Multidimensional Random Motion with Uniformly Distributed Changes of Direction and Erlang Steps

Transcription

1 Multdmensonal Random Moton wth Unformly Dstrbuted Changes of Drecton and Erlang Steps Anatoly A. Pogoru a,, Ramón M. Rodríguez-Dagnno b,, a Department of Mathematcs, Zhytomyr State Unversty. b Electrcal and Computer Engneerng, Tecnológco de Monterrey. Abstract In ths paper we study transport processes n R n, n, havng non-exponental dstrbuted sojourn tmes or non-markovan step duratons. We use the dea that the probablstc propertes of a random vector are completely determned by those of ts projecton on a fxed lne, and usng ths dea we avod many of the dffcultes appearng n the analyss of these problems n hgher dmensons. As a partcular case, we fnd the probablty densty functon n three dmensons for -Erlang dstrbuted sojourn tmes. Keywords: Random Evolutons, sem-markov processes, Erlang dstrbutons MSC: 6K35, 6K99, 6K5. Introducton One-dmensonal non-markovan generalzatons of the telegrapher s random process were obtaned n [, ] wth veloctes alternatng at Erlang-dstrbuted sojourn tmes. Unformly dstrbuted drecton of moton or sotropc moton has been studed by Pnsky [3] for transport processes on Remannan manfold and by Orsngher and De Gregoro n hgher dmensons [4]. However, most of the papers on multdmensonal random moton are devoted to analyss of Correspondng author Emal addresses: pogor@zu.edu.ua Anatoly A. Pogoru, rmrodrg@tesm.mx Ramón M. Rodríguez-Dagnno Valyka Berdychvska St., 4, 8, Zhytomyr, Ukrane 8. Av. Eugeno Garza Sada 5 Sur, C.P , Monterrey, N.L., Méxco. Preprnt submtted to Elsever November,

2 models n whch motons are drven by a homogeneous Posson process see [3]- [6] and references theren. The recent work of Le Caer [7] departs from ths trend snce he s studyng unformly dstrbuted orentaton random moton wth Pearson-Drchlet dstrbuted steps n a multdmensonal random walk settng. In ths work, we consder random motons wth unformly dstrbuted drectons on the multdmensonal space R n, n, wth Erlang dstrbuted step lengths. Our analyss s based on random evolutons on a sem-markov meda. Let us consder the renewal process ξt = max{m : τ m t}, t, where τ m = m k= θ k, τ = and θ k, k =,,..., are..d. random varables wth a dstrbuton functon Gt such that there exsts the probablty densty functon pdf gt = d dt Gt. We assume that a partcle startng from the coordnate orgn,,..., of the space R n, at tme t =, contnues ts moton wth a constant absolute velocty v along the drecton η n, where ηn = x, x,, x n s a random n-dmensonal vector unformly dstrbuted on the unt sphere Ω n = {x, x,..., x n : x + x + + x n = }. At nstant τ the partcle changes ts drecton to η n, where ηn and η n are..d. random vectors on Ω n. Then, at nstant τ the partcle changes ts drecton to η n, where ηn s also unformly dstrbuted on Ω n and ndependent of η n and η n, and so on. Denote by x n t, t, the partcle poston at tme t. We have that ξt x n t = v η n θ + v η n ξt t τ ξt. Bascally, Eq. determnes the random evoluton n the sem-markov medum ξt. Lemma. The probablty dstrbuton of the random vector x n t s determned by the probablty dstrbuton of ts projecton x n t = v ξt ηn θ + v η n ξt t τ ξt on a fxed lne, where η n s the projecton of η n on the lne. Proof. Let us consder the cumulatve dstrbuton functon cdf F x n ty = P x n t y. Then, the characterstc functon ϕ x n tα of x n t s gven

3 by { } ϕ x n t = E exp α, x n t { } = E exp α x n t = { } = E exp α e, x n t exp { α y} df x n ty, where α = α + α + + α n, x n t s the projecton of x n t onto the unt vector e and t has a cdf F x n ty. It s well known that f fx, x,..., x n L R n depends only on x = x + x + + x n,.e., fx, x,..., x n = gr, then the functon { ϕs, s,..., s n = fx exp R n } n s k x k dx depends only on s = s. Such functons are called radal functons and for these functons the Fourer transform n several varables goes over nto the Bessel transform n one varable as follows: ϕs = πn/ s n / k= gr r n/ J n / srdr, where J p x denotes the Bessel functon, of the frst knd, of order p []. Snce ϕ x α depends only on α = α, meanng that ϕ x α = ϕα then the pdf f xt y correspondng to the dstrbuton F xt y = P v ξt+ η n θ + v η n ξt t τ ξt y depends only on r = y, that s, f xt y = hr and we have ϕ xt α = πn/ α n / hr r n/ J n / αrdr. It also follows that f hr s contnuous on [, + and r n hrdr <, and f α n ϕαdα <, then []: f xt y = hr = π n/ r n / Now, let us defne ˆx n t = v ξt ηn ϕα α n/ J n / αrdα. θ and t = v η n ξt t τ ξt, and we wll denote as Fˆx n ty resp. F t y the cdf of ˆx n t resp. t. 3

