Applied Mathematics, 2010, 1, 24-28
doi:10.4236/am.2010.11004 Published Online May 2010 (http://www.SciRP.org/journal/am)
Stationary Distribution of Random Motion with Delay in
Reflecting Boundaries*
Anatoliy A. Pogorui1, Ramón M. Rodríguez-Dagnino2
Tecnológico de Monterrey (ITESM), Electrical and Computer Engineering. Sucursal de correos, Monterrey, México.
Received February 21, 2010; revised April 26, 2010; accepted May 8, 2010
Abstract
In this paper we study a continuous time random walk in the line with two boundaries [a,b], a < b. The par-
ticle can move in any of two directions with different velocities v1 and v2. We consider a special type of
boundary which can trap the particle for a random time. We found closed-form expressions for the stationary
distribution of the position of the particle not only for the alternating Markov process but also for a broad
class of semi-Markov processes.
Keywords: Random Motion, Reflecting Boundaries, Semi-Markov, Random Walk
1. Introduction
In this paper we study the stationary distribution of a
one-dimensional random motion performed with two
velocities, where the random times separating consecu-
tive velocity changes perform an alternating Markov
process. The sojourn times of this process are exponen-
tially distributed random variables. There are many pa-
pers on random motion devoted to analysis of models in
which motions are driven by a homogeneous Poisson
process [1-4], however we have not found any paper
investigating the stationary distribution of these
pro ce sses.
We assume that the particle moves on the line

in
the following manner: At each instant it moves according
to one of two velocities, namely
1
0v>
or
2
0v<
Starting at the position
0
x∈
the particle continues its
motion with velocity
10v>
during random time
1
τ
,
where
1
τ
is an exponential random variable with para-
meter
1
λ
,then the particle moves with velocity
2
0v<
during random time
2
τ
, where
2
τ
is an exponential
distributed random variable with parameter
2
λ
. Fur-
thermore, the particle moves with velocity
1
0v>
and
so on. When the particle reaches boundary a or b it will
stay at that boundary a random time given by the time
the particle remains in the same direction up to the time
such a particle changes direction. Similar partly reflect-
ing (or trapping) boundaries have been considered in [5],
and they may be found in optical photon propagation in
turbid medium or chemical processes with sticky layers
or boundaries.
We also consider a generalization of these results for
semi-Markov processes, i.e., when the random variables
1
τ
and
2
τ
are different from exponential. This paper is
divided in two main parts, namely the Markov case and
the generalization to the semi-Markov modeling. Our
main result, in the first part of this paper, consists on
finding the stationary distribution of the well-known
telegrapher process on the line with delays in reflecting
boundaries. In the second part, we find the stationary
distribution of a more general continuous time random
walk when the sojourn times are generally distributed.
2. Markov Case
2.1. Mathematical Modeling
Let us set the probability space (Ω,

, P). On the phase
space E = {1,2} consider an alternating Markov process
{
ξ
(t); t
≥
0} having the sojourn time
i
τ
correspond-
ing to the state
i∈E
, and transition probability matrix
of the embedded
Markov chain
01
10
P.

=

(1)
Denote by {x(t); t ≥ 0} the position of the particle at
time t. Consider the function
()Cx
on the space E
whi ch is defined as
*
We thank ITESM through the Research Chair in Telecommunications.

A. A. POGORUI ET AL. 25
( )
1
2
if 1
if 2
vx
Cx v x.
=

==

(2)
The position of the particle at any time t can be ex-
pressed as
( )() ( )
00,
t
xt xCξs ds=+
∫
(3)
where the starting point 0[ ,].x ab∈
Equation (3) determines the random evolution of the
particle in the alternating Markov medium {
( )
ξt
; t
≥
0} [6,7]. So, x(t) is the well-known one-dimensional
telegraph process [1,2]. We assume that a < b are two
delaying or adhesive boundaries on the line such that if a
particle reaches boundary a or b then it is delayed until
the instant that the process changes velocity.
Now, consider the two-component stochastic process
( )( )( )
{ (, }ζt xtξt=
on the phase space

