(R_{\mu})^{n}\ell_{\mu}-(R_{\lambda})^{n}\ell_{\lambda}\right]$
$\displaystyle=(-1)^{n}n!\left[(R_{\mu})^{n}\left(\frac{\ell_{\mu}-\ell_{\lambda}}{\mu-\lambda}\right)+\left(\frac{(R_{\mu})^{n}-(R_{\lambda})^{n}}{\mu-\lambda}\right)\ell_{\lambda}\right].$

Since $(R_{\lambda})^{\prime}=-(R_{\lambda})^{2}$, using the already checked
case $n=1$, we obtain

$\displaystyle\ell_{\lambda}^{(n+1)}$
$\displaystyle=(-1)^{n}n!(R_{\lambda})^{n}\ell_{\lambda}^{\prime}+(-1)^{n+1}n!n(R_{\lambda})^{n-1}(R_{\lambda})^{2}\ell_{\lambda}$
$\displaystyle=(-1)^{n+1}(n+1)!(R_{\lambda})^{n+1}\ell_{\lambda},$

as desired. ∎

To recall, a real-valued, non-negative function defined on $(0,\infty)$ is
said to be absolutely monotone iff it is infinitely differentiable and for
each $n$, its $(n+1)$st derivative has an opposite sign to its $n$th
derivative. The Bernstein Theorem says that a function is absolutely monotone
iff it is the Laplace transform of a Borel, possibly infinite, measure on
$[0,\infty)$ (see [feller]*p. 439). Thus, Proposition 3.1 implies that if
$\ell_{\lambda}$ is the Laplace transform of an exit law, then for each $x\in
S$, $\lambda\mapsto\ell_{\lambda}(x)$, being absolutely monotone and
satisfying $\lim_{\lambda\to 0+}\ell_{\lambda}(x)\leq 1$, is the Laplace
transform of a Borel sub-probability measure, say $m_{x}$, on $[0,\infty)$:

$\ell_{\lambda}(x)=\int_{0}^{\infty}\mathrm{e}^{-\lambda
t}\,m_{x}(\mathrm{d}t),\quad\lambda>0.$ (3.3)

We note that, for a Borel subset $\mathcal{B}$ of $[0,\infty)$,
$m_{x}(\mathcal{B})$ should be interpreted as the probability that the process
generated by $A$ and starting at $x$ will leave $S_{A}$ at time
$t\in\mathcal{B}$.

######  Proposition 3.2.

None of the measures $m_{x},x\in S_{A}$ has an atom at $t=0$.

###### Proof.

The resolvent $R_{\lambda},\lambda>0$ is regular, that is,
$\lim_{\lambda\to\infty}\lambda R_{\lambda}f=f,f\in\mbox{$C(S)$}$. This
implies that the left-hand side of (3.1) converges to $\ell_{\mu}$, as
$\lambda\to\infty$. On the other hand, by (3.3),
$\lim_{\lambda\to\infty}\ell_{\lambda}(x)=m_{x}(\\{0\\})$, showing that the
value of the right-hand side of (3.1) at a point $x\in S$ converges to
$\ell_{\mu}(x)-m_{x}(\\{0\\}).$ These two limits cannot be reconciled unless
$m_{x}(\\{0\\})=0.$ ∎

######  Corollary 3.3.

As a by-product of the proof of Proposition 3.2, we obtain
$\lim_{\lambda\to\infty}\ell_{\lambda}=0$ (in the norm of $C(S)$).

Our next theorem establishes existence of an exit law for any non-conservative
(not honest) Feller generator $A$. As already recalled, a Feller generator $A$
with resolvent $R_{\lambda},\lambda>0$ is conservative iff $\lambda
R_{\lambda}1_{S}=1_{S}$ for each $\lambda>0.$ Since $\|\lambda
R_{\lambda}\|\leq 1$ and $R_{\lambda}\geq 0$, we see that for nonconservative
$A$ the function $1_{S}-\lambda R_{\lambda}1_{S}\in\mbox{$C(S)$}$ is non-
negative and nontrivial (i.e., it does not vanish everywhere), for all
$\lambda>0$.

######  Theorem 3.4.

Suppose $A$ is not conservative. Then, there exists at least one exit law for
$A$.

###### Proof.

By assumption

$\ell_{\lambda}\coloneqq 1_{S}-\lambda R_{\lambda}1_{S}\geq 0,\qquad\lambda>0$

is non-zero and $\|\ell_{\lambda}\|\leq 1.$ Moreover,

$\ell_{\mu}-\ell_{\lambda}=\lambda R_{\lambda}1_{S}-\mu R_{\mu}1_{S},$

and, because of the resolvent equation for $R_{\lambda},\lambda>0$,

$\displaystyle(\lambda-\mu)R_{\lambda}\ell_{\mu}$
$\displaystyle=(\lambda-\mu)R_{\lambda}1_{S}+\mu(\mu-\lambda)R_{\lambda}R_{\mu}1_{S}$
$\displaystyle=(\lambda-\mu)R_{\lambda}1_{S}+\mu(R_{\lambda}-R_{\mu})1_{S}$
$\displaystyle=\lambda R_{\lambda}1_{S}-\mu R_{\mu}1_{S}.$

This establishes (3.1), and in particular implies representation (3.3). This
representation in turn, when combined with $\|\ell_{\lambda}\|\leq 1$, implies
that the limit $\lim_{\lambda\to 0+}\ell_{\lambda}(x)$ exists and does not
exceed $1$ for each $x\in S$. ∎

In what follows, we will write $\mathcal{L}_{\lambda}$ to denote the
particular Laplace transform of the exit law found in Theorem 3.4:

$\mathcal{L}_{\lambda}\coloneqq 1_{S}-\lambda R_{\lambda}1_{S}.$

######  Example 1.

Let $S=[0,1]$ be the unit interval, and let $Af=f^{\prime}$ on the domain
composed of continuously differentiable $f$ such that $f^{\prime}(1)+\alpha
f(1)=0,$ where $\alpha\geq 0$ is a given parameter. In the process related to
$A$, a particle starting at an $x\in[0,1)$ moves to the right with constant
velocity $v=1$; upon reaching $x=1$ it stays there for an exponential time
with parameter $\alpha$, and after this exponential time elapses, the process
is no longer defined (for $\alpha=0$ the process stays at $x=1$ for ever, and
in particular is well-defined for all $t\geq 0$). For this operator, the
semigroup may be given explicitly:

$\mathrm{e}^{tA}f(x)=\begin{cases}f(x+t),&t\leq 1-x,\\\
\mathrm{e}^{-\alpha(t-1+x)}f(1),&t>1-x.\end{cases}$

It follows that

$R_{\lambda}f(x)=\int_{x}^{1}\mathrm{e}^{\lambda(x-y)}f(y)\,\mathrm{d}\hskip
0.56917pty+\mathrm{e}^{\lambda(x-1)}\frac{f(1)}{\lambda+\alpha}.$ (3.4)

Thus, here, (if $\alpha>0$)

$\mathcal{L}_{\lambda}(x)=1-\lambda
R_{\lambda}1_{S}(x)=\mathrm{e}^{\lambda(x-1)}\frac{\alpha}{\lambda+\alpha},\qquad
x\in[0,1].$

The first factor in this product is the Laplace transform of the deterministic
time $T=1-x$ needed for the process starting at an $x\in[0,1]$ to reach $x=1$;
the other factor is the Laplace transform of the exponential time spent at the
boundary $x=1$. Thus, all in all, $\mathcal{L}_{\lambda}^{A}(x)$ is the
Laplace transform of the lifetime of the process governed by $A$ that starts
at $x\in[0,1].$ ∎

The example presented above is a particular case of the more general result.
Namely, we have the following observation.

######  Proposition 3.5.

Let $\tau$ be the lifetime of the process governed by a Feller resolvent
$R_{\lambda},\lambda>0$. Then $\mathcal{L}_{\lambda}$ is the Laplace transform
of $\tau$:

$\mathcal{L}_{\lambda}(x)=E_{x}\mathrm{e}^{-\lambda\tau},\qquad x\in S.$

###### Proof.

By (2.2), $\mathrm{e}^{tA}1_{S}(x)=\mathbb{P}_{x}(t\leq\tau)$. On the other
hand,

$\displaystyle E_{x}\mathrm{e}^{-\lambda\tau}$
$\displaystyle=\lambda\int_{0}^{\infty}\mathrm{e}^{-\lambda
t}\mathbb{P}_{x}(\tau\leq t)\,\mathrm{d}\hskip
0.56917ptt=1-\lambda\int_{0}^{\infty}\mathrm{e}^{-\lambda
t}\mathbb{P}_{x}(t\leq\tau)\,\mathrm{d}\hskip 0.56917ptt$
$\displaystyle=1-\lambda\int_{0}^{\infty}\mathrm{e}^{-\lambda
t}\mathrm{e}^{tA}1_{S}(x)\,\mathrm{d}\hskip 0.56917ptt=1-\lambda
R_{\lambda}1_{S}(x),$

as desired. ∎

######  Corollary 3.6.

We have

$\max_{x\in S}\mathcal{L}_{\lambda}(x)<1.$

###### Proof.

For no point $x\in S$ can we have $\mathcal{L}_{\lambda}(x)=1$ for this would
mean that the lifetime of the process starting at $x$ is $0$, contradicting
Proposition 3.2. Since $S$ is compact and $\mathcal{L}_{\lambda}$ is
continuous, we are done. ∎

######  Example 2.

Let $A$ be an honest Feller generator. Then, for any $\alpha>0,$
$A_{\alpha}\coloneqq A-\alpha I$ where $I$ is the identity operator, is a not
honest Feller generator. The process related to $A_{\alpha}$ is identical to
that related to $A$ up to an independent exponentially distributed time $T$
with parameter $\alpha$; from this point on, the process generated by
$A_{\alpha}$ is undefined. In other words, $T$ is the lifetime of the modified
process. Indeed, since $A$ is honest,

$1_{S}-\lambda\left(\lambda-
A_{\alpha}\right)^{-1}1_{S}=1_{S}-\lambda\left(\lambda+\alpha-A\right)^{-1}1_{S}={\textstyle\frac{\lambda}{\lambda+\alpha}}1_{S},$

and $\lambda\mapsto\frac{\lambda}{\lambda+\alpha}$ is the Laplace transform of
the distribution of $T$.

##  4\. Exit laws and excessive functions; Feller boundary

Let again $A$ be a Feller generator in $C(S)$ with resolvent
$R_{\lambda},\lambda>0.$ A nonnegative function $\phi\in C(S)$ such that
$\phi\leq 1_{S}$ and

$\lambda R_{\lambda}\phi\leq\phi,\qquad\lambda>0$ (4.1)

is termed _excessive_ , see [bass, beznea, chunglectures, doobc, revuz]. We
will say that an excessive $\phi$ is nontrivial if $\lambda
R_{\lambda}\phi\not=\phi$ for at least one $\lambda$ (and thus for all
$\lambda>0$).

######  Remark 1.

In most monographs, the requirement that $\phi\in\mbox{$C(S)$}$ is omitted in
the definition of excessive function; instead other, milder, regularity
conditions are imposed. In this paper, the assumption that the Laplace
transform of an exit law is a member of $C(S)$ seemed more pleasing.

As it turns out, many Laplace transforms of exit laws may be constructed from
nontrivial excessive functions. Namely, calculations presented in Theorem 3.4
show that

$\ell_{\lambda}\coloneqq\phi-\lambda R_{\lambda}\phi$ (4.2)

satisfies (3.1), and thus (being bounded by $1_{S}$) is the Laplace transform
of an exit law as long as $\phi$ is excessive and nontrivial.

Conversely, suppose we are given the Laplace transform of an exit law
$\ell_{\lambda},\lambda>0$. Since, for each $x\in S$, $\ell_{\mu}(x)$
increases as $\mu\searrow 0$, and $R_{\lambda}$s are integral operators, we
may consider

$\phi(x)\coloneqq\lim_{\mu\to 0}\ell_{\mu}(x)\leq 1.$ (4.3)

Then, letting $\mu\to 0$ in (3.1) we obtain (by the Monotone Convergence
Theorem),

$\ell_{\lambda}(x)=\phi(x)-\lambda R_{\lambda}\phi(x),\qquad x\in
S,\lambda>0.$ (4.4)

It should be stressed, though, that in this representation of
$\ell_{\lambda},\lambda>0$, in contrast to (4.2), $\phi$ need not be
continuous. Nevertheless, the right-hand side makes sense because the operator
$R_{\lambda}$, being related to a measurable kernel, has a natural extension
to the space of bounded measurable functions.

Even if we restrict ourselves to $\phi\in\mbox{$C(S)$}$, there still remains
the question of uniqueness of representation of exit laws in terms of
excessive functions. Fortunately, there is a straightforward test for such
uniqueness: to make sure that $\phi$ in (4.4) is determined by
$\ell_{\lambda},\lambda>0$ it suffices to check that there are no non-zero
$f\in\mbox{$C(S)$}$ such that $f=\lambda R_{\lambda}f$, that is, no non-zero
$f\in D(A)$ such that $Af=0$.

In what follows we restrict ourselves to exit laws of the form (4.2) with
$\phi\in\mbox{$C(S)$}$ and assume that the kernel of $A$ is trivial. Moreover,
since, as we shall see in the next section, our work is motivated by Feller
processes that have more than one exit law, we will in fact be concerned with
the situation described by the following definition (comp. [fellerbi] or
[fmc]*Section 3.7).

######  Definition 4.1.

_We will say that the process governed by $A$ has a regular Feller exit
boundary with ${\mathpzc k}$ exit points, if_

  * (i)

_the kernel of_ $A$ is trivial (that is, composed of zero function merely),

  * (ii)

_there are nontrivial excessive functions_ $\phi^{j},j\in\\{1,\dots,{\mathpzc
k}\\}$ for $A$ such that

$\sum_{j=1}^{\mathpzc k}\phi^{j}=1_{S}.$ (4.5)

The Laplace transform of the exit law $\ell_{\lambda}^{j}$ related to
$\phi^{j}$:

$\ell_{\lambda}^{j}\coloneqq\phi^{j}-\lambda R_{\lambda}\phi^{j}$

will be somewhat informally called the law for exit through the $j$th exit
(or: $j$th gate).

We stress again that $\phi^{j}$ are, by definition, continuous and that, by
assumption (i), representation in terms of excessive functions is unique. It
should be noted, however, that our definition says nothing about uniqueness of
representation (4.5): it is possible, in particular, that one of $\phi^{j}$s
is a sum of two smaller nontrivial excessive functions (compare discussion in
[fellerbi] or [fmc]*Section 3.7). Therefore, the definition should be
understood to mean that the process has _at least_ ${\mathpzc k}$ ways of
exiting its state-space, and it is in this sense that this definition is
applied in what follows. Nevertheless, in the examples of Section 6 it will be
clear from the probabilistic description of the processes involved that there
are precisely one, two or ${\mathpzc k}$ exits there.

