Multidimensional random motion with uniformly distributed changes of direction and Erlang steps
In this paper we study transport processes in Rⁿ, n≥1, having non-exponential distributed sojourn times or non-Markovian step durations. We use the idea that the probabilistic properties of a random vector are completely determined by those of its projection on a fixed line, and using this idea we a...
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irk-123456789-1660332020-02-19T01:25:39Z Multidimensional random motion with uniformly distributed changes of direction and Erlang steps Pogorui, A.O. Rodriguez-Dagnino, R.M. Короткі повідомлення In this paper we study transport processes in Rⁿ, n≥1, having non-exponential distributed sojourn times or non-Markovian step durations. We use the idea that the probabilistic properties of a random vector are completely determined by those of its projection on a fixed line, and using this idea we avoid many of the difficulties appearing in the analysis of these problems in higher dimensions. As a particular case, we find the probability density function in three dimensions for 2-Erlang distributed sojourn times. Дослiджено процеси переносу в Rⁿ, n≥1, що мають неекспоненцiально розподiлений час перебування або немарковську тривалiсть крокiв. Використано iдею про те, що ймовiрнiснi властивостi випадкового вектора цiлком визначаються такими самими властивостями його проекцiй на фiксовану пряму. Цей пiдхiд дозволив уникнути багатьох складностей, що з’являються при дослiдженнi цих проблем у вимiрностях вищого порядку. Як окремий випадок, знайдено функцiю щiльностi ймовiрностi у тривимiрному випадку для часу перебування з 2-розподiлом Ерланга. 2011 Article Multidimensional random motion with uniformly distributed changes of direction and Erlang steps / A.О. Pogorui, R.M. Rodriguez-Dagnino // Український математичний журнал. — 2011. — Т. 63, № 4. — С. 572–577. — Бібліогр.: 10 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/166033 519.21 en Український математичний журнал Інститут математики НАН України |
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Короткі повідомлення Короткі повідомлення Pogorui, A.O. Rodriguez-Dagnino, R.M. Multidimensional random motion with uniformly distributed changes of direction and Erlang steps Український математичний журнал |
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In this paper we study transport processes in Rⁿ, n≥1, having non-exponential distributed sojourn times or non-Markovian step durations. We use the idea that the probabilistic properties of a random vector are completely determined by those of its projection on a fixed line, and using this idea we avoid many of the difficulties appearing in the analysis of these problems in higher dimensions. As a particular case, we find the probability density function in three dimensions for 2-Erlang distributed sojourn times. |
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Article |
| author |
Pogorui, A.O. Rodriguez-Dagnino, R.M. |
| author_facet |
Pogorui, A.O. Rodriguez-Dagnino, R.M. |
| author_sort |
Pogorui, A.O. |
| title |
Multidimensional random motion with uniformly distributed changes of direction and Erlang steps |
| title_short |
Multidimensional random motion with uniformly distributed changes of direction and Erlang steps |
| title_full |
Multidimensional random motion with uniformly distributed changes of direction and Erlang steps |
| title_fullStr |
Multidimensional random motion with uniformly distributed changes of direction and Erlang steps |
| title_full_unstemmed |
Multidimensional random motion with uniformly distributed changes of direction and Erlang steps |
| title_sort |
multidimensional random motion with uniformly distributed changes of direction and erlang steps |
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Інститут математики НАН України |
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2011 |
| topic_facet |
Короткі повідомлення |
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http://dspace.nbuv.gov.ua/handle/123456789/166033 |
| citation_txt |
Multidimensional random motion with uniformly distributed changes of direction and Erlang steps / A.О. Pogorui, R.M. Rodriguez-Dagnino // Український математичний журнал. — 2011. — Т. 63, № 4. — С. 572–577. — Бібліогр.: 10 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT pogoruiao multidimensionalrandommotionwithuniformlydistributedchangesofdirectionanderlangsteps AT rodriguezdagninorm multidimensionalrandommotionwithuniformlydistributedchangesofdirectionanderlangsteps |
| first_indexed |
2025-07-14T20:31:13Z |
| last_indexed |
2025-07-14T20:31:13Z |
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1837655732838727680 |
| fulltext |
UDC 519.21
A. A. Pogorui (Zhytomyr State Univ., Ukraine),
R. M. Rodrı́guez-Dagnino (Monterrey Inst. Technol., Mexico)
MULTIDIMENSIONAL RANDOM MOTION
WITH UNIFORMLY DISTRIBUTED CHANGES
OF DIRECTION AND ERLANG STEPS
БАГАТОВИМIРНИЙ ВИПАДКОВИЙ РУХ
IЗ РIВНОМIРНО РОЗПОДIЛЕНИМИ ЗМIНАМИ НАПРЯМКУ
ТА КРОКАМИ ЕРЛАНГА
In this paper we study transport processes in Rn, n ≥ 1, having non-exponential distributed sojourn times
or non-Markovian step durations. We use the idea that the probabilistic properties of a random vector are
completely determined by those of its projection on a fixed line, and using this idea we avoid many of the
difficulties appearing in the analysis of these problems in higher dimensions. As a particular case, we find the
probability density function in three dimensions for 2-Erlang distributed sojourn times.