= [a,b]
×
{1,2}. The process
( )
ζt
is a homogeneous Markov
process with the following generating operator [6,7]:
( )( )( )( )( )
,,,,,,1, 2,
i
d
AxiC xixiλP xixii
dx
φφ φφ =+ −= 

(4)
where
( )()
,1 ,2Px x φφ =
and
()( )
,2 ,1Px x. φφ =
2.2. Stationary Distribution
Denote by
()π⋅
the stationary distribution of
()ζt
. The
analysis of the properties of the process
()ζt
leads up to
the conclusion that the stationary distribution
π
has
atoms at points (a, 2) and (b, 1), and we denote them as
[,2]πa
and [ ,1]πbrespectively. The continuous part of
π
is denoted as
( )
,,πxii ∈E
.
Since
π
is the stationary distribution of
()ζt
then
for any function
() φ ⋅
from the domain of the operator A
we have
( )( )
0A zdπz φ =
∫

(5)
Now, let *
A
be the conjugate or adjoint operator of
A. Then by changing the order of integration in (5) (inte-
grating by parts), we can obtain the following expres-
sions for the continuous part of
0
*
Aπ=
( )( )()
( )( )()
1 12
2 21
,1,1,20
,2,2,10
+− =



+ −=


d
vπxλπxλπx
dx
d
vπxλπxλπx.
dx
(6)
Similarly, from (5) we obtain
( )
( )
( )
()
( )
( )
( )
( )
11
12
22
21
,1,10
,1,2 0
,2,2 0
,2,1 0
λπbvπb
λπbvπb
λπavπa
λπavπa.
−
−
+
+

−=


+=


+=


−=


(7)
where
( )
( )
,: lim,
xb
πbiπxi
−
↑
=
and
( )
,:πai
+
=
lim
xa
π
↓
( )
,πxi
for i=1,2.
It follows from the set (6) that
( )()
12
,1, 20,
dd
vπxvπx
dx dx
+=
or equivalently
( )()
12
,1,2constantvπxvπxk .+==
By using (7), we get
( )
1
,1vπbvπb
−
+
( )
2
, 20,v πb
−
=
consequently0 andk=
( )()
12
,1,2 0vπxvπx+=
(8)
for all
[ ]
,x ab∈
.
By obtaining
( )
,2πx
from (8) and substituting such
a result into the first equation in the set (6) we have
( )( )( )
1
1 12
2
,1,1,1 0−− =
v
d
vπxλπxλπx.
dx v
(9)
Solving (9) we obtain for the continuous part of π
( )
,1
μx
πx Ce
−
=
(10)
And
( )
1
2
,2 ,
μx
v
πx Ce
v
−
= −
(11)
where
12
12
λλ
μ.
vv
= +
Now, from (7) we obtain for atoms
1
2
[, 2]
−
=
μa
v
πa Ce
λ
(12)
and
1
1
[ ,1]− =μb
v
πbCe .
λ (13)
The factor
C
can be calculated from the normaliza-
tion equation
( )
1,πz dz=
∫

(14)
or equivalently
( )() [ ]
,1,2,2[,1]1
bb
aa
πx dxπx dxπaπb.+++=
∫∫
(15)
It follows from (15) that
1
1 211 12
12 22
11
−
−−

  −−
= −+−

 

 

μbμa
v vvvvv
C ee.
λμvλμv
(16)