##  5\. Concatenation of $N$ processes

###  5.1. Set-up

Suppose we are given $N$ Feller processes in $N$ separate (i.e., disjoint)
compact spaces $S_{1},\dots,S_{N}$, generated by operators
$A_{1},\dots,A_{N}$, respectively. To avoid trivialities, we assume that at
least one of these generators is not conservative. Since $S_{i}$s can be
reordered if necessary, we assume without loss of generality that the first
$M\in\\{1,\dots,N\\}$ processes are not conservative, whereas the remaining
ones are conservative. Moreover, we assume that the process governed by
$A_{i},i=1,\dots,M$ has a regular Feller boundary with ${\mathpzc k}(i)$ exit
points: there are excessive functions $\phi^{i,j},j=1,\dots,{\mathpzc k}(i)$
such that

$\sum_{j=1}^{{\mathpzc k}(i)}\phi^{i,j}=1_{S_{i}},\qquad i=1,\dots,M.$ (5.1)

For the corresponding exit laws

$\ell_{\lambda}^{i,j}\coloneqq\phi^{i,j}-\lambda R_{\lambda,i}\phi^{i,j}$

this means that

$\sum_{j=1}^{{\mathpzc k}(i)}\ell_{\lambda}^{i,j}=1_{S_{i}}-\lambda
R_{\lambda,i}1_{S_{i}}\eqqcolon\mathcal{L}_{\lambda}^{i},\qquad i=1,\dots,M.$
(5.2)

We construct the resolvent of a Feller process on

$S_{\textnormal{\hskip 0.56917pt\tiny
u}}\coloneqq\dot{\bigcup_{i=1,\dots,N}}S_{i},$

(the dot stresses the fact that $S_{i}$ are disjoint; ‘u’ for ‘union’), which
is governed by the following rules.

  * (i)

Conditional on starting at an $x\in S_{i}$, the process is identical to that
related to $A_{i}$ up to the random time when the latter is no longer defined
(if $A_{i}$ is conservative, this random time is $\infty$ and thus both
processes are identical at all times).

  * (ii)

If and when the process related to $A_{i}$ is no longer defined, the process
we construct starts afresh at a random point $y\in S_{\textnormal{\hskip
0.56917pt\tiny u}}$, the distribution of its position at this moment depending
on the gate through which $S_{i}$ was exited. More specifically, this position
is described by a Borel sub-probability measure $\mathfrak{p}_{i,j}$ on
$S_{\textnormal{\hskip 0.56917pt\tiny u}}$. In most applications, the latter
measure is supported in $S_{\textnormal{\hskip 0.56917pt\tiny u}}\setminus
S_{i}$, but this assumption is not needed in this section.

  * (iii)

If $y$ belongs to $S_{j}$, the constructed process is identical to that
governed by $A_{j}$ up to the random time, when the latter is no longer
defined, and so on.

###  5.2. More notation

To proceed, we need to establish more notation. First of all, to shorten
formulae, we introduce

$\displaystyle\mathcal{N}$ $\displaystyle\coloneqq\\{1,\dots,N\\},$
$\displaystyle\mathcal{M}$ $\displaystyle\coloneqq\\{1,\dots,M\\},$
$\displaystyle\mathcal{IJ}$
$\displaystyle\coloneqq\\{(i,j);i\in\mathcal{M},j\in\\{1,\dots,{\mathpzc
k}(i)\\}\\}.$

Secondly, it will be convenient to think of an $f\in\mbox{$C(S_{i})$}$ as a
member of $C(S_{\textnormal{\hskip 0.56917pt\tiny u}})$ also: to this end we
extend $f$ to the entire $S_{\textnormal{\hskip 0.56917pt\tiny u}}$ by
agreeing that it equals zero outside of $S_{i}$. This allows thinking of sums
of $f_{i}\in C(S_{i})$ as members of $C(S_{\textnormal{\hskip 0.56917pt\tiny
u}})$:

$f\coloneqq\dot{\sum_{i\in\mathcal{N}}}f_{i};$

as above, the dot stresses the fact that we add together functions that are
defined on separate spaces. Conversely, for an
$f\in\mbox{$C(S_{\textnormal{\hskip 0.56917pt\tiny u}})$}$ we will denote its
part in $C(S_{i})$ by $f_{i}$, that is, $f_{i}\coloneqq f1_{S_{i}}$ or, with
the identification described above in mind, $f_{i}=f_{|S_{i}}.$

Let $R_{\lambda,i},\lambda>0$ denote the resolvent of the $i$th process:

$R_{\lambda,i}=\left(\lambda-
A_{i}\right)^{-1},\qquad\lambda>0,i\in\mathcal{N}.$

The formula

$R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
du}}f=\dot{\sum_{i\in\mathcal{N}}}R_{\lambda,i}f_{i},\qquad\lambda>0,f\in\mbox{$C(S_{\textnormal{\hskip
0.56917pt\tiny u}})$}$ (5.3)

describes the resolvent of the process called _disjoint union_ of the involved
process (see [sharpe]*p. 83). In this union there are no jumps between spaces
$S_{i},i\in\mathcal{N}$: if random evolution is started at $x\in S_{i}$, the
rules of $A_{i}$ are obeyed throughout the process’s lifetime. When a particle
exists $S_{i}$, the process is left undefined; it never jumps to any other
$S_{j}$ or back to $S_{i}$. We note the following connection between the exit
laws of the right-hand side of (5.2) and $R_{\lambda}^{\textnormal{\hskip
0.56917pt\tiny du}},\lambda>0$:

$\dot{\sum_{i\in\mathcal{M}}}{\mathcal{L}_{\lambda}^{i}}=1_{S_{\textnormal{\hskip
0.56917pt\tiny u}}}-\lambda R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
du}}1_{S_{\textnormal{\hskip 0.56917pt\tiny
u}}}\eqqcolon\mathcal{L}_{\lambda}.$ (5.4)

###  5.3. The definition: first attempt

By the _concatenation of processes_ governed by $A_{i},i\in\mathcal{N}$ we
understand the process with the following resolvent:

$R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
co}}f=\dot{\sum_{i\in\mathcal{N}}}[R_{\lambda}^{i}f_{i}+\sum_{j=1}^{{\mathpzc
k}(i)}\left(\int_{S}R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
co}}f\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j}\right)\ell_{\lambda}^{i,j}],\qquad\lambda>0,f\in\mbox{$C(S_{\textnormal{\hskip
0.56917pt\tiny u}})$};$ (5.5)

for $i\in\mathcal{N}\setminus\mathcal{M}$, the inner sum above is understood
to be zero. The Borel measure $\mathfrak{p}_{i,j}$ which, by assumption,
satisfies $0\leq\mathfrak{p}_{i,j}(S_{\textnormal{\hskip 0.56917pt\tiny
u}})\leq 1$, is the distribution of the point from which the process starts
anew after exiting from the $i$th region through the $j$th gate. For some of
$\mathfrak{p}_{i,j}$ we may have $\mathfrak{p}_{i,j}(S_{\textnormal{\hskip
0.56917pt\tiny u}})<1$, some of $\mathfrak{p}_{i,j}$ might in fact be zero. Of
course, if all of them are zero, $R_{\lambda}^{\textnormal{\hskip
0.56917pt\tiny co}},\lambda>0$ coincides with the resolvent
$R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny du}},\lambda>0$ of the
disjoint union process.

We note also that, because of the convention concerning the inner sum, formula
(5.5) may equivalently be written as

$R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
co}}f=R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
du}}f+\dot{\sum_{i\in\mathcal{M}}}\sum_{j=1}^{{\mathpzc
k}(i)}\left(\int_{S}R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
co}}f\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j}\right)\ell_{\lambda}^{i,j},$ (5.6)

where $R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny du}},\lambda>0$ is the
resolvent of the disjoint union, defined in (5.3).

###  5.4. Consistency condition

The first and obvious problem with (5.5) or its equivalent form (5.6) is that
these formulae define $R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}}$
via yet undefined integrals

$\int_{S}R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
co}}f\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j},\qquad(i,j)\in\mathcal{IJ}.$ (5.7)

Fortunately, these may be calculated beforehand from consistency condition for
(5.5), which we will now derive.

To this end, we change summation indexes $i$ and $j$ in (5.6) to $k$ and $l$,
respectively, then fix $i\in\mathcal{M}$ and $j\in\\{1,\dots,{\mathpzc
k}(i)\\}$, and integrate both sides of equation (5.6) with respect to
$\mathfrak{p}_{i,j}.$ Since $\ell_{\lambda}^{k,l}$ is supported in $S_{k}$, we
obtain

$\int_{S_{\textnormal{\hskip 0.56917pt\tiny
u}}}R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}}f\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j}=\int_{S_{\textnormal{\hskip 0.56917pt\tiny
u}}}R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny du}}f\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j}+{\sum_{k\in\mathcal{M}}}\sum_{l=1}^{{\mathpzc
k}(k)}\int_{S_{\textnormal{\hskip 0.56917pt\tiny
u}}}R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}}f\,\mathrm{d}\hskip
0.56917pt\mu_{k,l}\int_{S_{k}}\ell_{\lambda}^{k,l}\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j}.$ (5.8)

We claim that this consistency condition determines integrals (5.7) as
functions of

$\int_{S_{\textnormal{\hskip 0.56917pt\tiny
u}}}R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny du}}f\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j},\quad(i,j)\in\mathcal{IJ}\quad\text{ and
}\quad\int_{S_{k}}\ell_{\lambda}^{k,l}\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j},\quad(i,j),(k,l)\in\mathcal{IJ},$

and these may be treated as known prior to defining
$R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}}f$.

In the proof of our claim, we consider any $u\in\mathbb{R}^{\mathpzc k}$ where

${\mathpzc k}={\mathpzc k}(1)+\dots+{\mathpzc k}(M),$

as the concatenation (in the linear algebra sense) of vectors

$\begin{matrix}&(u_{1,1}&u_{1,2}&\dots&u_{1,{\mathpzc
k}(1)})\in\mathbb{R}^{{\mathpzc k}(1)},\\\
&\vdots&\vdots&\dots&\vdots\phantom{\in\mathbb{R}^{{\mathpzc k}(1)},}\\\
&(u_{M,1}&u_{M,2}&\dots&u_{M,{\mathpzc k}(M)})\in\mathbb{R}^{{\mathpzc
k}(M)},\end{matrix}$

and equip $\mathbb{R}^{\mathpzc k}$ with the maximum norm
$\|u\|=\max_{(i,j)\in\mathcal{IJ}}|u_{i,j}|$.

######  Lemma 5.1.

Fix $\lambda>0$. For any
$v=\left(v_{i,j}\right)_{(i,j)\in\mathcal{IJ}}\in\mathbb{R}^{\mathpzc k}$
there is precisely one
$u=\left(u_{i,j}\right)_{(i,j)\in\mathcal{IJ}}\in\mathbb{R}^{\mathpzc k}$ such
that

$u_{i,j}=v_{i,j}+{\sum_{k\in\mathcal{M}}}\sum_{l=1}^{{\mathpzc
k}(k)}u_{k,l}\int_{S_{k}}\ell_{\lambda}^{k,l}\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j},\qquad(i,j)\in\mathcal{IJ}.$ (5.9)

Furthermore, $u\geq 0$ whenever $v\geq 0$.

###### Proof.

It suffices to show that the linear map, say, $N_{\lambda}$, that assigns

$\left({\sum_{k\in\mathcal{M}}}\sum_{l=1}^{{\mathpzc
k}(k)}w_{k,l}\int_{S_{k}}\ell_{\lambda}^{k,l}\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j}\right)_{(i,j)\in\mathcal{IJ}}\in\mathbb{R}^{\mathpzc
k}$

to a $w\coloneqq(w_{i,j})_{(i,j)\in\mathcal{IJ}}$, has norm smaller than $1$.
For, then the series $\sum_{n=0}^{\infty}N_{\lambda}^{n}$ converges and
$u\coloneqq(\sum_{n=0}^{\infty}N_{\lambda}^{n})v$ is the unique solution to
(5.9). Moreover, $u\geq 0$ provided that $v\geq 0$ because $N_{\lambda}$ is a
non-negative operator.

On the other hand, obviously,

$\|N_{\lambda}w\|\leq\|w\|\max_{(i,j)\in\mathcal{IJ}}\sum_{k\in\mathcal{M}}\sum_{l=1}^{{\mathpzc
k}(k)}\int_{S_{k}}\ell_{\lambda}^{k,l}\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j},$

whereas, by (5.2) in the first step and (5.4) in the last,

$\sum_{k\in\mathcal{M}}\sum_{l=1}^{{\mathpzc
k}(k)}\int_{S_{k}}\ell_{\lambda}^{k,l}\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j}=\sum_{k\in\mathcal{M}}\int_{S_{k}}\mathcal{L}_{\lambda}^{k}\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j}=\int_{S_{\textnormal{\hskip 0.56917pt\tiny
u}}}\dot{\sum_{k\in\mathcal{M}}}\mathcal{L}_{\lambda}^{k}\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j}=\int_{S_{\textnormal{\hskip 0.56917pt\tiny
u}}}\mathcal{L}_{\lambda}\,\mathrm{d}\hskip 0.56917pt\mathfrak{p}_{i,j}.$
(5.10)

This means that

$\|N_{\lambda}\|\leq\max_{(i,j)\in\mathcal{IJ}}\int_{S_{\textnormal{\hskip
0.56917pt\tiny u}}}\mathcal{L}_{\lambda}\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j}.$

Now, since $\mathfrak{p}_{i,j}$s are sub-probability measures, Corollary 3.6
tells us that $0\leq\int_{S_{\textnormal{\hskip 0.56917pt\tiny
u}}}\mathcal{L}_{\lambda}\,\mathrm{d}\hskip 0.56917pt\mathfrak{p}_{i,j}<1$ for
each $(i,j)\in\mathcal{IJ}.$ This completes the proof, the set $\mathcal{IJ}$
being finite. ∎

In what follows we write $M_{\lambda}\coloneqq I-N_{\lambda}$ where $I$ is the
identity operator in $\mathbb{R}^{\mathpzc k}$. Our lemma says that the
operator $M_{\lambda}$ is invertible with

$M_{\lambda}^{-1}\coloneqq\sum_{n=0}^{\infty}N_{\lambda}^{n}.$

###  5.5. Back to the definition

Lemma 5.1 allows filling the hole in the definition given in Section 5.3 as
follows. For $\lambda>0$ and $f\in\mbox{$C(S_{\textnormal{\hskip
0.56917pt\tiny u}})$}$ we define vectors $v=v(f,\lambda)$ and $u=u(f,\lambda)$
in $\mathbb{R}^{\mathpzc k}$ by

$v=\left(\int_{S}R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
du}}g\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j}\right)_{(i,j)\in\mathcal{IJ}}\quad\text{ and
}\quad u=(u_{i,j}(f,\lambda))_{(i,j)\in\mathcal{IJ}}=M_{\lambda}^{-1}v,$

and then let $R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}}f$ be defined
by (cf. (5.6))

$R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
co}}f=R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
du}}f+\dot{\sum_{i\in\mathcal{M}}}\sum_{j=1}^{{\mathpzc
k}(i)}u_{i,j}(f,\lambda)\ell_{\lambda}^{i,j}.$ (5.11)

Since $\ell^{i,j}_{\lambda}$s are continuous functions,
$R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}}$ maps
$C(S_{\textnormal{\hskip 0.56917pt\tiny u}})$ to
$\mbox{$C(S_{\textnormal{\hskip 0.56917pt\tiny u}})$},$ and it is clear that
it is a linear operator.

In the next section, we will show that the so-defined operators
$R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}},\lambda>0$ form a Feller
resolvent. Here we would like to argue merely that, for the so-defined
$R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}}f$, the vector of
integrals $\int_{S_{\textnormal{\hskip 0.56917pt\tiny
u}}}R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}}f\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j},(i,j)\in\mathcal{IJ}$ coincides with
$u(f,\lambda):$

$\widetilde{u}\coloneqq\left(\int_{S_{\textnormal{\hskip 0.56917pt\tiny
u}}}R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}}f\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j}\right)_{(i,j)\in\mathcal{IJ}}=\left(u_{i,j}(f,\lambda)\right)_{(i,j)\in\mathcal{IJ}}$
(5.12)

Indeed, integrating (5.11) over $S_{\textnormal{\hskip 0.56917pt\tiny u}}$
with respect to $\mathfrak{p}_{i,j}$, we check that

$\widetilde{u}=v+N_{\lambda}u.$

But $u$ is the only solution to the equation $u=v+N_{\lambda}u$, where $v$ is
given, and so, in particular, $v+N_{\lambda}u$ equals $u$. This establishes
the claim.

