 Applied Mathematics, 2010, 1, 24-28 doi:10.4236/am.2010.11004 Published Online May 2010 (http://www.SciRP.org/journal/am) Copyright © 2010 SciRes. 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. E-mail: {pogorui, rmrod rig }@itesm.mx 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 10v> or 2 0v< Starting at the position 0x∈ the particle continues its motion with velocity 10v> during random time1τ, where 1τ is an exponential random variable with para-meter 1λ,then the particle moves with velocity 2 0v< during random time 2τ, where 2τ is an exponential distributed random variable with parameter 2λ. Fur-thermore, the particle moves with velocity 10v> 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 , 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 1τ and 2τ 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 iτ correspond-ing to the state i∈E, and transition probability matrix of the embedded Markov chain 0110P.= (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 Copyright © 2010 SciRes. AM ( )12if 1if 2vxCx v x.=== (2) The position of the particle at any time t can be ex-pressed as ( )()( )00,txt 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,idAxiC xixiλP xixiidxφφ φφ=+ −= (4) where ( )(),1 ,2Px xφφ=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 ()ζtleads up to the conclusion that the stationary distribution π has atoms at points (a, 2) and (b, 1), and we denote them as [,2]πaand [ ,1]πbrespectively. The continuous part of π is denoted as ( ),,πxii ∈E. Since π is the stationary distribution of ()ζtthen for any function ()φ⋅from the domain of the operator A we have ( )( )0A zdπzφ=∫ (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= ( )( )()( )( )()1 122 21,1,1,20,2,2,10+− =+ −=dvπxλπxλπxdxdvπxλπxλπx.dx (6) Similarly, from (5) we obtain ( )( )( )()( )( )( )( )11122221,1,10,1,2 0,2,2 0,2,1 0λπbvπbλπbvπbλπavπaλπavπa.−−++−=+=+=−= (7) where ( )( ),: lim,xbπbiπxi−↑=and ( ),:πai+=limxaπ↓ ( ),πxi for i=1,2. It follows from the set (6) that ( )()12,1, 20,ddvπxvπxdx dx+= or equivalently ( )()12,1,2constantvπxvπxk .+== By using (7), we get ( )1,1vπbvπb−+( )2, 20,vπb−= 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 ( )( )( )11 122,1,1,1 0−− =vdvπxλπxλπx.dx v (9) Solving (9) we obtain for the continuous part of π ( ),1μxπx Ce−= (10) And ( )12,2 ,μxvπx Cev−= − (11) where 1212λλμ.vv= + Now, from (7) we obtain for atoms 12[, 2]−=μavπa Ceλ (12) and 11[ ,1]−=μbvπbCe .λ (13) The factor C can be calculated from the normaliza-tion equation ( )1,πz dz=∫ (14) or equivalently ( )()[ ],1,2,2[,1]1bbaaπx dxπx dxπaπb.+++=∫∫ (15) It follows from (15) that 11 211 1212 2211−−− −−= −+−  μbμav vvvvvC ee.λμvλμv (16) 26 A. A. POGORUI ET AL. Copyright © 2010 SciRes. AM We should notice that the stationary distribution ()xπ of the process ()xt over the interval (,)ab is ( )xπ ( )(),1 ,2πxπx.= + 2.3. Balanced and One-Boundary Cases 2.3.1. Balanced Case Let us call the balanced case when 12120λλμ.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 BC, reduces to 221,( )()BvCv vbaδ=− −+ (17) where 1212vvδ.λλ== − Therefore, the stationary distribution can be expressed as ( )()12,1and, 2BBvπxCπxC. v== − (18) Thus, ( )()()1,1 ,2,xπxπxbaδ=+=−+π (19) and the atoms are given by [ ]12,2and [,1] BBvπaδCπbδC.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 OC, given by 2121 22()μaOveC.vvv vλμ=−− (21) Hence, ( )()12 ,1and ,2,μxμxOOvπx Ceπx Cev−−== − (22) with the atom [ ]1221 212,2 ()=−−vvπa.λvvvv μ (23) 3. Semi-Markov Case 3.1. Mathematical Model The particle movement is given by the equation ( )00(( ))txt xCψs ds.= +∫ (24) where 0[,]x ab∈ is the particle starting point inside the two reflecting boundariesab<, 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) ( ),iGt i∈E. We assume that ( )1Gt and ( )2Gt are not degenerated, and that their probability density func-tion (pdf) and first moment, say ()()iidG tgt dt= and 0()iimtgtdt∞=∫ respectively, exist. Now, the hazard rates are given by ( )()() 1iiigtrt Gt=−, and assume ( )110Cv= > and ( )220Cv= <. 