26 A. A. POGORUI ET AL.
We should notice that the stationary distribution
()x π
of the process
()xt
over the interval
(,)
ab is
( )
x π
( )()
,1 ,2πxπx.= +
2.3. Balanced and One-Boundary Cases
2.3.1. Balanced Case
Let us call the balanced case when
12
12
0
λλ
μ.vv
=+=
In
this case we can observe that
( )
,1πx
and
( )
,2πx
do
not depend on x. Hence, the continuous part of the sta-
tionary distribution of the process
()xt
is uniform over
the open interval
(,)
ab . Now, the factor
C
, say
B
C
,
reduces to
2
21
,
( )()
B
v
Cv vbaδ =− −+
(17)
where
12
12
vv
δ.λλ == −
Therefore, the stationary distribution can be expressed
as
( )()
1
2
,1and, 2
BB
v
πxCπxC. v == −
(18)
Thus,
( )()()
1
,1 ,2,xπxπxbaδ =+=
−+
π
(19)
and the atoms are given by
[ ]
1
2
,2and [,1]
BB
v
πaδCπbδC.
v
= =
(20)
2.3.2. One-Boundary Case
Now, suppose that there is just the left boundary a, and
the starting position of the process
()xt
is
[
)
0
,xa .∈ +∞
Then for
0μ>
we have the factor
C
,
say O
C
, given by
2
121 2
2
()
μa
O
ve
C.
vvv v
λμ
=−
−
(21)
Hence,
( )()
1
2
,1and ,2,
μxμx
OO
v
πx Ceπx Cev
−−
== − (22)
with the atom
[ ]
12
21 2
12
,2 () =−
−
vv
πa.
λvv
vv μ
(23)
3. Semi-Markov Case
3.1. Mathematical Model
The particle movement is given by the equation
( )
0
0
(( ))
t
xt xCψs ds.= +
∫
(24)
where 0
[,]
x ab∈ is the particle starting point inside the
two reflecting boundaries
ab<
, and
()ψs
is an alter-
nating semi-Markov process with phase space E = {1,2}
and embedded transition probability matrix P given in
(1). The sojourn time at state is a random variable
with a common cumulative distribution function (cdf)
( )
,
i
Gt i∈E
. We assume that
( )
1
Gt
and
( )
2
Gt
are
not degenerated, and that their probability density func-
tion (pdf) and first moment, say
()
()
i
i
dG t
gt dt =
and
0
()
ii
mtgtdt
∞
=∫
respectively, exist.
Now, the hazard rates are given by
( )
()
() 1
i
i
i
gt
rt Gt =−
, and
assume
( )
1
10Cv= >
and
( )
2
20Cv= <
.
Define
( )
sup{0:()( )}τt:=tutψuψt−≤≤≠
and con-
sider the three-component process
( )( )( )
(, ,χtτt xt=
( )
)ψt
on the phase space
[
)
[ ]
0 ,,{1,2}ab=∞××W
. It
is well-known that
( )
χt
is a Markov process with the
following infinitesimal operator [8,9]
( )( )
( )()()( )
,, ,,
0,,,, (,),,
i
Aφτxiφτxi τ
rτPφxiφτbiC xiφτxi x
∂
= +
∂
∂
−+ 

∂
(25)
with boundary conditions say
(,, 2)φτb
τ
∂=
∂
( ,,1)φτa
τ
∂
∂
0=
and
( )
,,τxi∈W
. The function
( )
,,φτxi
is conti-
nuously differentiable on
τ
and
x
. We also have that
() ()
0, ,10, ,2Pφxφx=
and
( )
0,,2Pφx=
( )
0, ,1φx.
3.2. Stationary Distribution
Denote by
( )
ρ⋅
the stationary distribution of the sto-
chastic process
( )
xt.
This stationary distribution has
atoms at points
( )
, ,2τa
and
( )
, ,1τb
, and we denote
them as
[,, 2]ρτa
and
[ ,,1]ρτb
, respectively. The con-
tinuous part of
ρ
is denoted as
( )
,,ρτxi
,
i.∈E
For any function
( )
φ⋅
belonging to the domain of the
operator
A
we have