###  5.6. $R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}},\lambda>0$ is a
Feller resolvent in $C(S_{\textnormal{\hskip 0.56917pt\tiny u}})$

For the proof that $R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
co}},\lambda>0$ is a Feller resolvent, we need two lemmas. The first of these
will help us prove that $R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
co}},\lambda>0$ satisfies the Hilbert equation, the second will be
instrumental in proving $\lambda R_{\lambda}^{\textnormal{\hskip
0.56917pt\tiny co}}1_{S_{\textnormal{\hskip 0.56917pt\tiny u}}}\leq
1_{S_{\textnormal{\hskip 0.56917pt\tiny u}}}.$

######  Lemma 5.2.

For $\lambda,\mu>0$ and $f\in\mbox{$C(S_{\textnormal{\hskip 0.56917pt\tiny
u}})$}$,

$(\lambda-\mu)u(R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
co}}f,\mu)=u(f,\mu)-u(f,\lambda).$

In other words,

$\int_{S_{\textnormal{\hskip 0.56917pt\tiny
u}}}(\lambda-\mu)R_{\mu}^{\textnormal{\hskip 0.56917pt\tiny
co}}R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}}f\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j}=\int_{S_{\textnormal{\hskip 0.56917pt\tiny
u}}}R_{\mu}^{\textnormal{\hskip 0.56917pt\tiny co}}f\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j}-\int_{S_{\textnormal{\hskip 0.56917pt\tiny
u}}}R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}}f\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j},\qquad(i,j)\in\mathcal{IJ}.$

###### Proof.

We have

$\displaystyle u(R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}}f,\mu)$
$\displaystyle=M_{\mu}^{-1}\left(\int_{S}R_{\mu}^{\textnormal{\hskip
0.56917pt\tiny du}}(R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
co}}f)\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j}\right)_{(i,j)\in\mathcal{IJ}}$
$\displaystyle=M_{\mu}^{-1}\left(\int_{S}[R_{\mu}^{\textnormal{\hskip
0.56917pt\tiny du}}R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
du}}f+R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
du}}\dot{\sum_{i\in\mathcal{M}}}\sum_{j=1}^{{\mathpzc
k}(i)}u_{i,j}(f,\lambda)\ell_{\lambda}^{i,j}]\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j}\right)_{(i,j)\in\mathcal{IJ}}$
$\displaystyle=M_{\mu}^{-1}\left(\int_{S}[R_{\mu}^{\textnormal{\hskip
0.56917pt\tiny du}}R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
du}}f+\dot{\sum_{i\in\mathcal{M}}}\sum_{j=1}^{{\mathpzc
k}(i)}u_{i,j}(f,\lambda)R_{\lambda,i}\ell_{\lambda}^{i,j}]\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j}\right)_{(i,j)\in\mathcal{IJ}},$

with the last equality by (5.3). Therefore, the resolvent equation and the
definition of the Laplace transform of an exit law show that
$(\lambda-\mu)u(R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}}f,\mu)$
equals

$\displaystyle M_{\mu}^{-1}\left(\int_{S}R_{\mu}^{\textnormal{\hskip
0.56917pt\tiny du}}f\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j}\right)_{(i,j)\in\mathcal{IJ}}$ $\displaystyle-
M_{\mu}^{-1}\left(\int_{S}[R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
du}}f+\dot{\sum_{i\in\mathcal{M}}}\sum_{j=1}^{{\mathpzc
k}(i)}u_{i,j}(f,\lambda)(\ell_{\lambda}^{i,j}-\ell_{\mu}^{i,j})]\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j}\right)_{(i,j)\in\mathcal{IJ}}$ $\displaystyle=$
$\displaystyle\,u(f,\mu)-M_{\mu}^{-1}\left(\int_{S}R_{\lambda}^{\textnormal{\hskip
0.56917pt\tiny co}}f\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j}-\sum_{i\in\mathcal{M}}\sum_{j=1}^{{\mathpzc
k}(i)}u_{i,j}(f,\lambda)\int_{S_{i}}\ell_{\mu}^{i,j}\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j}\right)_{(i,j)\in\mathcal{IJ}}.$

Since, as established at the end of Section 5.5,
$\int_{S}R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
co}}f\,\mathrm{d}\hskip 0.56917pt\mathfrak{p}_{i,j}$ is the same as
$u_{i,j}(f,\lambda)$, the second term here equals
$M_{\mu}^{-1}M_{\mu}u(f,\lambda)=u(f,\lambda)$. This completes the proof. ∎

######  Lemma 5.3.

For $\lambda>0$,

$\lambda u(1_{S_{\textnormal{\hskip 0.56917pt\tiny
u}}},\lambda)\leq\left(\mathfrak{p}_{i,j}(S_{\textnormal{\hskip 0.56917pt\tiny
u}})\right)_{(i,j)\in\mathcal{IJ}}.$

Moreover, the inequality can be replaced by equality provided that
$\mathfrak{p}_{i,j}(S_{\textnormal{\hskip 0.56917pt\tiny u}})=1$ for all
$(i,j)\in\mathcal{IJ}.$

###### Proof.

Let

$v=v(1_{S_{\textnormal{\hskip 0.56917pt\tiny
u}}},\lambda)=\left(\int_{S_{\textnormal{\hskip 0.56917pt\tiny
u}}}R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
du}}1_{S_{\textnormal{\hskip 0.56917pt\tiny u}}}\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j}\right)_{(i,j)\in\mathcal{IJ}},$

so that, by definition $u(1_{S_{\textnormal{\hskip 0.56917pt\tiny
u}}},\lambda)=(\sum_{n=0}^{\infty}N_{\lambda}^{n})v.$ We have, in other words,
$u(1_{S_{\textnormal{\hskip 0.56917pt\tiny
u}}},\lambda)=\lim_{n\to\infty}u^{n}$, where $u^{0}=v$ and
$u^{n+1}=v+N_{\lambda}u^{n},n\geq 1.$ It suffices, therefore, to show that

$\lambda u^{n}\leq\left(\mathfrak{p}_{i,j}(S_{\textnormal{\hskip
0.56917pt\tiny u}})\right)_{(i,j)\in\mathcal{IJ}},$ (5.13)

and we will do this by induction argument.

This condition holds for $n=0$, because $\lambda
R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny du}}1_{S_{\textnormal{\hskip
0.56917pt\tiny u}}}\leq 1_{S_{\textnormal{\hskip 0.56917pt\tiny u}}}.$
Moreover, writing

$u^{n}=\left(u_{i,j}^{n}\right)_{(i,j)\in\mathcal{IJ}},\qquad n\geq 0,$

and assuming that (5.13) holds for an $n$, we obtain

$\displaystyle\lambda u^{n+1}_{i,j}$
$\displaystyle=\int_{S_{\textnormal{\hskip 0.56917pt\tiny u}}}\lambda
R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny du}}1_{S_{\textnormal{\hskip
0.56917pt\tiny u}}}\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j}+\sum_{k\in\mathcal{M}}\sum_{l=1}^{{\mathpzc
k}(k)}u_{i,j}^{n}\int_{S_{k}}\ell_{\lambda}^{k,l}\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j}$ $\displaystyle\leq\int_{S_{\textnormal{\hskip
0.56917pt\tiny u}}}\lambda R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
du}}1_{S_{\textnormal{\hskip 0.56917pt\tiny u}}}\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j}+\sum_{k\in\mathcal{M}}\sum_{l=1}^{{\mathpzc
k}(k)}\int_{S_{k}}\ell_{\lambda}^{k,l}\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j},$ (5.14)

because, clearly, $u^{n}_{i,j}\leq\mathfrak{p}_{i,j}(S_{\textnormal{\hskip
0.56917pt\tiny u}})\leq 1.$ As proved in (5.10), the double sum here equals
$\int_{S_{\textnormal{\hskip 0.56917pt\tiny
u}}}\mathcal{L}_{\lambda}\,\mathrm{d}\hskip 0.56917pt\mathfrak{p}_{i,j}$.
Hence,

$\lambda u^{n+1}_{i,j}\leq\int_{S_{\textnormal{\hskip 0.56917pt\tiny
u}}}[\lambda R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
du}}1_{S_{\textnormal{\hskip 0.56917pt\tiny
u}}}+\mathcal{L}_{\lambda}]\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j}=\int_{S_{\textnormal{\hskip 0.56917pt\tiny
u}}}1_{S_{\textnormal{\hskip 0.56917pt\tiny u}}}\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j}=\mathfrak{p}_{i,j}(S_{\textnormal{\hskip
0.56917pt\tiny u}}),$

as desired.

We are thus left with proving the last assertion. To this end, we note that,
as long as all $\mathfrak{p}_{i,j}(S_{\textnormal{\hskip 0.56917pt\tiny u}})$
are equal to $1$, the vector
$u\coloneqq\left(\lambda^{-1}\right)_{(i,j)\in\mathcal{IJ}}$ satisfies

$u=v+N_{\lambda}u,$ (5.15)

for $v$ defined at the beginning of our proof. Indeed, the $(i,j)$th
coordinate of the right hand side is equal to the expression in the second
line of (5.14), divided by $\lambda.$ As proved below (5.14), this expression
further reduces to $\frac{1}{\lambda}\mathfrak{p}_{i,j}(S_{\textnormal{\hskip
0.56917pt\tiny u}})=\frac{1}{\lambda}$. But $u(1_{S_{\textnormal{\hskip
0.56917pt\tiny u}}},\lambda)$ is the unique solution to equation (5.15), and
so we must have $u(1_{S_{\textnormal{\hskip 0.56917pt\tiny u}}},\lambda)=u$.
This shows the claim. ∎

We are finally ready to prove our main theorem in this section, and a
fundamental result for all that follows.

######  Theorem 5.4.

The family $R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}},\lambda>0$
defined by (5.11) is a Feller resolvent on $C(S_{\textnormal{\hskip
0.56917pt\tiny u}})$. It is conservative if all $\mathfrak{p}_{i,j}$s are
probability measures.

###### Proof.

We start with the Hilbert equation for $R_{\lambda}^{\textnormal{\hskip
0.56917pt\tiny co}},\lambda>0$. Fix $\lambda,\mu>0$ and
$f\in\mbox{$C(S_{\textnormal{\hskip 0.56917pt\tiny u}})$}$. By definition
(5.11),

$\displaystyle R_{\mu}^{\textnormal{\hskip 0.56917pt\tiny
co}}R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}}f$
$\displaystyle=R_{\mu}^{\textnormal{\hskip 0.56917pt\tiny
du}}(R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
co}}f)+\sum_{i\in\mathcal{M}}\sum_{j=1}^{{\mathpzc
k}(i)}u_{i,j}(R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
co}}f,\mu)\ell_{\mu}^{i,j}$ $\displaystyle=R_{\mu}^{\textnormal{\hskip
0.56917pt\tiny du}}R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
du}}f+\sum_{i\in\mathcal{M}}\sum_{j=1}^{{\mathpzc
k}(i)}u_{i,j}(f,\lambda)R_{\mu}^{\textnormal{\hskip 0.56917pt\tiny
du}}\ell_{\mu}^{i,j}+\sum_{i\in\mathcal{M}}\sum_{j=1}^{{\mathpzc
k}(i)}u_{i,j}(R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
co}}f,\mu)\ell_{\mu}^{i,j}.$

Therefore, by the resolvent equation and the definition of an exit law,

$\displaystyle(\lambda-\mu)R_{\mu}^{\textnormal{\hskip 0.56917pt\tiny
co}}R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}}f$
$\displaystyle=R_{\mu}^{\textnormal{\hskip 0.56917pt\tiny
du}}f-R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
du}}f+\sum_{i\in\mathcal{M}}\sum_{j=1}^{{\mathpzc
k}(i)}u_{i,j}(f,\lambda)(\ell_{\mu}^{i,j}-\ell_{\lambda}^{i,j})$
$\displaystyle\phantom{=}+(\lambda-\mu)\sum_{i\in\mathcal{M}}\sum_{j=1}^{{\mathpzc
k}(i)}u_{i,j}(R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
co}}f,\mu)\ell_{\mu}^{i,j}.$

This differs from $R_{\mu}^{\textnormal{\hskip 0.56917pt\tiny
co}}f-R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}}f$ by the sum

$\sum_{i\in\mathcal{M}}\sum_{j=1}^{{\mathpzc
k}(i)}[u_{i,j}(f,\lambda)-u_{i,j}(f,\mu)+(\lambda-\mu)u_{i,j}(R_{\lambda}^{\textnormal{\hskip
0.56917pt\tiny co}}f,\mu)]\ell_{\mu}^{i,j}.$

However, all summands here are zero, by Lemma 5.2, and so we obtain
$(\lambda-\mu)R_{\mu}^{\textnormal{\hskip 0.56917pt\tiny
co}}R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
co}}f=R_{\mu}^{\textnormal{\hskip 0.56917pt\tiny
co}}f-R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}}f.$ Since
$\lambda,\mu$ and $f$ are arbitrary, this shows that
$R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}},\lambda>0$ is a
pseudoresolvent.

Next, by Lemma 5.1, $u\geq 0$ whenever $f\geq 0$. It follows that the
operators $R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}}f,\lambda>0$ are
non-negative. Moreover, by Lemma 5.3,

$R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}}1_{S_{\textnormal{\hskip
0.56917pt\tiny u}}}=R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
du}}1_{S_{\textnormal{\hskip 0.56917pt\tiny
u}}}+\dot{\sum_{i\in\mathcal{M}}}\sum_{j=1}^{{\mathpzc
k}(i)}u_{i,j}(1_{S_{\textnormal{\hskip 0.56917pt\tiny
u}}},\lambda)\ell_{\lambda}^{i,j}\leq R_{\lambda}^{\textnormal{\hskip
0.56917pt\tiny du}}1_{S_{\textnormal{\hskip 0.56917pt\tiny
u}}}+\dot{\sum_{i\in\mathcal{M}}}\sum_{j=1}^{{\mathpzc
k}(i)}\ell_{\lambda}^{i,j}$

and the right hand side coincides, by (5.2) and (5.4), with
$R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny du}}1_{S_{\textnormal{\hskip
0.56917pt\tiny u}}}+\mathcal{L}_{\lambda}=1_{S_{\textnormal{\hskip
0.56917pt\tiny u}}}.$ Hence,

$\lambda R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
co}}1_{S_{\textnormal{\hskip 0.56917pt\tiny u}}}\leq 1_{S_{\textnormal{\hskip
0.56917pt\tiny u}}},$ (5.16)

and if all $\mathfrak{p}_{i,j}$s are probability measures, the inequality
above can be replaced by equality — this is a consequence of the other part of
Lemma 5.3.

We are thus left with showing that $\lim_{\lambda\to\infty}\lambda
R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}}f=f$. Since
$R_{\lambda,i},\lambda>0$ are regular, we have $\lim_{\lambda\to\infty}\lambda
R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny du}}f=f$ and thus it suffices
to show that

$\lim_{\lambda\to\infty}\dot{\sum_{i\in\mathcal{M}}}\sum_{j=1}^{{\mathpzc
k}(i)}\left(\int_{S}\lambda R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
co}}f\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,j}\right)\ell_{\lambda}^{i,j}=0$ (5.17)

However, inequality (5.16) combined with non-negativity of the operators
$R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}},\lambda>0$ shows that

$\|\lambda R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}}\|\leq
1,\qquad\lambda>0.$

Therefore, the norm of the sum in (5.17) does not exceed

$\|f\|\sum_{i\in\mathcal{M}}\sum_{j=1}^{{\mathpzc
k}(i)}\|\ell_{\lambda}^{i,j}\|.$

This converges to zero, as $\lambda\to\infty$, because, by Corollary 3.3,
$\lim_{\lambda\to\infty}\|\ell_{\lambda}^{i,j}\|=0$ for all
$(i,j)\in\mathcal{IJ}$. ∎

###  5.7. In (5.6) Laplace transforms of exit laws need to be used

As we have see above, the fact that $\ell^{i,j}_{\lambda},\lambda>0$ is the
Laplace transform of an exit law was crucial in proving the resolvent equation
for $R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}},\lambda>0$. In fact,
the defining equation (3.1) was used both in the proof of Theorem 5.4 and in
the proof of Lemma 5.2, preceding the theorem. The following result may be
thought of as a converse to Theorem 5.4; we prove namely that formula (5.6)
will not work, unless $\ell^{i,j}_{\lambda}$s are Laplace transforms of exit
laws.