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,,,, (,),,iAφτxiφτxi τrτPφxiφτbiC xiφτxi x∂= +∂∂−+∂ (25) with boundary conditions say (,, 2)φτbτ∂=∂( ,,1)φτaτ∂∂ 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 Copyright © 2010 SciRes. AM ( )()w0Aφzρdz. =∫ (26) By changing the order of integration (integration by parts), we obtain expressions for*Aρ, where *A is the adjoint operator of A, namely ()() (),,,,,,0,1,2,iiρτxirρτxi vρτxiiτx∂∂++ ==∂∂ (27) and ( )()()0, ,0, ,;,,1,2,irτρτxi dτρx jiji j∞= ≠=∫ (28) with the limiting behavior ( ), ,0,ρxi+∞ =for all [ ],x ab∈. For the atoms we have [ ]( )[ ]( )22, ,2, ,2,,20ρτarτρτavρτaτ+∂+ +=∂ (29) [ ]( )[ ]()11, ,1, ,1,,10ρτbrτρτbvρτbτ−∂+ −=∂ (30) where ( )( ),,:lim, ,xbρτbiρτxi−↑= and ( )( ),,:lim, ,,xaρτaiρτxi+↓=for 1, 2i.= We also have [] [][], ,20, ,2,,1ρaρaρb+∞== +∞=[ ]0, ,10ρb= Now, by taking into account boundary conditions we have ( )[ ]1200,,1( ,,, 2)rτρτbdτvρτbdτ∞∞−= −∫∫ (31) and ( )[ ]2100,,2(,,1)rτρτadτvρτadτ.∞∞+=∫∫ (32) By solving (27) we obtain ( )( )()0(,,)exp,1, 2,τii iρτxif xvτrtdti=−− =∫ (33) where 1if∈. By substituting (33) into (28) and by noting that ( )()( )0exp 1τiirtdtGτ−=−∫ we obtain ( )( )()0,,,1, 2i iijfxvτgτdτfxiji j.∞−=≠=∫ (34) It follows from (34) that ()()( )()1 21200 iifxvτvt gτg tdτdtfx .∞∞ −− =∫∫ (35) From (34) and (35) we can assume that the functions ( )ifx are of the form ( )λ,1,2xiif xcei.= = (36) Now, by substituting (36) into (35), we obtain () ()1 122ˆˆ 1=gλvgλv (37) where ()( )0ˆstiigsgtedt∞−=∫is the Laplace transform of ( ),1, 2igti .=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 1m= ( )0,itgtdt∞∫ and there exist 12121, ,0ppσ<< > and 20σ> such that ( )[ ]( )1112 2, ,,gtσt gt≥∈ 2,σ≥ [ ]12,t pp∈and ( )11 2211 210 ,0v vpv vp< +<−+. Then, there exists 00λ≠ which satisfies (37). Proof Let us define ( )()()1 122ˆˆ ,pλgλvgλv= so ( )()112 200pvmvm.′=−+ ≠ Now, suppose ( )()112 200pvmvm.′=−+ < then ()( )( )22121112,pvλtvλtppλegt dtegt dt−−≥∫∫ hence ( )( )()1112212 21212asλ.vλvλvλpvλpσσpλeeeevv−− − −≥− −→ +∞→+∞ The case ( )()112 200pvm vm.′=−+ > can be reduced to the previous one by assuming st= − and using 122 20v vp<+. Theorem 3.2.1 A) If 112 20vm vm+≠ and 00λ≠ is the solution for (37), and( )( )01101λvτeGτdτ∞−−<+∞∫, then there exists a stationary distribution of ( )xt with the following con-tinuous part: ( )( )( )( )0111, ,11,λxvτρτx ceGτ−= − (38) ( )( )( )( )( )021 1012ˆ, ,21−= −λxvτρτx cgλveGτ (39) and atoms ( )( )01011 1011[ ,,1]1,λvτλbeρτbcveGτλv−−= − (40) ( )( )( )( )02012 1012021ˆ, ,21−−= −λvτλaeρτxcv gλve Gτ.λv (41) The normalization factor c1 can be calculated from ( )w1ρdz .=∫ B) If 112 20vm vm+= and there exists the second moment ( )20it gt dt∞∫,i∈E, then the stationary measure of ( )xt is as follows 28 A. A. POGORUI ET AL. Copyright © 2010 SciRes. AM ()( )( )()()( )21 22,,11, ,,21ρτxcGτρτx cGτ=−=− (42) with atoms [ ]( )( )[ ]( )( )21122 2, ,11,, ,21ρτb cvτGτρτa cvτGτ=−=− (43) whe re ( )1(2) (2)122 1212() 22mmcm m bavv.−= +−+− Proof It is easy to see that ( )0λ11xf xce= and ( )( )021101ˆ=λxf 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 20vm vm+≠ then the value 00,λ≠ such that () ()1 01202ˆˆ 1,=gλvgλv also satisfies (31) and (32). Similarly, for 112 20vm vm+= we obtain (42) and (43) in the same manner as for the case 112 20vm vm+≠ when it is considered that 00λ.= 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;0iλti iigtλeλit.−=>= ≥ The n, () ()121 012 021 01202ˆˆ 1λλgλvgλv.λλvλλv = = −−  Therefore, 12012,λλλvv= + and this case is the same as the one in the first part of this paper. Example Erlang Case Let ( )( )121 121,,0, 0,λtptgtλegtpteλp−−==>> and 0t.≥ Then, ()()211 012 021 0102ˆˆ 1,  = =  −−  λpgλvgλvλλvpλv (46) where we have the conditions 1 0102an d λλvpλv.>> Now, by solving (46) and taking into account the pre-vious conditions, we obtain a unique solution for (46) ()( )212112 11012242λvpvv vpλpλλvv− ++−+=− Since( )( )22 110, ,,1ptrtpas tand rtλpt=→>→+∞=+ 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 . 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  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.  M. Kac, “A Stochastic Model Related to the Telegraph-er’s Equation,” Rocky Mountain Journal of Mathematics, Vol. 4, No. 3, 1974, pp. 497-509.  E. Orsingher, “Hyperbolic Equations Arising in Random Models,” Stochastic Processes and their Applications, Vol. 21, No. 1, 1985, pp. 93-106.  A. F. Turbin, “Mathematical Model of Einstein, Wiener, Levy,” in Russian, Fractal Analysis and Related Fields, Vol. 2, 1998, pp. 47-60.  J. Masoliver, J. M. 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