A. A. POGORUI ET AL. 27
( )()
w
0Aφzρdz. =
∫
(26)
By changing the order of integration (integration by
parts), we obtain expressions for
*
A ρ
, where
*
A
is the
adjoint operator of
A
, namely
()() ()
,,,,,,0,1,2,
ii
ρτxirρτxi vρτxii
τx
∂∂
++ ==
∂∂
(27)
and
( )()()
0
, ,0, ,;,,1,2,
i
rτρτxi dτρx jiji j
∞
= ≠=
∫
(28)
with the limiting behavior
( )
, ,0,ρxi+∞ =
for all
[ ]
,x ab∈
.
For the atoms we have
[ ] ( ) [ ]
( )
22
, ,2, ,2,,20ρτarτρτavρτa
τ
+
∂+ +=
∂
(29)
[ ] ( ) [ ]
()
11
, ,1, ,1,,10ρτbrτρτbvρτb
τ
−
∂+ −=
∂
(30)
where
( )
( )
,,:lim, ,
xb
ρτbiρτxi
−
↑
=
and
( )
( )
,,:lim, ,,
xa
ρτaiρτxi
+
↓
=for
1, 2i.=
We also have
[] [][]
, ,20, ,2,,1ρaρaρb+∞== +∞=
[ ]
0, ,10ρb=
Now, by taking into account boundary conditions we
have
( )
[ ]
12
00
,,1( ,,, 2)rτρτbdτvρτbdτ
∞∞
−
= −∫∫
(31)
and
( )
[ ]
21
00
,,2(,,1)rτρτad τvρτadτ.
∞∞
+
=
∫∫
(32)
By solving (27) we obtain
( ) ( )
()
0
(,,)exp,1, 2,
τ
ii i
ρτxif xvτrtdti=−− =
∫
(33)
where
1
i
f∈
.
By substituting (33) into (28) and by noting that
( )
()
( )
0
exp 1
τ
ii
rtdtGτ−=−
∫
we obtain
( ) ( )()
0
,,,1, 2
i iij
fxvτgτdτfxiji j.
∞
−=≠=
∫
(34)
It follows from (34) that
()()( )()
1 212
00 ii
fxvτvt gτg tdτdtfx .
∞∞ −− =
∫∫
(35)
From (34) and (35) we can assume that the functions
( )
i
fx
are of the form
( )
λ,1,2 x
ii
f xcei.= =
(36)
Now, by substituting (36) into (35), we obtain
() ()
1 122
ˆˆ 1=gλvgλv
(37)
where
()( )
0
ˆ
st
ii
gsgtedt
∞−
=∫
is the Laplace transform of
( )
,1, 2
i
gti .=
The set of pdf’s for which (37) exists is
similar to the set of functions which satisfies the Cramér
condition.
Lemma 3.2.1 If
112 20,vm vm+≠
where 1
m=
( )
0
,
i
tgtdt
∞
∫
and there exist
12121
, ,0ppσ<< >
and
20σ> such that
( )
[ ]
( )
1112 2
, ,,gtσt gt≥∈
2,σ≥
[ ]
12
,t pp∈
and
( )
11 2211 21
0 ,0v vpv vp< +<−+
.
Then, there exists
0
0λ≠
which satisfies (37).
Proof Let us define
( )()()
1 122
ˆˆ ,pλgλvgλv=
so
( )()
112 2
00pvmvm.
′=−+ ≠
Now, suppose
( )()
112 2
00pvmvm.
′=−+ <
then
()( )( )
22
12
11
12
,
p
vλtvλt
p
pλegt dtegt dt
−−
≥
∫∫


hence
( )
( )()
1112212 2
12
12
asλ.
vλvλvλpvλp
σσ
pλeeee
vv
−− − −
≥− −
→ +∞→+∞

The case
( )()
112 2
00pvm vm.
′=−+ >
can be reduced
to the previous one by assuming
st= −
and using
122 2
0v vp<+
.
Theorem 3.2.1
A) If
112 20vm vm+≠
and
0
0λ≠
is the solution for
(37), and
( )
( )
01
1
0
1
λvτ
eGτdτ
∞−
−<+∞
∫
, then there exists a
stationary distribution of
( )
xt
with the following con-
tinuous part:
( )
( )
( )
( )
01
11
, ,11,
λxvτ
ρτx ceGτ
−
= −
(38)
( ) ( )
( )
( )
( )
02
1 1012
ˆ
, ,21
−
= −
λxvτ
ρτx cgλveGτ
(39)
and atoms
( ) ( )
01
0
11 1
01
1
[ ,,1]1,
λvτ
λbe
ρτbcveGτλv
−
−
= −
(40)
( )
( )
( )
( )
02
0
12 1012
02
1
ˆ
, ,21
−
−
= −
λvτ
λa
e
ρτxcv gλve Gτ.λv
(41)
The normalization factor c1 can be calculated from
( )
w
1ρdz .=
∫
B) If
112 2
0vm vm+=
and there exists the second
moment
( )
2
0i
t gt dt
∞
∫
,
i∈E
, then the stationary measure
of
( )
xt
is as follows