Suppose thus that, as in Section 5.1, we have $N$ Feller processes, of which
$M$ first are not conservative, and that
$\ell_{\lambda}^{i,j}\in\mbox{$C(S_{i})$},(i,j)\in\mathcal{IJ}$ are given
functions such that $0\leq\ell_{\lambda}^{i,j}\leq 1_{S_{i}}$ and (5.6)
defines a resolvent in $C(S_{\textnormal{\hskip 0.56917pt\tiny u}})$ for all
measures $\mathfrak{p}_{i,j}$. We claim that then, for each
$(i,j)\in\mathcal{IJ},$

$(\lambda-\mu)R_{\mu,i}\ell_{\lambda}^{i,j}=\ell_{\mu}^{i,j}-\ell_{\lambda}^{i,j},\qquad\lambda,\mu>0.$
(5.18)

To prove this, we fix $(i,j)\in\mathcal{IJ}$, choose a $k\not=i$, a point
$x\in S_{k}$, and take all measures involved to be zero, except for
$\mathfrak{p}_{i,j}$ which we define to be the Dirac measure at $x$. Then
(5.6) takes the form

$R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
co}}f=R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
du}}f+[R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
co}}f(x)]\ell_{\lambda}^{i,j}.$

Since $\ell^{i,j}_{\lambda}$ is supported in $S_{i}$ and $i\not=k$, it follows
that $R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
co}}f(x)=R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny du}}f(x)$, and the
formula simplifies to

$R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
co}}f=R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
du}}f+[R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
du}}f(x)]\ell_{\lambda}^{i,j}.$

(The resulting resolvent is related to the following concatenation of
processes: a particle that moves about initially in $S_{i}$ and eventually
exits this region by the $j$th gate, starts all over again at the $x\in S_{k}$
and moves according to the law known in $S_{k}$, until the process is possibly
no longer defined. However, if the initial position is not in $S_{i}$ or the
process does not exit through the $j$th gate, after its lifetime is over the
process is not continued at all.)

A short calculation yields

$(\lambda-\mu)R_{\mu}^{\textnormal{\hskip 0.56917pt\tiny
co}}R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
co}}f=(\lambda-\mu)\bigl{[}R_{\mu}R_{\lambda}^{\textnormal{\hskip
0.56917pt\tiny du}}f+[R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
du}}f(x)]R_{\mu,i}\ell_{\lambda}^{i,j}+[R_{\mu}^{\textnormal{\hskip
0.56917pt\tiny du}}R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
co}}f(x)]\ell_{\lambda}^{i,j}\bigr{]}.$ (5.19)

Furthermore, since $\ell_{\lambda}^{i,j}$ is supported in $S_{i}$, so is
$R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny du}}\ell_{\lambda}^{i,j}$, and
hence the relation

$R_{\mu}^{\textnormal{\hskip 0.56917pt\tiny
du}}R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
co}}f=R_{\mu}^{\textnormal{\hskip 0.56917pt\tiny
du}}R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
du}}f+[R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
du}}f(x)]R_{\mu}^{\textnormal{\hskip 0.56917pt\tiny du}}\ell_{\lambda}^{i,j}$

proves that $R_{\mu}^{\textnormal{\hskip 0.56917pt\tiny
du}}R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
co}}f(x)=R_{\mu}^{\textnormal{\hskip 0.56917pt\tiny
du}}R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny du}}f(x)$ (because $x\in
S_{k}$ and $S_{k}$ is disjoint with $S_{i}$). Thus, the right-hand side of
(5.19) differs from $R_{\mu}^{\textnormal{\hskip 0.56917pt\tiny
co}}f-R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}}f$ by

$[R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
du}}f(x)]\left((\lambda-\mu)R_{\mu,i}\ell_{\lambda}^{i,j}+\ell_{\lambda}^{i,j}-\ell_{\mu}^{i,j}\right),$

$R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny du}},\lambda>0$ being a
pseudoresolvent.

Since, $R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}},\lambda>0$ is a
pseudoresolvent also, to complete the proof it suffices to find an
$f\in\mbox{$C(S_{i})$}$ such that $R_{\mu}^{\textnormal{\hskip 0.56917pt\tiny
du}}f(x)$ is non-zero for all $\mu>0$. To this end, let $f$ be non-negative
and such that $f(x)>0$. Since the paths of the process $X_{k}$ with generator
$A_{k}$ are right-continuous on a set of probability $1$, if we restrict
ourselves to the paths starting at $x$, we will have $f(X_{k}(t))>0$ for
sufficiently small $t\in[0,\infty)$, and for other $t$ we will have
$f(X_{k}(t))\geq 0$. It follows that $\int_{0}^{\infty}\mathrm{e}^{-\lambda
t}f(X_{k}(t))\,\mathrm{d}\hskip 0.56917ptt>0$ for all $\lambda>0.$ Therefore,
by the Fubini Theorem,

$R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
du}}f(x)=E_{x}\int_{0}^{\infty}\mathrm{e}^{-\lambda
t}f(X_{k}(t))\,\mathrm{d}\hskip 0.56917ptt>0,\qquad\lambda>0,$

as desired.

##  6\. Examples of regular Feller boundary

In this section we discuss three examples of processes with regular Feller
exit boundary composed of finite number of gates (identified with excessive
functions), as defined in Definition 4.1. These processes are Brownian motions
on $[0,\infty]$, $[0,r]$ where $r>0$, and star-like graphs, respectively, with
rather general boundary conditions describing the gates. We will be able to
provide explicit formulae for exit laws and related excessive functions that
feature in these examples: in the first of them, there is one gate, in the
second — two gates, and in the third — ${\mathpzc k}\geq 2$ gates. Moreover,
this section exemplifies that fact that Theorem 5.4 together with the result
of Section 5.7 can be instrumental in finding formulae for exit laws and
excessive functions.

###  6.1. Example A: $S=[0,\infty]$; one exit law.

Consider the space $C[0,\infty]$ of continuous functions on
$\mathbb{R}^{+}=[0,\infty)$ with limits at infinity. Let $A$ be the operator
$Af=\frac{1}{2}f^{\prime\prime}$ with domain composed of twice continuously
differentiable functions with $f^{\prime\prime}\in C[0,\infty]$, satisfying
the boundary condition

$af^{\prime\prime}(0)-bf^{\prime}(0)+cf(0)=0,$ (6.1)

where $a,b$ and $c$ are given non-negative constants with $a+b+c>0.$

It is well-known that, as long as $a+b>0$, $A$ generates a Feller semigroup on
$C[0,\infty]$, and that the resolvent
$R_{\lambda}=\left(\lambda-A\right)^{-1},\lambda>0$ of $A$ is given by (see
e.g. [knigaz]*pp. 17-18)

$R_{\lambda}f(x)=C\mathrm{e}^{\sqrt{2\lambda}x}+D\mathrm{e}^{-\sqrt{2\lambda}x}-\sqrt{\frac{2}{\lambda}}\int_{0}^{x}\sinh\sqrt{2\lambda}(x-y)f(y)\,\mathrm{d}\hskip
0.56917pty,\qquad x\geq 0,$

for all $f\in C[0,\infty]$, where $C=C(\lambda,f)$ and $D=D(\lambda,f)$ are
constants defined as follows:

$\displaystyle
C=\frac{1}{\sqrt{2\lambda}}\int_{0}^{\infty}\mathrm{e}^{-\sqrt{2\lambda}y}f(y)\,\mathrm{d}\hskip
0.56917pty,\quad\text{ }\quad
D=\frac{(b\sqrt{2\lambda}-2a\lambda-c)C+2af(0)}{2a\lambda+b\sqrt{2\lambda}+c}.$

For $f=1_{S}$, the constants are $C=\frac{1}{2\lambda}$ and
$D=\frac{1}{2\lambda}\frac{2a\lambda+b\sqrt{2\lambda}-c}{2a\lambda+b\sqrt{2\lambda}+c}$.
This renders

$\lambda
R_{\lambda}1_{S}=1_{S}-\frac{c}{2a\lambda+b\sqrt{2\lambda}+c}e_{\sqrt{2\lambda}},$

where $e_{\sqrt{2\lambda}}(x)=\mathrm{e}^{-\sqrt{2\lambda}x},x\geq 0.$

Therefore, as long as $c>0$, we have $\lambda R_{\lambda}1_{S}<1_{S}$, that
is, the resolvent is not conservative. In this case, the Laplace transform of
the lifetime of the related process is

$\mathcal{L}_{\lambda}=1_{S}-\lambda
R_{\lambda}1_{S}=\frac{c}{2a\lambda+b\sqrt{2\lambda}+c}e_{\sqrt{2\lambda}}.$
(6.2)

Moreover, as it is easy to see, the kernel of $A$ is trivial, and thus we have
the first example of a regular Feller boundary with (at least) one exit. An
intuitive description of the related process tells us furthermore that this
process can exit the state-space only through $0$, and therefore, there is
precisely one exit here.

We note also that in this example, the excessive function for the exit law is
$1_{[0,\infty]}$: this corresponds to the fact that the process considered
here with probability $1$ eventually hits $0$, regardless of its starting
point.

Two sub-cases are of interest here. First of all, if $b=0$, the origin is a
so-called _elementary exit_ (comp. [fellera1]*p. 3 or [knigaz]*p. 19). The
related process is a standard Brownian motion on the right half axis which,
however, after reaching the origin is captured there for an exponential time
with parameter $\frac{c}{2a};$ after this time is over, the process is no
longer defined.

Formula (6.2) confirms this description: $\frac{c}{2a\lambda+c}$ is the
Laplace transform of the exponential time with parameter $\frac{c}{2a}$,
whereas $e_{\sqrt{2\lambda}}(x)$ is the Laplace transform of the time needed
for the standard Brownian motion starting at $x>0$ to reach the origin — see
[ito]* Eq. 5) p. 26, [karatzas]*Eq. (8.6) p. 96 or the original [levybook]*pp.
221-223.

Secondly, in the case of $a=0$, the boundary condition (6.1) describes the
_elastic Brownian motion_. This is a process in the right half axis which
initially behaves as if it were a reflected Brownian motion. However, the time
it spends at the origin is measured by means of the local time of P. Lévy;
when an exponential time with parameter $\frac{c}{b}$ passes with respect to
this local time, the process is left undefined. This description is again
confirmed by (6.2), because $\frac{c}{b\sqrt{2\lambda}+c}$ is the Laplace
transform of the distribution of exponential time with parameter $\frac{c}{b}$
with respect to the Lévy local time — see [ito]*p. 45, Eq. 1, that is, of the
time elastic Brownian motion needs to exit the space after reaching the origin
for the first time.

If $a=b=0\not=c$, we encounter the _minimal_ Brownian motion — here the
Laplace transform of the time needed for the process starting at $x>0$ to exit
the space is $\mathrm{e}^{-\sqrt{2\lambda}x}$, that is, the Laplace transform
of the time needed to reach the origin for the first time. Strictly speaking,
though, $0$ is not in the state-space of the minimal Brownian motion, and the
related semigroup is defined in the subspace $C_{0}(0,\infty]$ of
$C[0,\infty]$ of $f$ from the latter space which satisfy $f(0)=0$.

For $a>0$, the origin is more ‘sticky’ than in the case of $a=0$, and so the
time process spends at the boundary is longer; in the case of $c=0$ the
process is known as _sticky_ or _slowly reflecting_ Brownian motion — see
[kostrykin2010brownian], [liggett]*p. 127 or [revuz]*p. 421. For non-zero $c$,
$\frac{c}{2a\lambda+b\sqrt{2\lambda}+c}$ is the Laplace transform of the time
needed for the related process to exit the space after reaching the boundary
for the first time.

###  6.2. Example B: $S=[0,r]$; two exits (with exceptions)

Let $S=[0,r]$ be a closed interval of length $r>0$, and let $A$ be the
operator in $C(S)$ given by $Af=\frac{1}{2}f^{\prime\prime}$ on the domain
composed of $f\in\mbox{$C(S)$}$ that are twice continuously differentiable and
such that

$pf^{\prime\prime}(0)-(1-p)f^{\prime}(0)+cf(0)=0\quad\text{and}\quad
qf^{\prime\prime}(r)+(1-q)f^{\prime}(r)+df(r)=0,$ (6.3)

where $p,q\in[0,1]$ and $c,d\geq 0.$ To explain, the condition on the left is
a new incarnation of (6.1): take $p=\frac{a}{a+b}$ and redefine $c$ as
$\frac{c}{a+b}$. The condition on the right is of similar nature.

From this description it is clear that, as long as $c$ and $d$ are larger than
$0$, there are two exits here. We will find the corresponding exit laws and
excessive functions with the help of Theorem 5.4 and the result of Section
5.7. Since these excessive functions turn out to be continuous and add up to
$1_{[0,r]}$, the process considered here has a Feller exit boundary.

Before continuing, we note that a standard argument (see e.g. [knigaz]*p. 17)
can be used to show that $A$ satisfies the positive maximum principle and is
densely defined. The calculations presented below prove also that it satisfies
the range condition, and thus is a generator of a Feller semigroup. The same
remark applies to $A^{\textnormal{\hskip 0.56917pt\tiny co}}$ introduced
below.

####  6.2.1. Auxiliary concatenation

To find exit laws for $A$, we think of $S$ as of $S_{1}$ in Theorem 5.4 and
add an auxiliary space $S_{2}$ composed of two points:
$S_{2}\coloneqq\\{o_{1},o_{2}\\}$ ($o_{1},o_{2}\not\in[0,r]$) and equip
$S_{2}$ with discrete topology. An auxiliary process in $S_{2}$ is trivial,
that is, both points are absorbing: the process starting at one of these
points stays there for ever. In other words, its generator is $A_{2}=0$. We
concatenate these two processes by requiring that (a) when the process related
to $A_{1}=A$ exists $[0,r]$ through the boundary point $0$, it jumps to
$o_{1}$ to stay there for ever, and (b) when the process exists through $r$,
it jumps to $o_{2}$ to stay there for every.

The generator $A^{\textnormal{\hskip 0.56917pt\tiny co}}$ of the concatenated
process in $S_{\textnormal{\hskip 0.56917pt\tiny
u}}=[0,r]\cup\\{o_{1},o_{2}\\}$ is defined as follows. Its domain is composed
of $f$ which, as restricted to $[0,r]$, are twice continuously differentiable,
and satisfy the following transmission conditions:

$\displaystyle pf^{\prime\prime}(0)-(1-p)f^{\prime}(0)+cf(0)$
$\displaystyle=cf(o_{1}),$ $\displaystyle
qf^{\prime\prime}(r)+(1-q)f^{\prime}(r)+df(r)$ $\displaystyle=df(o_{2}).$
(6.4)

Also, $A^{\textnormal{\hskip 0.56917pt\tiny
co}}f(x)=\frac{1}{2}f^{\prime\prime}(x),x\in[0,r],$ whereas
$A^{\textnormal{\hskip 0.56917pt\tiny co}}f(o_{1})=A^{\textnormal{\hskip
0.56917pt\tiny co}}f(o_{2})=0$.