4 It s easy to verfy that ˆx n t and t are ndependent. F x n ty = Fˆx n ty F t y. Hence, we have Therefore, by usng Lemma we can study the cdf of x n t but we need to know the cdf of ˆx n t and t. Lemma. Let F n t be the cdf of η n θ and t s of the followng form n t n G x n 3/ dx, f t, + Γ F n t = Γ n n x G t x x n 3/ dx, f t <. Proof. Let us denote by f η x the pdf of the projecton η n η n of the vector onto a fxed lne. It s showed n [8] that f η x s of the followng form Γ n n x n 3/, f x [, ], f n η x = 3, f x / [, ]. Snce η n the form. and θ are ndependent t s easy to verfy that the cdf of η n θ s of The process γt = t τ ξt s a Markov process and t has the followng generator operator A [9] where ϕ C R. Aϕs = ϕ s + Lemma 3. The cdf F t s = P F t s = + Γ n n Γ n n gs ϕ ϕs, s, Gs v η n ξt t τ ξt s s gven by s F γt x n 3/ dx, f s, vx F γt s vx x n 3/ dx, f s <. 4

5 Proof. The cdf F γt u = P γt u satsfes the followng Markov renewal equaton [9] F γt u = V t, u + t where V t, u = P γt u, τ > t = Gt I {t u}. gsf γt s uds, 4 Let us defne the functon Rt = k= g k t, where the symbol n denotes the k-fold convoluton of gt wth tself. Then, Eq. 4 can be rewrtten as F γt u = V Rt, u = t V t s, udrs. Snce v η n ξt and γt are ndependent that concludes the proof.. Evoluton n odd-dmensonal spaces Now, let us assume that n = l+3, l =,,,... and θ k has a n -Erlang dstrbuton, that s gt = λn Γn tn e λt. It follows from Lemma that the pdf f n t of the random varable η n θ has the form f n t = or equvalently, n Γ n λ Γn λγ l + 3 f n t = l + Γl + l k= λt l+ x l+ e λt/x x l dx l k λt k s l k e s ds, k λt for t. Furthermore, the followng equvalent expresson can be found after some algebrac smplfcatons f n t = λe λt l! l+ l k l k! k!l k! k= We have f n t = f n t for the case when t <. l k m= λt k+m m! Evoluton n three dmensons Let us consder the partcular case when n = 3. Thus, by takng nto account Lemma, we have that η 3 s unformly dstrbuted on [, ]. 5

6 Let random varables θ k, k =,,,... be -Erlang dstrbuted,.e., gt = λ te λt, λ >, t. form For ths partcular case, the Laplace transform of Rt, say Rs, s of the Rs = Rte st dt = k= g k te st dt = k= and the Laplace transform V s, u of V t, u can be wrtten as u > V s, u = λ + s λus + λ us + s + e λ+su λ + s. k λ λ + s = λ + s s + λs, Therefore, the Laplace transform F γ s, u of F γt u s gven by F γ s, u = Rs V s, u = λ + s λus + λ us + s + e λ+su. 6 ss + λ After applyng the nverse Laplace transform to F γ s, u, we obtan for t > F γt u = e λt +λue λt snh λt u+e λt snhλt λu+e λt u. Thus, we have the lmt result lm F γtu = e λu λu t + e λu. Takng nto account Lemma 3 we can obtan the correspondng expresson for F t s. It follows from Eq. 5 that η θ has the Laplace dstrbuton wth pdf f 3 t = λe λ t. Therefore, the Fourer transform of P e λy dp v v k η θ y s gven by k λ k η θ y = λ + v α. On the other hand, snce ξt Fˆx ty = P v θ 3 y = P v η 3 k= k η 3 θ y P ξt = k 6

7 then the characterstc functon of ˆx 3 t, say, can be calculated as follows Let us defne Φ = ϕˆx 3 tα = E[e αˆx3 t ] = ϕˆx 3 tα = e αy dfˆx 3 ty λ k P ξt = k λ + v α k= = e λt λ k λt k λ + v α k! k= λ λ + v α, then ϕˆx 3 tα = e λt [ cosh Φt + λ + v α λ + λtk+ k +! ] snh Φt. Therefore, by usng the nverse Fourer transform, we can obtan Fˆx 3 ty. References [] A. D Crescenzo, On random motons wth veloctes alternatng at Erlangdstrbuted random tmes, Adv. Appl. Prob [] A.A. Pogoru, R.M. Rodrguez-Dagnno, One-dmensonal sem-markov evolutons wth general Erlang sojourn tmes, Random Operators and Stoch. Equat [3] M. Pnsky, Isotropc transport process on a Remann manfold, Trans. Amer. Math. Soc [4] E. Orsngher, A. De Gregoro, Random flghts n hgher spaces, J. Theoret. Prob [5] A.D. Kolesnk, Random motons at fnte speed n hgher dmensons, J. Stat. Phys [6] W. Stadje, Exact soluton for non-correlated random walk models, J. Stat. Phys

8 [7] G. Le Caer, A Pearson-Drchlet random walk, J. Stat. Phys [8] A.A. Pogoru, Fadng evoluton n multdmensonal spaces, Ukranan Mathematcal Journal n Ukranan. [9] V.S. Korolyuk, N. Lmnos, Stochastc Systems n Mergng Phase Space, World Scentfc Publshng, 5. [] S. Bochner, K. Chandrasekharan, Fourer Transforms, Annals of Mathematcs Studes, No. 9, Prnceton Unversty Press,