28 A. A. POGORUI ET AL.
()( )
( )
()()
( )
21 22
,,11, ,,21ρτxcGτρτx cGτ=−=−
(42)
with atoms
[ ] ( )
( )
[ ] ( )
( )
21122 2
, ,11,, ,21ρτb cvτGτρτa cvτGτ=−=−
(43)
whe re
( )
1
(2) (2)
12
2 1212
() 22
mm
cm m bavv.
−

= +−+−


Proof It is easy to see that
( )
0
λ
11
x
f xce= and
( )
( )
0
21101
ˆ
=
λx
f xcgλve
satisfy (34). Substituting these
functions fi into (33) we obtain (38) and (39). Therefore
we substitute (38) and (39) into (29) and (30), then by
solving these equations we obtain (40) and (41).
It can be easily verified that if
112 2
0vm vm+≠
then
the value 00,λ≠ such that
() ()
1 01202
ˆˆ 1,=gλvgλv
also
satisfies (31) and (32).
Similarly, for
112 2
0vm vm+=
we obtain (42) and (43)
in the same manner as for the case
112 2
0vm vm+≠
when it is considered that
0
0λ.=
We should notice that the stationary measure of the
particle position
( )
xt
is determined by the following
relations
( )()()
( )
( )
0
,,1,,2,for ,,ρxρτxρτxdτx ab
∞
=+∈
∫
(44)
[][][][]
00
,2,,2,,1, ,1ρaρτadτρbρτbdτ.
∞∞
= =
∫∫
(45)
Example Markov Case
Suppose
( )
,0,1, 2;0
i
λt
i ii
gtλeλit.
−
=>= ≥
The n,
() ()
12
1 012 02
1 01202
ˆˆ 1
λλ
gλvgλv.
λλvλλv
 
= =
 
−−
 
Therefore,
12
0
12
,
λλ
λvv = +
and this case is the same as
the one in the first part of this paper.
Example Erlang Case
Let
( )( )
1
2
1 121
,,0, 0,
λtpt
gtλegtpteλp
−−
==>>
and
0t.≥
Then,
()()
2
1
1 012 02
1 0102
ˆˆ 1,
  
= =
  
−−
  
λp
gλvgλvλλvpλv
(46)
where we have the conditions
1 0102
an d λλvpλv.>>
Now, by solving (46) and taking into account the pre-
vious conditions, we obtain a unique solution for (46)
()( )
2
12112 11
0
12
24
2
λvpvv vpλpλ
λvv
− ++−+
=−
Since
( )( )
2
2 11
0, ,,
1
pt
rtpas tand rtλ
pt
=→>→+∞=
+
then the Theorem 3.2.1 is applicable.
4. Conclusions
The two-state continuous time random walk has been
studied by many researchers for the Markov case and
only a few have studied for non-Markovian processes
[10]. This basic model has many applications in physics,
biology, chemistry, and engineering. Most of the former
models were oriented to solve the boundary-free particle
motion. Recently this basic model has been extended in
several directions, such as two and three dimensions,
with reflecting and absorbing boundaries. Only a few of
these works consider partly reflecting boundaries [5,10],
and references therein. However, in none of these pre-
vious works a stationary distribution for the particle po-
sition is presented, as we did in this paper. We have in-
cluded the Markov case since it is illustrative and it mo-
tivates our analysis of the semi-Markov process.
5. Referen ces
[1] S. Goldstein, “On Diffusion by Discontinuous Move-
ments and on the Telegraph Equation,” The Quarterly
Journal of Mechanics and Applied Math ematics, Vol. 4,
No. 2, 1951, pp. 129-156.
[2] M. Kac, “A Stochastic Model Related to the Telegraph-
er’s Equation,” Rocky Mountain Journal of Mathematics,
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