Next, we solve the resolvent equation for $A^{\textnormal{\hskip
0.56917pt\tiny co}}$: given $g\in C(S_{1}\dot{\cup}S_{2})$ and $\lambda>0$ we
find an $f\in D(A^{\textnormal{\hskip 0.56917pt\tiny co}})$ such that $\lambda
f-A^{\textnormal{\hskip 0.56917pt\tiny co}}f=g.$ The part of this equation in
$C(S_{2})$ is very simple and we immediately realize that $\lambda
f(o_{i})=g(o_{i}),i=1,2.$ To solve the other part we search for $f$ of the
form

$f(x)=C\mathrm{e}^{\sqrt{2\lambda}x}+D\mathrm{e}^{-\sqrt{2\lambda}x}+h_{\lambda}(x),\qquad
x\in[0,r]$ (6.5)

where
$h_{\lambda}(x)\coloneqq\frac{1}{\sqrt{2\lambda}}\int_{0}^{r}\mathrm{e}^{-\sqrt{2\lambda}|x-y|}g(y)\,\mathrm{d}\hskip
0.56917pty$, such that transmission conditions (6.4) are satisfied. In fact,
for reasons that will become clear in a moment, we will find a unique $f$
satisfying slightly more general conditions:

$\displaystyle pf^{\prime\prime}(0)-(1-p)f^{\prime}(0)+cf(0)$
$\displaystyle=\widetilde{c}f(o_{1}),$ $\displaystyle
qf^{\prime\prime}(r)+(1-q)f^{\prime}(r)+df(r)$
$\displaystyle=\widetilde{d}f(o_{2}),$ (6.6)

where $0\leq\widetilde{c}\leq c$ and $0\leq\widetilde{d}\leq d.$

####  6.2.2. A linear system for $C$ and $D$

We note that (6.6) is fulfilled iff $C$ and $D$ of (6.5) are solutions to the
system of linear equations:

$\displaystyle a_{1,1}C+a_{1,2}D$ $\displaystyle=b_{1}+\widetilde{c}f(o_{1}),$
$\displaystyle a_{2,1}C+a_{2,2}D$ $\displaystyle=b_{2}+\widetilde{d}f(o_{2}),$

where

$\displaystyle a_{1,i}$ $\displaystyle\coloneqq 2\lambda
p+(-1)^{i}\sqrt{2\lambda}(1-p)+c,$ (6.7) $\displaystyle a_{2,i}$
$\displaystyle\coloneqq\mathrm{e}^{(-1)^{i+1}\sqrt{2\lambda}r}(2\lambda
q-(-1)^{i}\sqrt{2\lambda}(1-q)+d),\qquad i=1,2,$

and (since $h_{\lambda}^{\prime}(0)=\sqrt{2\lambda}h_{\lambda}(0)$ and
$h_{\lambda}^{\prime}(r)=-\sqrt{2\lambda}h_{\lambda}(r)$)

$\displaystyle b_{1}$ $\displaystyle\coloneqq 2pg(0)-(2\lambda
p-\sqrt{2\lambda}(1-p)+c)h_{\lambda}(0),$ $\displaystyle b_{2}$
$\displaystyle\coloneqq 2qg(r)-(2\lambda
q-\sqrt{2\lambda}(1-q)+d)h_{\lambda}(r).$ (6.8)

It is clear that $|a_{1,1}|\leq a_{1,2}$ (with equality for $p=1$),
$|a_{2,2}|<a_{2,1}$ and that neither of $a_{1,2}$ and $a_{2,1}$ is zero.
Hence, the main determinant $\operatorname{Det}$ of the system is negative,
and the system has a unique solution.

####  6.2.3. Two exit laws

It follows that

$\displaystyle R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}}g(x)$
$\displaystyle=\frac{1}{\operatorname{Det}}\begin{vmatrix}b_{1}&a_{1,2}\\\
b_{2}&a_{2,2}\end{vmatrix}\mathrm{e}^{\sqrt{2\lambda}x}+\frac{1}{\operatorname{Det}}\begin{vmatrix}a_{1,1}&b_{1}\\\
a_{2,1}&b_{2}\end{vmatrix}\mathrm{e}^{-\sqrt{2\lambda}x}+h_{\lambda}(x)$
$\displaystyle\phantom{=}+\frac{1}{\operatorname{Det}}\begin{vmatrix}\widetilde{c}f(o_{1})&a_{1,2}\\\
\widetilde{d}f(o_{2})&a_{2,2}\end{vmatrix}\mathrm{e}^{\sqrt{2\lambda}x}+\frac{1}{\operatorname{Det}}\begin{vmatrix}a_{1,1}&\widetilde{c}f(o_{1})\\\
a_{2,1}&\widetilde{d}f(o_{2})\end{vmatrix}\mathrm{e}^{-\sqrt{2\lambda}x}.$
(6.9)

The first three terms here correspond to the case of
$\widetilde{c}=\widetilde{d}=0$, that is, provide the formula for the
resolvent of $A$. The remaining two may be written as

$\frac{\widetilde{c}}{\operatorname{Det}}(a_{2,2}\mathrm{e}^{\sqrt{2\lambda}x}-a_{2,1}\mathrm{e}^{-\sqrt{2\lambda}x})f(o_{1})+\frac{\widetilde{d}}{\operatorname{Det}}(a_{1,1}\mathrm{e}^{-\sqrt{2\lambda}x}-a_{1,2}\mathrm{e}^{\sqrt{2\lambda}x})f(o_{2}).$

Since

$f(o_{i})=\frac{g(o_{i})}{\lambda}=R_{\lambda,i}g(o_{i})=R_{\lambda}^{\textnormal{\hskip
0.56917pt\tiny co}}g(o_{i}),\qquad i=1,2,$

formula (6.9) is a particular case of (5.6) with $N=2,M=1,{\mathpzc
k}(1)=2,\mathfrak{p}_{1,1}=\frac{\widetilde{c}}{c}\delta_{o_{1}}$ and
$\mathfrak{p}_{1,2}=\frac{\widetilde{d}}{d}\delta_{o_{2}}$:

$R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
co}}g=R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
du}}g+{\textstyle\frac{\widetilde{c}}{c}}[R_{\lambda}^{\textnormal{\hskip
0.56917pt\tiny
co}}g(o_{1})]\ell_{\lambda}^{1}+{\textstyle\frac{\widetilde{d}}{d}}[R_{\lambda}^{\textnormal{\hskip
0.56917pt\tiny co}}g(o_{2})]\ell_{\lambda}^{2}$

where

$\displaystyle\ell_{\lambda}^{1}(x)$
$\displaystyle\coloneqq\frac{c}{\operatorname{Det}}(a_{2,2}\mathrm{e}^{\sqrt{2\lambda}x}-a_{2,1}\mathrm{e}^{-\sqrt{2\lambda}x}),$
$\displaystyle\ell_{\lambda}^{2}(x)$
$\displaystyle\coloneqq\frac{d}{\operatorname{Det}}(a_{1,1}\mathrm{e}^{-\sqrt{2\lambda}x}-a_{1,2}\mathrm{e}^{\sqrt{2\lambda}x}),\qquad
x\in[0,r].$ (6.10)

Above, we write $\ell_{\lambda}^{1}$ and $\ell_{\lambda}^{2}$ instead of
$\ell_{\lambda}^{1,1}$ and $\ell_{\lambda}^{1,2}$ for simplicity and, at the
same time, to comply with notation of Section 4.

The argument of Section 5.7 is in force here and we conclude that
$\ell_{\lambda}^{j},j=1,2$ are exit laws for $A$. We also note that the
resolvent of $A$ can be expressed in terms of these laws as follows:

$\left(\lambda-A\right)^{-1}g=\frac{b_{1}}{c}\ell_{\lambda}^{1}+\frac{b_{2}}{d}\ell_{\lambda}^{2}+h_{\lambda}.$
(6.11)

####  6.2.4. Two excessive functions

To find the excessive functions corresponding to exit laws (6.10), we let

$\phi^{j}(x)\coloneqq\lim_{\lambda\to 0+}\ell_{\lambda}^{j}(x),\qquad j=1,2,$

and write

$\ell_{\lambda}^{1}=c\frac{\operatorname{Det}_{1}}{\operatorname{Det}}\quad\text{
and
}\quad\ell_{\lambda}^{2}=d\frac{\operatorname{Det}_{2}}{\operatorname{Det}}$
(6.12)

where

$\operatorname{Det}_{1}\coloneqq\begin{vmatrix}\mathrm{e}^{\sqrt{2\lambda}x}&\mathrm{e}^{-\sqrt{2\lambda}x}\\\
a_{2,1}&a_{2,2}\end{vmatrix}\quad\text{ and
}\quad\operatorname{Det}_{2}\coloneqq\begin{vmatrix}a_{1,1}&a_{1,2}\\\\[4.0pt]
\mathrm{e}^{\sqrt{2\lambda}x}&\mathrm{e}^{-\sqrt{2\lambda}x}\end{vmatrix}.$
(6.13)

Then formula (10.1) in Appendix shows that

$\displaystyle\phi^{1}(x)$
$\displaystyle=c\frac{d(r-x)+1-q}{cdr+(1-q)c+(1-p)d},$
$\displaystyle\phi^{2}(x)$
$\displaystyle=d\frac{cx+1-p}{cdr+(1-q)c+(1-p)d},\qquad x\in[0,r].$ (6.14)

For $p=q=1$ (this is the case of elementary exits from both interval’s
endpoints) these quantities are familiar probabilities that a Brownian motion
starting at $x\in[0,r]$ hits $0$ before hitting $r$, and vice versa,
respectively. In view of the well-known relationship between excessive
functions and exit probabilities (see [doobc]*p. 565), in general these are
apparently the probabilities of exiting through $0$ and $r$, conditional on
the process generated by $A$ starting at $x$. Notably,

$\phi^{1}+\phi^{2}=1_{[0,r]},$

and both $\phi^{j},j=1,2$ are continuous functions: with probability one the
process eventually exits either through the left-end or the right-end of the
interval. It is also quite easy to see that the kernel of $A$ is trivial.
Thus, as heralded before, the process considered here has Feller exit boundary
with (at least) two exits — it is the intuitive description of the related
process that tells us that there are precisely two exits here.

####  6.2.5. A particular case

For $p=q=1$ (two elementary exits) formulae for excessive functions simplify
as follows:

$\displaystyle\ell_{\lambda}^{1}(x)$
$\displaystyle=\frac{r-x}{r}\left(\frac{r}{r-x}\frac{\sinh\sqrt{2\lambda}(r-x)}{\sinh\sqrt{2\lambda}r}\right)\frac{c}{2\lambda+c},\qquad
x\in[0,r),$ $\displaystyle\ell_{\lambda}^{2}(x)$
$\displaystyle=\frac{x}{r}\left(\frac{r}{x}\frac{\sinh\sqrt{2\lambda}x}{\sinh\sqrt{2\lambda}r}\right)\frac{d}{2\lambda+d},\qquad
x\in(0,r],$

and these relations have a clear interpretation. To wit, the first factor in
the formula for $\ell_{\lambda}^{1}(x)$ is the probability that a standard
Brownian starting at $x\in[0,r]$ will reach $0$ before reaching $r$. The
second is the Laplace transform of the time needed for the standard Brownian
motion to reach $0$, conditional on reaching it before reaching $r$ — see
[karatzas]*p. 100. The third is the Laplace transform of the exponential time
the process generated by $A$ spends at $0$ before exiting the state-space. The
formula for $\ell_{\lambda}^{2}$ is interpreted similarly.

####  6.2.6. Another particular case

Another case of interest is $p=1$ and $q=0$ (elementary exit on the left,
elastic boundary at the right). Here,

$\displaystyle\ell_{\lambda}^{1}(x)$
$\displaystyle=\frac{\sqrt{2\lambda}\cosh\sqrt{2\lambda}(r-x)}{d\sinh\sqrt{2\lambda}r+\sqrt{2\lambda}\cosh\sqrt{2\lambda}r}\frac{c}{2\lambda+c},$
$\displaystyle\ell_{\lambda}^{2}(x)$
$\displaystyle=\frac{\sinh\sqrt{2\lambda}x}{d\sinh\sqrt{2\lambda}r+\sqrt{2\lambda}\cosh\sqrt{2\lambda}r},\qquad
x\in[0,r].$

It is interesting to note how the change in the nature of the boundary at $r$
influences the exit at the left. In the new scenario, a particle reaching $r$
for the first time is not captured there for an exponential time as in Section
6.2.5, but is reflected and can thus exit through $0$. Hence, the probability
of exiting through $0$, equaling $\phi^{1}(x)=\frac{1+d(r-x)}{1+dr}$ is larger
than that obtained in Section 6.2.5.

####  6.2.7. The ‘degenerate’ case: one exit law

If either $c=0$ or $d=0$ (but not both), the related process has one exit law.
For instance, we consider the case of elastic boundary at $0$ and reflecting
boundary at $r$, that is, the case where $p=q=d=0$ whereas $c\not=0.$ Then,
$\ell_{\lambda}^{2}(x)$ reduces to zero, reflecting the fact that the process
cannot exit $[0,r]$ through the interval’s right end. Moreover,

$\ell_{\lambda}^{1}(x)=\frac{\cosh\sqrt{2\lambda}(x-r)}{\cosh\sqrt{2\lambda}r}\frac{c\cosh\sqrt{2\lambda}r}{c\cosh\sqrt{2\lambda}r+\sqrt{2\lambda}\sinh\sqrt{2\lambda}r},\qquad
x\in[0,r].$

The first factor here is the Laplace transform of the distribution of the time
needed for the Brownian motion starting at $x\in(0,r)$ and reflected at $x=r$
to reach $x=0$ for the first time (see Appendix 10.1). Also, the second factor
is completely monotone (see Appendix 10.2) and its value for $\lambda=0$ is
$1$, proving that it is the Laplace transform of a Borel probability measure
on $[0,\infty)$. The second factor thus should be interpreted as the Laplace
transform of the time needed to exit $[0,r]$ after reaching $x=0$ for the
first time.

Since $\lim_{\lambda\to 0+}\ell_{\lambda}^{1}(x)=1$, this exit law corresponds
to the excessive function $1_{[0,r]}$, that is, $\ell_{\lambda}^{1}$ coincides
with $\mathcal{L}_{\lambda}.$

####  6.2.8. A black sheep: representation in terms of excessive functions is
not unique

The case of $p=q=1$, $d=0$ and $c>0$ is somewhat exceptional (together with
the analogous case of $p=q=1$, $c=0$ and $d>0$). Here, we have again only one
exit, but representation in terms of excessive functions is not unique.

