Stationary Distribution of Random Motion with Delay in Reflecting Boundaries*

In this paper we study a continuous time random walk in the line with two boundaries [a,b], a < b. The particle can move in any of two directions with different velocities v 1 and v 2. 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.


Introduction
In this paper we study the stationary distribution of a one-dimensional random motion performed with two velocities, where the random times separating consecutive velocity changes perform an alternating Markov process.The sojourn times of this process are exponentially distributed random variables.There are many papers on random motion devoted to analysis of models in which motions are driven by a homogeneous Poisson process [1][2][3][4], however we have not found any paper investigating the stationary distribution of these processes.
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 0 v > or 2 0 v < Starting at the position 0 x ∈  the particle continues its motion with velocity 1 0  v > 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 0 v < during random time 2 τ , where 2 τ is an exponential distributed random variable with parameter 2 λ .Furthermore, the particle moves with velocity 1 0 v > 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 reflecting (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.

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 0 1 Denote by {x(t); t ≥ 0} the position of the particle at time t.Consider the function ( )  C x on the space E which is defined as *We thank ITESM through the Research Chair in Telecommunications.
Copyright © 2010 SciRes.AM ( ) The position of the particle at any time t can be expressed as where the starting point 0 [ , ].
x a b ∈ 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 x .φ φ =

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] πb respectively.The continuous part of π is denoted as ( ) Since π is the stationary distribution of ( ) ζt then for any function ( ) φ ⋅ from the domain of the operator A we have Now, let * A be the conjugate or adjoint operator of A. Then by changing the order of integration in (5) (integrating by parts), we can obtain the following expressions for the continuous part of 0 Similarly, from (5) we obtain ( ) It follows from the set ( 6) that ( ) ( ) By using ( 7), we get By obtaining ( ) , 2 πx from ( 8) and substituting such a result into the first equation in the set (6) we have Solving ( 9) we obtain for the continuous part of π ( ) , 2 , where Now, from (7) we obtain for atoms The factor C can be calculated from the normaliza- tion equation ( ) It follows from (15) that We should notice that the stationary distribution ( ) x π of the process ( ) x t over the interval ( , ) )( ) where Therefore, the stationary distribution can be expressed as ( ) ( ) ( ) ,1 and , 2 , with the atom [ ]

Mathematical Model
The particle movement is given by the equation where 0 [ , ] x a b ∈ is the particle starting point inside the two reflecting boundaries a b < , and ( ) ψs is an alternating 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 G t i∈ E .We assume that ( ) Now, the hazard rates are given by ( ) , and assume ( ) with boundary conditions say ( , , 2) φτb τ nuously differentiable on τ and x .We also have that

Stationary Distribution
Denote by ( ) ρ ⋅ the stationary distribution of the sto- chastic process ( ) By changing the order of integration (integration by parts), we obtain expressions for * A ρ , where * A is the adjoint operator of A , namely with the limiting behavior ( ) For the atoms we have , , 2 , , 2 , , 2 0 ρτa r τρτa v ρτa τ , , : lim , , Now, by taking into account boundary conditions we have By solving ( where By substituting (33) into (28) and by noting that It follows from (34) that From ( 34) and (35) we can assume that the functions Now, by substituting (36) into (35), we obtain ( ) ( ) where ( ) ( ) is the Laplace transform of ( ), ( ) [ ] ( ) , t p p ∈ and ( ) Then, there exists 0 0 λ ≠ which satisfies (37).

∫
, then there exists a stationary distribution of ( ) x t with the following con- tinuous part: and atoms ( ) ( ) The normalization factor c 1 can be calculated from ( )

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 previous works a stationary distribution for the particle position is presented, as we did in this paper.We have included the Markov case since it is illustrative and it motivates our analysis of the semi-Markov process.
In this case we can observe that ( ) x. Hence, the continuous part of the stationary distribution of the process ( )x t is uniform over the open interval ( , ) a b .Now, the factor C , say B C not degenerated, and that their probability density function (pdf) and first moment, say ( and this case is the same as the one in the first part of this paper.