First of all, as in the previous section, in this scenario
$\ell_{\lambda}^{2}(x)$ reduces to zero. Moreover,

$\ell_{\lambda}^{1}(x)=\frac{c\sinh\sqrt{2\lambda}(r-x)}{(2\lambda+c)\sinh\sqrt{2\lambda}r},\qquad
x\in[0,r].$

Since $\lim_{\lambda\to 0+}\ell_{\lambda}^{1}(x)=1$ for all $x\in[0,r]$, we
see that, again, $\ell_{\lambda}^{1}=\mathcal{L}_{\lambda}.$ However, for any
$\alpha\in\mathbb{R}$, the function $f(x)=\alpha x,x\in[0,r]$ belongs to the
kernel of $A$. As a result, for any $\alpha\in[0,\frac{1}{r}]$,

$\ell_{\lambda}^{1}=\phi^{\alpha}-\lambda R_{\lambda}\phi^{\alpha}$

where

$0\leq\phi^{\alpha}(x)\coloneqq 1-\alpha x\leq 1,\qquad x\in[0,r].$

####  6.2.9. A remarkable property of exit laws

Coming back to the general case and $c,d>0$ we note the following remarkable
properties of exit laws. To begin with, we note that (derivatives with respect
to $x$)

$(\ell_{\lambda}^{j})^{\prime\prime}=2\lambda\ell_{\lambda}^{j},\qquad j=1,2.$
(6.15)

Next, we calculate:

$\displaystyle p(\ell_{\lambda}^{1})^{\prime\prime}(0)$
$\displaystyle-(1-p)(\ell_{\lambda}^{1})^{\prime}(0)+c\ell_{\lambda}^{1}(0)$
$\displaystyle=\frac{c}{\operatorname{Det}}((2\lambda
p+c)(a_{2,2}-a_{2,1})-(1-p)\sqrt{2\lambda}(a_{2,2}-a_{2,1}))$
$\displaystyle=\frac{c}{\operatorname{Det}}(a_{1,1}a_{2,2}-a_{2,1}a_{1,2})=c.$

At the same time

$\displaystyle p(\ell_{\lambda}^{2})^{\prime\prime}(0)$
$\displaystyle-(1-p)(\ell_{\lambda}^{2})^{\prime}(0)+c\ell_{\lambda}^{1}(0)=\frac{c}{\operatorname{Det}}(a_{1,1}a_{1,2}-a_{1,2}a_{1,1})=0.$

Introducing $F_{1}:C^{2}[0,r]\to\mathbb{R}$ given by $F_{1}f\coloneqq
pf^{\prime\prime}(0)-(1-p)f^{\prime}(0)+cf(0)$ we summarize the ‘algebraic’
nature of exits laws at the interval’s left end as follows:

$F_{1}\ell_{\lambda}^{j}=\delta_{1,j}c,\qquad j=1,2,$ (6.16)

where $\delta_{1,j}$ is the Kronecker delta. There is also a counterpart of
these relations at the right end:

$\displaystyle F_{2}\ell_{\lambda}^{j}=\delta_{2,j}d,\qquad j=1,2$ (6.17)

where $F_{2}f\coloneqq qf^{\prime\prime}(r)+(1-q)f^{\prime}(r)+df(r).$ It is
worth noting that, (6.15)–(6.17) in fact determine $\ell_{\lambda}^{1}$ and
$\ell_{\lambda}^{2}$ uniquely.

We also remark that quantities $b_{1}$ and $b_{2}$ of (6.8) are equal to
$-F_{1}(h_{\lambda})$ and $-F_{2}(h_{\lambda})$, respectively. This together
with $\lambda h_{\lambda}-\frac{1}{2}h_{\lambda}=g$ and (6.15)–(6.17) yields
an easy proof of (6.11).

We hasten to add that the properties of exit laws discussed above are not
characteristic to this particular example merely. For the process considered
in Section 6.1 we similarly have
$(\mathcal{L}_{\lambda})^{\prime\prime}=2\lambda\mathcal{L}_{\lambda}$ and
$F\mathcal{L}_{\lambda}=c$ where $Ff\coloneqq
af^{\prime\prime}(0)-bf^{\prime}(0)+c.$ Likewise, for
$\ell_{\lambda}^{1},\lambda>0$ of Section 6.2.7 we have
$(\ell_{\lambda}^{1})^{\prime\prime}=2\lambda\ell_{\lambda}^{1}$ and
$-(\ell_{\lambda}^{1})^{\prime}(0)+c\ell_{\lambda}^{1}(0)=c.$

###  6.3. Example C: Walsh-type Brownian motion on a star-like graph

Let $r>0$ and a natural number ${\mathpzc k}\geq 2$ be given. Moreover, let
$S$ be the union of ${\mathpzc k}$ copies of the interval $[0,r]$, with all
left ends identified. In other words, $S$ is the star-like graph
$K_{1,{\mathpzc k}}$. Continuous functions on $S$ can be identified with
continuous functions $f$ on

$\widetilde{S}\coloneqq\bigcup_{j\in{\mathpzc
K}}\\{j\\}\times[0,r],\qquad\text{where }{\mathpzc
K}\coloneqq\\{1,\dots,{\mathpzc k}\\},$

that satisfy $f(j,0)=f(k,0),j,k\in{\mathpzc K}$; their common value at $0$
will be denoted $f(0)$. It will also be handy to write $f\in\mbox{$C(S)$}$ as
a sum of $f_{j}\in C[0,r]$ such that $f_{j}(0)=f_{k}(0),j,k\in{\mathpzc K}:$

$f=\dot{\sum_{j\in{\mathpzc K}}}f_{j}.$

Let $\beta\in[0,1)$ and $\alpha_{j},j\in{\mathpzc K}$ be non-negative numbers
such that

$\beta+\sum_{j\in{\mathpzc K}}\alpha_{j}=1.$

Also, let $q\in[0,1]$ and $d>0$ be given. We define an operator $A$ in $C(S)$
as follows. Its domain is composed of $f\in\mbox{$C(S)$}$ such that

  * (a)

$f\in C^{2}(S);$ that is, for each $j\in{\mathpzc K}$, $f_{j}$ is twice
continuously differentiable on $[0,r]$ and

$f^{\prime\prime}\coloneqq\dot{\sum_{j\in{\mathpzc K}}}f_{j}^{\prime\prime}$
(6.18)

belongs to $C(S)$,

  * (b)

for each $j\in{\mathpzc K}$,

$qf_{j}^{\prime\prime}(r)+(1-q)f^{\prime}_{j}(r)+df_{j}(r)=0,$

  * (c)

and

$\beta f^{\prime\prime}(0)=\sum_{j\in{\mathpzc K}}\alpha_{j}f_{j}^{\prime}(0)$

(note that $f^{\prime\prime}(0)$ is well-defined by (a), even though
$f^{\prime}_{j}(0)$ depends on $j$).

Moreover, we agree that

$Af={\textstyle\frac{1}{2}}f^{\prime\prime}.$

Boundary condition (b) is of the same form as the second relation in (6.4),
and thus says that each ‘outer’ node of the graph is an exit point for the
process generated by $A$. Transmission condition (c) describes the law
governing the process at the graph’s center. If $\beta$ is zero, this
condition is known to describe the Skew Brownian Motion of J. Walsh — see
[lejayskew]*p. 45 eq. (57), consult also [yor97]*p. 107 and [manyor]*pp.
115-117. In this process, a particle reaching the center of the graph chooses
to continue its further motion on $j$th vertex with ‘probability’ $\alpha_{j}$
(although not quite) — see the already cited [lejayskew] for a more precise
description and many equivalent constructions of this process; in particular,
transmission condition (c) with $\beta=0$ can be obtained in the limit
procedure of Friedlin and Wentzell’s averaging principle [fwbook] — see
[fw]*Thm. 5.1, comp. [emergence]*Eq. (3.1). In general, when $\beta$ is non-
zero, the center is more ‘sticky’ and so the process stays at this point
‘longer’, but no probability mass is lost there (if $\beta$ were allowed to be
$1$, the graph’s center would be an absorbing point). In its more general form
(i.e., with $\beta\not=0$), condition (c) appears also e.g. in
[kostrykin2012]*Eq. (2.5), with yet one more term, describing loss of
probability mass.

From this intuitive description it is clear that there are precisely
${\mathpzc k}$ exits here. We will argue that $A$ is a Feller generator, and
calculate the corresponding Laplace transforms of exit laws and the related
excessive functions.

####  6.3.1. The positive maximum principle for $A$

It is rather easy to be persuaded that $A$ is densely defined. Turning to the
question of the positive maximum principle, we note that if $f\in D(A)$
attains its positive maximum at a point $x$ different from the nodes of $S$,
we obviously have $Af(x)=\frac{1}{2}f^{\prime\prime}(x)\leq 0$. The already
recalled argument from [knigaz]*p. 17 shows also that $Af(x)\leq 0$ if $x$ is
one of the ‘outer’ nodes. Finally, if $x$ is the graph’s center, we clearly
have $f^{\prime}_{j}(0)\leq 0$ for all $j\in{\mathpzc K}$. Hence, if
$\beta>0$, we have
$Af(0)=\frac{1}{2}f^{\prime\prime}(0)=\frac{1}{2\beta}\sum_{j\in{\mathpzc
K}}\alpha_{j}f_{j}^{\prime}(0)\leq 0.$ If $\beta=0$, (c) requires that
$f_{j}^{\prime}(0)=0$ as long as $\alpha_{j}>0$. Since there is at least one
$\alpha_{j}>0$, for at least one $j$ we have $f_{j}^{\prime}(0)=0$, and it
follows that for this $j$, $f^{\prime\prime}_{j}(0)\leq 0$. But $f$ belongs to
$D(A)$, and so $f_{j}^{\prime\prime}(0)$ does not depend on $j$ and we have
$f^{\prime\prime}(0)=f_{j}^{\prime\prime}(0),j\in{\mathpzc K}$, proving that
$Af(0)\leq 0$. To check that $A$ is a Feller generator, it suffices therefore
to prove that it satisfies the range condition. This will be a by-product of
the analysis presented in the next section.

####  6.3.2. The resolvent equation for $A$ and $A^{\textnormal{\hskip
0.56917pt\tiny co}}$

We proceed as in Section 6.2.1: we concatenate the process generated by $A$
with the trivial process on

$S_{2}\coloneqq\\{o_{1},\dots,o_{\mathpzc k}\\}$

(for $o_{j}\not\in S,j\in{\mathpzc K}$), and from an explicit form of the
resolvent of the concatenated process read off the exit laws we search for. As
a by-product, our calculations (with $\widetilde{d}_{j}=0,j\in{\mathpzc K}$,
see below) provide the proof that $A$ satisfies the range condition and is
thus a Feller generator.

Given additionally $\widetilde{d}_{j}\in[0,d],j\in{\mathpzc K}$, we define an
operator $A^{\textnormal{\hskip 0.56917pt\tiny co}}$ in
$C(S_{\textnormal{\hskip 0.56917pt\tiny u}})$, where $S_{\textnormal{\hskip
0.56917pt\tiny u}}\coloneqq S_{1}\dot{\cup}S_{2}$ and $S_{1}\coloneqq S$, as
follows. Its domain is composed of

$f=\dot{\sum_{j\in{\mathpzc
K}}}f_{j}\dot{+}f_{|S_{2}}\in\mbox{$C(S_{\textnormal{\hskip 0.56917pt\tiny
u}})$}$

such that

  * (a)

$f_{|S}\in C^{2}(S)$,

  * (b)

for each $j\in{\mathpzc K}$,

$qf_{j}^{\prime\prime}(r)+(1-q)f^{\prime}_{j}(r)+df_{j}(r)=\widetilde{d}_{j}f(o_{j}),$

  * (c)

and

$\beta f^{\prime\prime}(0)=\sum_{j\in{\mathpzc
K}}\alpha_{j}f_{j}^{\prime}(0).$

For such $f$, we define $A^{\textnormal{\hskip 0.56917pt\tiny
co}}f=\frac{1}{2}f_{|S}^{\prime\prime}$ (in particular, $A(f_{|S_{2}})=0$).

Our claim is that $A^{\textnormal{\hskip 0.56917pt\tiny co}}$ is a Feller
generator in $C(S_{\textnormal{\hskip 0.56917pt\tiny u}})$. The related
process, if started in $S$, obeys the rules dictated by $A$; when it exits $S$
through the $j$th gate, it jumps to $o_{j}$ to stay there for ever.

Arguing as in Section 6.3.1, we check that $A^{\textnormal{\hskip
0.56917pt\tiny co}}$ (is densely defined and) satisfies the positive maximum
principle (this is where condition $0\leq\widetilde{d}_{j}\leq d$ is used).
Hence, we are left with showing that $A^{\textnormal{\hskip 0.56917pt\tiny
co}}$ satisfies the range condition.

To this end, given $\lambda>0$ and $g\in\mbox{$C(S_{\textnormal{\hskip
0.56917pt\tiny u}})$}$ we look for $f\in D(A^{\textnormal{\hskip
0.56917pt\tiny co}})$ satisfying $\lambda f-A^{\textnormal{\hskip
0.56917pt\tiny co}}f=g$ of the form:

$\displaystyle f(o_{j})$ $\displaystyle=\lambda^{-1}g(o_{j}),$ $\displaystyle
f_{j}(x)$
$\displaystyle=C_{j}\mathrm{e}^{\sqrt{2\lambda}x}+D_{j}\mathrm{e}^{-\sqrt{2\lambda}x}+h_{\lambda,j}(x),\qquad
x\in[0,r],j\in{\mathpzc K},$ (6.19)

where

$h_{\lambda,j}(x)\coloneqq\frac{2\beta
g(0)}{\sqrt{2\lambda}(\beta-1)}\sinh\sqrt{2\lambda}x-\sqrt{\frac{2}{\lambda}}\int_{0}^{x}\sinh\sqrt{2\lambda}(x-y)g_{j}(y)\,\mathrm{d}\hskip
0.56917pty,$ (6.20)

for $j\in{\mathpzc K},x\in[0,r]$, and the constants $C_{j}$ and $D_{j}$ are to
be found. We also introduce functionals $F,F_{j}$ given by

$\displaystyle Ff$ $\displaystyle\coloneqq\beta
f^{\prime\prime}(0)-\sum_{j\in{\mathpzc K}}\alpha_{j}f_{j}^{\prime}(0)$
$\displaystyle F_{j}f$ $\displaystyle\coloneqq
qf_{j}^{\prime\prime}(r)+(1-q)f_{j}^{\prime}(r)+df_{j}(r),\qquad j\in{\mathpzc
K},$

on the common domain of $f$ such that $f_{|S}\in C^{2}(S)$. Hence, $f\in
D(A^{\textnormal{\hskip 0.56917pt\tiny co}})$ iff $f_{|S}\in C^{2}(S)$, $Ff=0$
and $F_{j}f=\widetilde{d}_{j}f(o_{j}),j\in{\mathpzc K}$.

To continue, $\dot{\sum}_{j\in{\mathpzc K}}f_{j}\in C(\widetilde{S})$ belongs
to $C(S)$ iff

$C_{j}+D_{j}\quad\text{ does not depend on }j\in{\mathpzc K}.$ (6.21)

Since $\lambda f_{j}-\frac{1}{2}f_{j}^{\prime\prime}=g_{j},j\in{\mathpzc K}$,
this is also a necessary and sufficient condition for $f^{\prime\prime}$ to
belong to $C(S)$. If (6.21) holds, the quantity featured in it is $f(0)$, and
we obtain

$C_{j}=f(0)-D_{j},\qquad j\in{\mathpzc K}.$ (6.22)

Next, as in Section 6.2.2 we check that $F_{j}f=\widetilde{d}_{j}f(o_{j})$ iff

$\displaystyle
a_{2,1}C_{j}+a_{2,2}D_{j}=\widetilde{d}_{j}f(o_{j})-F_{j}h_{\lambda},\qquad
j\in{\mathpzc K}$

where $a_{2,i},i=1,2$ are defined in (6.7), and

$h_{\lambda}\coloneqq\dot{\sum}_{j\in{\mathpzc K}}h_{\lambda,j}.$ (6.23)

These relations together yield

$(a_{2,2}-a_{2,1})D_{j}=\widetilde{d}_{j}f(o_{j})-F_{j}h_{\lambda}-a_{2,1}f(0),\qquad
j\in{\mathpzc K}.$ (6.24)

Finally, (c) of the definition of $D(A^{\textnormal{\hskip 0.56917pt\tiny
co}})$ holds iff

$\beta(2\lambda f(0)-2g(0))=\sum_{j\in{\mathpzc
K}}\sqrt{2\lambda}\alpha_{j}(C_{j}-D_{j}).$ (6.25)

To solve the system (6.22), (6.24)–(6.25) for $C_{j}$ and $D_{j},j\in{\mathpzc
K}$, we note first that the right hand-side in (6.25) equals

$\displaystyle\sqrt{2\lambda}\sum_{j\in{\mathpzc
K}}\alpha_{j}(f(0)-2D_{j})=\sqrt{2\lambda}(1-\beta)f(0)-2\sqrt{2\lambda}\sum_{j\in{\mathpzc
K}}\alpha_{j}D_{j},$

and that, by (6.24),

$(a_{2,2}-a_{2,1})\sum_{j\in{\mathpzc K}}\alpha_{j}D_{j}=\sum_{j\in{\mathpzc
K}}\alpha_{j}(\widetilde{d}_{j}f(o_{j})-F_{j}h_{\lambda})-a_{2,1}(1-\beta)f(0).$

Therefore,

$f(0)=\frac{2\beta}{m_{\lambda}}g(0)+\frac{2\sqrt{2\lambda}}{m_{\lambda}(a_{2,1}-a_{2,2})}\sum_{j\in{\mathpzc
K}}\alpha_{j}(\widetilde{d}_{j}f(o_{j})-F_{j}h_{\lambda}),$ (6.26)

where

$m_{\lambda}\coloneqq
2\beta\lambda+\sqrt{2\lambda}(1-\beta)\frac{a_{2,1}+a_{2,2}}{a_{2,1}-a_{2,2}}.$

Once $f(0)$ is known, $D_{j}$s can be calculated from (6.24) and then $C_{j}$s
can be calculated from (6.22). The fact that the so-derived $C_{j}$s and
$D_{j}$s solve the system (6.21)–(6.25) with
$\widetilde{d}_{j}=d,j\in{\mathpzc K}$ shows that (6.19) defines a solution
(which is necessarily unique) to the resolvent equation for
$A^{\textnormal{\hskip 0.56917pt\tiny co}}$. The same calculation with all
$\widetilde{d}_{j}$s equal to $0$ proves that $A$ satisfies the range
condition also.

####  6.3.3. Exit laws for $A$

Fix $j\in{\mathpzc K}.$ The $f_{j}$ of (6.19) with constants calculated from
(6.22), (6.24)–(6.25) differs from the $f_{j}$ with constants calculated from
this system with $\widetilde{d}_{k}=0,k\in{\mathpzc K}$ by

$\widetilde{d}_{j}\frac{2\sinh\sqrt{2\lambda}x}{a_{2,1}-a_{2,2}}f(o_{j})+\frac{2\sqrt{2\lambda}(a_{2,1}\mathrm{e}^{-\sqrt{2\lambda}x}-a_{2,2}\mathrm{e}^{\sqrt{2\lambda}x})}{m_{\lambda}(a_{2,1}-a_{2,2})^{2}}\sum_{k\in{\mathpzc
K}}\alpha_{k}\widetilde{d}_{k}f(o_{k}).$

Hence, the resolvent of $A^{\textnormal{\hskip 0.56917pt\tiny co}}$ has the
form

$R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
co}}g=R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny du}}g+\sum_{j\in{\mathpzc
K}}{\textstyle\frac{\widetilde{d}_{j}}{d}}[R_{\lambda}^{\textnormal{\hskip
0.56917pt\tiny co}}g(o_{j})]\ell_{\lambda}^{j},$

where $\ell_{\lambda}^{j}\in\mbox{$C(S)$}$ is defined by its parts
$(\ell_{\lambda}^{i})_{k}\in C[0,r],k\in{\mathpzc K}$ as follows

$(\ell_{\lambda}^{j})_{k}(x)=\frac{2d\sinh\sqrt{2\lambda}x}{a_{2,1}-a_{2,2}}\delta_{j,k}+\frac{2d\sqrt{2\lambda}(a_{2,1}\mathrm{e}^{-\sqrt{2\lambda}x}-a_{2,2}\mathrm{e}^{\sqrt{2\lambda}x})}{m_{\lambda}(a_{2,1}-a_{2,2})^{2}}\alpha_{j},$
(6.27)

and $\delta_{j,k}$ is the Kronecker delta.

As in Section 6.2.9, we note that

$(\ell_{\lambda}^{j})^{\prime\prime}=2\lambda\ell_{\lambda}^{j},\qquad
j\in{\mathpzc K},$

and

$\displaystyle F_{k}\ell_{\lambda}^{j}$
$\displaystyle=\frac{d(a_{2,1}-a_{2,2})}{a_{2,1}-a_{2,2}}\delta_{j,k}+\frac{2d\sqrt{2\lambda}(a_{2,2}a_{2,1}-a_{2,2}a_{2,1})}{m_{\lambda}(a_{2,1}-a_{2,2})^{2}}\alpha_{k}$
$\displaystyle=d\delta_{j,k},\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
j,k\in{\mathpzc K}.$

A bit longer calculation (see Appendix 10.4) shows that

$F\ell_{\lambda}^{j}=0,\qquad j\in{\mathpzc K}.$ (6.28)

Together with the observation that $Fh_{\lambda}=0$ ($h_{\lambda}$ was defined
in (6.23)) these relations lead to the following formula for
$\left(\lambda-A\right)^{-1}$ in terms of exit laws (comp. (6.11)):

$\left(\lambda-A\right)^{-1}g=h_{\lambda}-d^{-1}\sum_{j\in{\mathpzc
K}}(F_{j}h_{\lambda})\ell_{\lambda}^{j}.$ (6.29)

####  6.3.4. Excessive functions for $A$

Using the formula in Appendix 10.3, it can be seen that

$\displaystyle\lim_{\lambda\to
0+}\frac{a_{2,1}\mathrm{e}^{-\sqrt{2\lambda}x}-a_{2,2}\mathrm{e}^{\sqrt{2\lambda}x}}{2\sqrt{2\lambda}}$
$\displaystyle=d(r-x)+1-q,$ $\displaystyle\lim_{\lambda\to 0+}{m_{\lambda}}$
$\displaystyle=\frac{d(1-\beta)}{rd+1-q}.$

Therefore, as long as $\beta<1$, the excessive function corresponding to the
exit through the $j$th outer node is
$\phi^{j}\coloneqq\dot{\sum}_{k\in{\mathpzc K}}(\phi^{j})_{k}$ where

$(\phi^{j})_{k}(x)\coloneqq\lim_{\lambda\to
0+}(\ell_{\lambda}^{j})_{k}(x)=\frac{dx}{rd+1-q}\delta_{j,k}+\frac{d(r-x)+1-q}{rd+1-q}\frac{\alpha_{j}}{1-\beta},$
(6.30)

for $x\in[0,r]$. It is clear that $(\phi^{j})_{k}(0)$ does not depend on $k$,
proving that $\phi^{j}$ belongs to $\mbox{$C(S)$}.$ Since, as we shall see in
the next section, in the case of $\beta<1$, the kernel of $A$ is trivial, we
conclude that

$\ell_{\lambda}^{j}=\phi^{j}-\lambda R_{\lambda}\phi^{j},\qquad j\in{\mathpzc
K}.$

To interpret (6.30), we note that the quantity $\frac{dx}{rd+1-q}$ coincides
with the right-hand side in the second equation in (6.14) with $p=1$. Since in
the case of $p=1$ the left interval’s end in the example of Section 6.2 is
absorbing, this quantity is the probability that a particle performing a
Brownian motion on $[0,r]$ with right-end boundary described by the second
equation in (6.3) exits through this boundary without touching $0$ in the
meantime. Since the nature of the boundary at the $i$th node is the same as in
the process just described (see (b) in the definition of $D(A)$),
$\frac{dx}{rd+1-q}$ is the probability that a Walsh-type Brownian motion
starting at the point $x$ at the $j$th node will exit through this node’s end
without ever going through the graph’s center. On the other hand,
$\frac{d(r-x)+1-q}{rd+1-q}$ is the probability that a particle starting from
this $x$ will reach the graph’s center before exiting from the state-space
(necessarily through the $i$th node). Finally, $\frac{\alpha_{k}}{1-\beta}$ is
the probability that a process starting from the graph’s center will exit the
graph through its $k$th node.

It is an interesting feature of this example that even though all the outer
nodes, that is, all the exits, have the same ‘local’ nature (being described
by a boundary condition with the same coefficients), the related exit laws
differ — see the formulae for $\ell_{\lambda}^{j},\lambda>0$ and
$\phi^{j},j\in{\mathpzc K}.$ The reason for that lies in the graph’s center —
because of that point the process has a tendency to sojourn in some vertices
more often than in other vertices, and thus exits through one node more often
than through the other.

####  6.3.5. Triviality of the kernel

Finally, we show that, as long as $\beta\in[0,1)$, the kernel of $A$ is
trivial so that, in particular, the representation of exit laws in terms of
excessive functions is unique. This together with the considerations presented
above establishes that the process related to $A$ has a regular Feller exit
boundary with ${\mathpzc k}\geq 2$ exits.

To this end, we note that if $f\in\operatorname{Ker}A$, then
$f_{j}(x)=m_{j}x+b,x\in[0,r],j\in{\mathpzc K}$ for certain real numbers $b$
and $m_{j},j\in{\mathpzc K}$, because $f_{j}(0)$ is to be independent of $j$.
Next, condition (b) tells us that we should have

$(1-q)m_{j}+d(m_{j}r+b)=0.$ (6.31)

It follows that $m_{j}$ does not depend on $j$ either, and thus can be denoted
by $m$. Also, by (c), we need to have $0=\sum_{j\in{\mathpzc
K}}\alpha_{j}m=(1-\beta)m$, implying $m=0$. Since, by (6.31), this forces
$b=0$ and, consequently, $f=0$, we are done.

For $\beta=1$, the kernel of $A$ is not trivial. It contains all functions of
the form $f=\dot{\sum}_{j\in{\mathpzc K}}f_{j}$ where

$f_{j}(x)\coloneqq mx-d^{-1}(1-q+dm)r,\qquad x\in[0,1],j\in{\mathpzc K}.$

##  7\. A sample generation theorem

In the foregoing sections we described concatenated stochastic processes by
two apparently different means. In Section 5 we used resolvents, whereas in
Section 6 each example was commenced by an account in terms of generator and
corresponding boundary and transmission conditions. Although is was more or
less obvious from the calculations presented in the examples that these are
two descriptions of the same stochastic process, we never proved this
formally. This is what we aim to do in this section.

To this end, we come back to generation theorems for diffusions on graphs of
the type considered in [bobmor, nagrafach, gregosiewicz, gregosiewiczsem,
banfal2, gregnew, binz]. To recall, in these papers, it is imagined that
graph’s vertices are semi-permeable membranes. Therefore, it is convenient to
think of the graph as the union of, say, $N$, disjoint intervals:

$S_{\textnormal{\hskip 0.56917pt\tiny
u}}=\dot{\bigcup_{i\in\mathcal{N}}}\\{i\\}\times[0,r_{i}],$

where, as before, $\mathcal{N}\coloneqq\\{1,\dots,N\\}$ and
$r_{i},i\in\mathcal{N}$ are positive numbers (edge’s lengths). Each interval’s
(i.e., edge’s) end is considered as different even from the ends of adjacent
edges, since it is seen as lying on ‘this side’ of the membrane, as opposed to
‘the other side’ of the membrane. Moreover, the membrane’s permeability
depends both on the edge from which a diffusing particle filters and on the
edge to which the diffusing particle filters.

A more precise definition involves $N$ quadruples of parameters

$(p_{i},c_{i},q_{i},d_{i}),i\in\mathcal{N}$

such that $p_{i},q_{i}\in[0,1]$ and $c_{i},d_{i}>0$, and $2N$ sub-probability
measures $\mathfrak{p}_{i,j},i\in\mathcal{N}$ and $j=1,2$ on $S$. The domain
of the generator $A$ of the process considered is composed of
$f=\dot{\sum}_{i\in\mathcal{N}}f_{i}$ such that $f_{i}$ is twice continuously
differentiable on $[0,r_{i}]$, and the following boundary and transmission
conditions are satisfied (cf. (6.4))

$\displaystyle
p_{i}f_{i}^{\prime\prime}(0)-(1-p_{i})f_{i}^{\prime}(0)+c_{i}f_{i}(0)$
$\displaystyle=c_{i}\int f\,\mathrm{d}\hskip 0.56917pt\mathfrak{p}_{i,1},$
$\displaystyle
q_{i}f_{i}^{\prime\prime}(r_{i})+(1-q_{i})f_{i}^{\prime}(r_{i})+d_{i}f_{i}(r_{i})$
$\displaystyle=d_{i}\int f\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,2},\qquad i\in\mathcal{N}$ (7.1)

(these are particular examples of non-local Feller–Wentzel conditions). For
such $f$,

$Af\coloneqq{\textstyle\frac{1}{2}}f^{\prime\prime}\coloneqq{\textstyle\frac{1}{2}}\dot{\sum_{i\in\mathcal{N}}}f_{i}^{\prime\prime}.$

In [nagrafach] (and in [bobmor], where $N=3$), $p_{i}$s are zeros; the case of
$p_{i}\not=0$ is considered in [gregnew]; the measures $\mathfrak{p}_{i,1}$
and $\mathfrak{p}_{i,2}$ are convex combinations of Dirac measures at the
endpoints of edges that are adjacent to $(i,0)$ and $(i,r_{i}),$ respectively.
In [banfal2] and [marjeta, binz] more general operators and transmission
conditions are discussed, including those without a clear probabilistic
interpretation. Moreover, analysis is carried out also in spaces other than
the space of continuous functions. The requirement that $\mathfrak{p}_{i,j}$s
are convex combinations of Dirac measures at the ends of adjacent edges is
natural and means that after filtering through the end of the $i$th vertex, a
particle starts at one of the ends of the adjacent edges, choosing an edge
with probability dictated by the convex combination. In what follows, however,
we will work with general $\mathfrak{p}_{i,j}$s: all we assume is that they
are sub-probability measures.

The latter assumption is necessary for the conclusion that the operator $A$
defined above satisfies the positive maximum principle. More precisely, it
guarantees that if a positive maximum of a function $f\in D(A)$ is attained at
one of the vertices, the value of $Af$ at this vertex is not positive (cf.
[knigaz]*p. 17 or our Section 6.3.1). Since it is rather clear that $A$ is
densely defined, to prove that $A$ is a Feller generator, one needs to check
the range condition. As mentioned in the introduction, this is by far the
hardest task; in the previous papers, various methods, rarely with clear
probabilistic meaning, were used to accomplish it, including Greiner-like
perturbations (see [greiner] and [banasiak, banfal2, 3Dlayers], and references
given therein, comp. also [arkuk2])), passing to isomorphic semigroups (see
[bobmor, nagrafach]), and lifting techniques (see [banfal1]*Section 3.2) that
are often used in singular perturbation theory involving less standard
boundary conditions — see, for example, [banasiak]*Lemma 2.3, or [banbob2]*pp.
230–232.

Meanwhile, the range condition is a direct consequence of our Theorem 7.1, a
bit further down, which, in turn, is a corollary to Theorem 5.4.

Before presenting the theorem, we need some notation. To begin with, for
$i\in\mathcal{N}$, let $A_{i}$ be the operator in $C[0,r_{i}]$ with domain
composed of $C^{2}[0,r_{i}]$ functions, that is, of twice continuously
differentiable functions, satisfying the boundary conditions

$\displaystyle p_{i}f^{\prime\prime}(0)-(1-p_{i})f^{\prime}(0)+c_{i}f(0)$
$\displaystyle=0,$ $\displaystyle
q_{i}f^{\prime\prime}(r_{i})+(1-q_{i})f^{\prime}(r_{i})+c_{i}f(r_{i})$
$\displaystyle=0,$ (7.2)

and defined by $A_{i}f=\frac{1}{2}f^{\prime\prime}.$ In Section 6.2 we have
established that there are two exit laws, say, $\ell_{\lambda}^{i,j},j=1,2$
for $A_{i}$; they are given by (6.10) and (6.7) with $p,q,c$ and $d$ replaced
by $p_{i},q_{i},c_{i}$ and $d_{i}$, respectively.

Also, let $R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny du}},\lambda>0$ be
the resolvent of the disjoint union of the related processes (see (5.3)).
Finally, let $R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}},\lambda>0$
defined by

$R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
co}}g=R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
du}}g+\sum_{i\in\mathcal{N}}\sum_{j=1,2}u_{i,j}\ell_{\lambda}^{i,j},\qquad
g\in\mbox{$C(S)$},\lambda>0,$

be the resolvent of the process concatenated by means of the measures
$\mathfrak{p}_{i,j}$ featured in (7.1); this means that $u_{i,j}$s are unique
solutions to the counterpart of the system (5.9), and in fact, see (5.12),

$u_{i,j}=\int R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
co}}g\,\mathrm{d}\hskip 0.56917pt\mathfrak{p}_{i,j},\qquad
i\in\mathcal{N},j=1,2.$ (7.3)

######  Theorem 7.1.

For any $\lambda>0$ and $g\in\mbox{$C(S)$}$ there is a unique $f\in D(A)$
solving the resolvent equation $\lambda f-Af=g$. This $f$ is

$f=R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}}g.$

###### Proof.

Let $C^{2}(S_{\textnormal{\hskip 0.56917pt\tiny u}})$ be the subspace of
$C(S_{\textnormal{\hskip 0.56917pt\tiny u}})$ of
$f=\dot{\sum}_{i\in\mathcal{N}}f_{i}$ such that $f_{i}\in C^{2}[0,r_{i}]$
(that is, $f_{i}$ is twice continuously differentiable), and let
$\Delta:C^{2}(S_{\textnormal{\hskip 0.56917pt\tiny
u}})\to\mbox{$C(S_{\textnormal{\hskip 0.56917pt\tiny u}})$}$ be given by
$\Delta
f\coloneqq\frac{1}{2}f^{\prime\prime}\coloneqq\dot{\sum}_{i\in\mathcal{N}}f_{i}^{\prime\prime}.$
Also, let $F_{i,j},i\in\mathcal{N},j=1,2$ be the following functionals on
$C^{2}(S_{\textnormal{\hskip 0.56917pt\tiny u}})$:

$\displaystyle F_{i,1}f$
$\displaystyle=p_{i}f_{i}^{\prime\prime}(0)-(1-p_{i})f_{i}^{\prime}(0)+c_{i}f_{i}(0),$
$\displaystyle F_{i,2}f$
$\displaystyle=q_{i}f^{\prime\prime}_{i}(r_{i})+(1-q_{i})f^{\prime}_{i}(r_{i})+d_{i}f(r_{i}),\qquad
i\in\mathcal{N}.$

Since the kernel of $F_{i,j}$ includes all functions that are equal to zero on
the $i$th edge and because of the definition of $A_{k}$, we see that if
$f=\dot{\sum}_{k\in\mathcal{N}}f_{k}$ is such that $f_{k}\in
D(A_{k}),k\in\mathcal{N}$, we have $F_{i,j}f=0$. In particular,
$F_{i,j}R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny du}}g=0$ for all
$g\in\mbox{$C(S_{\textnormal{\hskip 0.56917pt\tiny u}})$}.$

Moreover, see (6.15)–(6.17),

$\Delta\ell_{\lambda}^{i,j}=\lambda\ell_{\lambda}^{i,j},\qquad
i\in\mathcal{N},j=1,2,$ (7.4)

and

$\displaystyle F_{i,1}\ell_{\lambda}^{i,j}=\delta_{1,j}c_{i}\quad\text{ and
}\quad F_{i,2}\ell_{\lambda}^{i,j}=\delta_{2,j}d_{i},\qquad j=1,2.$ (7.5)

Therefore, $F_{i,1}R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
co}}g=F_{i,1}R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
du}}+c_{i}u_{i,1}=c_{i}u_{i,1},$ and, similarly, $F_{i,2}=d_{i}u_{i,2}$. When
combined with (7.3), this means that

$F_{i,1}R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}}=c_{i}\int
R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}}\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,1}\quad\text{ and }\quad
F_{i,2}R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}}=d_{i}\int
R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}}\,\mathrm{d}\hskip
0.56917pt\mathfrak{p}_{i,2},$

that is, by (7.1), that $R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}}g$
belongs to $D(A)$. Moreover, since $A$ is a restriction of $\Delta$ and so is
the generator of the process of disjoint union,

$(\lambda-A)R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
co}}g=(\lambda-\Delta)R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny du}}g=g$
(7.6)

where we also used (7.4).

We have shown that $R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}}g$ is a
solution to $\lambda f-Af=g$. To complete the proof, we note that, since $A$
satisfies the positive maximum principle, it is dissipative by [ethier]*Lemma
2.1, p. 165: for any $f\in D(A)$, $\|\lambda f-Af\|\geq\|f\|.$ Therefore, the
solution is unique. ∎

######  Corollary 7.2.

The operator $A$ coincides with the generator $A^{\textnormal{\hskip
0.56917pt\tiny co}}$ of the process related to
$R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}},\lambda>0$.

###### Proof.

We have proved that $D(A^{\textnormal{\hskip 0.56917pt\tiny co}})\subset
D(A)$. Moreover, by (7.6), $\lambda R_{\lambda}^{\textnormal{\hskip
0.56917pt\tiny co}}g-A^{\textnormal{\hskip 0.56917pt\tiny
co}}R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}}g=g=\lambda
R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
co}}g-AR_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}}g$, proving that
$AR_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny co}}g=A^{\textnormal{\hskip
0.56917pt\tiny co}}R_{\lambda}^{\textnormal{\hskip 0.56917pt\tiny
co}}g,g\in\mbox{$C(S_{\textnormal{\hskip 0.56917pt\tiny u}})$}$. In other
words, $A^{\textnormal{\hskip 0.56917pt\tiny co}}$ is a restriction of $A$ to
$D(A^{\textnormal{\hskip 0.56917pt\tiny co}})$. On the other hand, both
operators are generators of Feller semigroups, and thus are dissipative. This
is impossible unless $D(A)=D(A^{\textnormal{\hskip 0.56917pt\tiny co}})$ (see
[ethier]*Proposition 4.1, p. 21), as desired. ∎

Before closing this section, we note that the argument presented in the proof
of Theorem 7.1 applies to more general scenarios. First of all, for some $i$s,
either $c_{i}$ or $d_{i}$ may be zero, as long as we avoid the case described
in Section 6.2.8. In such a scenario there is only one exit for the process
related to $A_{i}$ and one of transmission conditions in (7.1) should be
changed appropriately. Nevertheless, the argument is the same because the
remaining exit law has the properties listed at the end of Section 6.2.9
(these properties allow writing counterparts of (7.4)–(7.5)). Likewise, some
of the intervals can be replaced by half-lines and $A_{i}$s can be replaced by
generators of the type discussed in Section 6.1. Finally, each interval can be
replaced by the star-like graph of Section 6.3 and the corresponding generator
can be replaced by the generator of Walsh-type Brownian Motion. In the last
scenario, we would perhaps face more than two transmission conditions for each
component of $S_{\textnormal{\hskip 0.56917pt\tiny u}}$, but again the
argument would be the same, modulo natural changes. The same remark applies to
more general graphs like those in [kostrykin2012].

Secondly, even though we restrict our applications to stochastic processes on
graphs (which are our main motivation), it is worth stressing that the results
of Section 5 apply also to processes with exit laws that are not described by
boundary and transmission conditions.

Finally, a comment is also in order on the remarkable properties of exit laws
(7.4)–(7.5) that played such a crucial role in the proof of Theorem 7.1.
Functions with such or similar properties are typically used as handy tools in
lifting techniques (see the references cited above). Thus, it appears that the
analysis presented in this paper is another reflection of the fact that
classical analysis has its probabilistic counterpart, with the latter
providing intuitions that are not easily visible in the former (cf. [doobc]).

##  8\. Convergence of semigroups

Our last section (Section 9) is devoted to an applications of the results
obtained in the first part of the paper: we prove an averaging principle for
fast processes, of the kind considered before in [banfal1, bobmor, nagrafach,
zmarkusem]. These results hinge on the theory of convergence of semigroups
which is recalled briefly here.

The classical Trotter–Kato Theorem (see e.g. [goldstein, pazy]) says that
strongly continuous equibounded semigroups
$\mbox{$\left(\mathrm{e}^{t{B_{\varepsilon}}}\right)_{t\geq
0}$},\varepsilon\in(0,1]$ in a Banach space $E$ converge as $\varepsilon\to 0$
to a strongly continuous semigroup $\left(\mathrm{e}^{t{B}}\right)_{t\geq 0}$,
that is,

$\lim_{\varepsilon\to 0}\mathrm{e}^{tB_{\varepsilon}}f=\mathrm{e}^{tB}f,\qquad
t\geq 0,f\in E,$ (8.1)

iff

$\lim_{\varepsilon\to 0}\left(\lambda-
B_{\varepsilon}\right)^{-1}f=\left(\lambda-B\right)^{-1}f,\qquad f\in E,$

for some/all $\lambda>0$; moreover, then the limit (8.1) is uniform in $t$ in
compact subsets of $[0,\infty)$. In other words, such regular convergence of
semigroups is completely characterized (see also [kniga, knigaz, ethier] for
the Sova–Kurtz version [kurtz, sova] of this characterization).

However, in the theory of singular perturbations and in the particular example
we are studying in this paper the limit semigroup is strongly continuous only
on a subspace of $E$: we are facing a limit theorem of the form

$\lim_{\varepsilon\to
0}\mathrm{e}^{tB_{\varepsilon}}f=\mathrm{e}^{tB}Pf,\qquad t>0,f\in E$ (8.2)

where $\left(\mathrm{e}^{t{B}}\right)_{t\geq 0}$ is a strongly continuous
semigroup on a subspace $E_{0}$ of $E$ and $P$ is a projection on $E_{0}$ (in
the sense that $P^{2}=P$ and $Pf=f,f\in E_{0}$). Needless to say, in this case
the classical theory does not work and, in particular, condition

$\lim_{\varepsilon\to 0}\left(\lambda-
B_{\varepsilon}\right)^{-1}f=\left(\lambda-B\right)^{-1}Pf,\qquad f\in E,$
(8.3)

for all/some $\lambda>0$ is necessary but not sufficient for (8.2) (see [deg]
or [knigaz]).

Condition (8.3) can imply (8.2) provided that the semigroups involved possess
additional regularity properties (like, for example, uniform holomorphicity —
see e.g. [knigaz]*Chapters 31 and 41 for details). A different set of
conditions guaranteeing that (8.3) implies (8.2) has been given by T. G. Kurtz
[ethier, kurtz, kurtzper, kurtzapp]. While Kurtz’s singular convergence
theorem is usually expressed in terms of the so-called extended limit of
generators, for our subsequent analysis the following resolvent-version will
be more practical. This result can be easily deduced e.g. from combined Lemma
7.1 and Theorem 42.2 in [knigaz].

######  Theorem 8.1.

Suppose $B_{\varepsilon},\varepsilon\in(0,1]$ are generators of strongly
continuous equibounded semigroups. Suppose also that for some $\lambda>0$

$\lim_{\varepsilon\to 0}\left(\lambda-\varepsilon
B_{\varepsilon}\right)^{-1}=\left(\lambda-A_{0}\right)^{-1}$ (8.4)

where $A_{0}$ is the generator of a strongly continuous semigroup
$\left(\mathrm{e}^{t{A_{0}}}\right)_{t\geq 0}$ such that

$Pf\coloneqq\lim_{t\to\infty}\mathrm{e}^{tA_{0}}f,\qquad f\in E$

exists. Then condition (8.3) (for some $\lambda>0$, with the same $P$) implies
(8.2), and the limit is uniform in $t$ in compact subsets of $(0,\infty)$; for
$f\in E_{0}$ the limit is uniform in $t$ in compact subsets of $[0,\infty).$

##  9\. An averaging principle

###  9.1. Asymptotically splittable processes

Before we begin our main subject, let us first think what happens when a non-
honest process is accelerated. In other words, how multiplying the generator
$A$ of a non-honest process by $\varepsilon^{-1}$, where $\varepsilon\ll 1$
influences the process. It is quite easy to see that a $\phi$ is excessive for
$\varepsilon^{-1}A$ iff it is excessive for $A$. Also, a
$\widetilde{\ell}_{\lambda},\lambda>0$ is an exit law for $\varepsilon^{-1}A$
iff it is of the form $\tilde{\ell}_{\lambda}=\ell_{\varepsilon\lambda}$ where
$\ell_{\lambda},\lambda>0$ is an exit law for $A$. Since, by the
representation of (3.3), $\lim_{\varepsilon\to
0}\ell_{\varepsilon\lambda}(x)=m_{x}([0,\infty))$ we conclude that in the
limit of accelerated processes the probabilities of exiting through various
exits do not change, but the process’s lifetime is $0$. ($\lambda\mapsto
m_{x}([0,\infty))$ is the Laplace transform of the Dirac measure at $0$
multiplied by $m_{x}([0,\infty))$.)

A more interesting limit is obtained if while the process accelerates, the
speed with which probability mass escapes from the state-space is tuned so
that the fluxes through the boundary’s exit points are more or less constant.
These considerations lead to the following definition.

Let $\mathsf{v}\in\mathbb{R}^{\mathpzc k}$ be a non-zero vector and let
$A_{\varepsilon\mathsf{v}},\varepsilon\in[0,1]$ be Feller generators in the
space $C(S)$ of continuous functions on a compact space $S$. We think of all
these generators as describing ‘one stochastic process’ with regular Feller
boundary and ${\mathpzc k}$ exit points, and of $\varepsilon\mathsf{v}$ as a
vector of parameters that describe the speeds with which probability mass
escapes the state-space through exit points. In particular, we assume that
$A_{\mathsf{0}}$ is a conservative Feller process. Moreover, to each
$\varepsilon$ there correspond ${\mathpzc k}$ excessive functions describing
exits, which we denote

$\phi_{\varepsilon}^{j},\qquad j\in{\mathpzc K}(=\\{1,\dots,{\mathpzc k}\\}).$

######  Definition 9.1.

_We will say that $A_{\mathsf{v}}$ is a generator of an asymptotically
splittable process if the following conditions are satisfied._

  * (a)

_There are constants_ $\varrho^{j}\in C(S)$ such that $\lim_{\varepsilon\to
0}\phi_{\varepsilon}^{j}=\varrho^{j}1_{S},j\in{\mathpzc K}$ (necessarily
$\varrho^{j}\geq 0$ and $\sum_{j\in{\mathpzc K}}\varrho^{j}=1$).

  * (b)

$\lim_{\varepsilon\to 0}\left(\lambda-
A_{\varepsilon\mathsf{v}}\right)^{-1}f=\left(\lambda-
A_{\mathsf{0}}\right)^{-1}f,f\in C(S)$_._

  * (c)

$A_{\mathsf{0}}$_is a conservative Feller generator and_
$\lim_{t\to\infty}\mathrm{e}^{tA_{\mathsf{0}}}f=Pf,f\in C(S)$ where $P$ is a
projection of $C(S)$ onto the subspace
$\mbox{$C^{\flat}(S)$}\subset\mbox{$C(S)$}$ of functions that are constant on
$S.$

  * (d)

_There is a_ $\gamma>0$ such that $\lim_{\varepsilon\to
0}\left(\lambda-\varepsilon^{-1}A_{\varepsilon\mathsf{v}}\right)^{-1}f=\frac{1}{\lambda+\gamma}Pf,f\in\mbox{$C(S)$}$.

We note that in view of representation of exit laws in terms of excessive