Дослiджено процеси переносу в Rn, n ≥ 1, що мають неекспоненцiально розподiлений час перебування
або немарковську тривалiсть крокiв. Використано iдею про те, що ймовiрнiснi властивостi випадкового
вектора цiлком визначаються такими самими властивостями його проекцiй на фiксовану пряму. Цей
пiдхiд дозволив уникнути багатьох складностей, що з’являються при дослiдженнi цих проблем у вимiр-
ностях вищого порядку. Як окремий випадок, знайдено функцiю щiльностi ймовiрностi у тривимiрному
випадку для часу перебування з 2-розподiлом Ерланга.
1. Introduction. One-dimensional non-Markovian generalizations of the telegrapher’s
random process were obtained in [1, 2] with velocities alternating at Erlang-distributed
sojourn times. Uniformly distributed direction of motion or isotropic motion has been
studied by Pinsky [3] for transport processes on Riemannian manifold and by Orsingher
and De Gregorio in higher dimensions [4]. However, most of the papers on multidimen-
sional random motion are devoted to analysis of models in which motions are driven by
a homogeneous Poisson process (see [3 – 6] and references therein). The recent work of
Le Caer [7] departs from this trend since he is studying uniformly distributed orientation
random motion with Pearson – Dirichlet distributed steps in a multidimensional random
walk setting. In this work, we consider random motions with uniformly distributed di-
rections on the multidimensional space Rn, n ≥ 1, with Erlang distributed step lengths.
Our analysis is based on random evolutions on a semi-Markov media.
Let us consider the renewal process ξ(t) = max{m ≥ 0: τm ≤ t}, t ≥ 0, where
τm =
∑m
k=1
θk, τ0 = 0 and θk ≥ 0, k = 1, 2, ..., are i.i.d. random variables with a
distribution function G(t) such that there exists the probability density function (pdf)
g(t) =
d
dt
G(t).
We assume that a particle starting from the coordinate origin (0, 0, . . . , 0) of the
space Rn, at time t = 0, continues its motion with a constant absolute velocity v along
the direction η
(n)
0 , where η
(n)
0 = (x1, x2, . . . , xn) is a random n-dimensional vector
uniformly distributed on the unit sphere Ωn−1
1 = {(x1, x2, . . . , xn) : x2
1+x2
2+. . .+x2
n =
= 1}.
At instant τ1 the particle changes its direction to η
(n)
1 , where η
(n)
0 and η
(n)
1 are i.i.d.
random vectors on Ωn−1
1 . Then, at instant τ2 the particle changes its direction to η
(n)
2 ,
c© A. A. POGORUI, R. M. RODRÍGUEZ-DAGNINO, 2011
572 ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 4
MULTIDIMENSIONAL RANDOM MOTION WITH UNIFORMLY DISTRIBUTED CHANGES . . . 573
where η
(n)
2 is also uniformly distributed on Ωn−1
1 and independent of η
(n)
0 and η
(n)
1 ,
and so on.
Denote by x(n)(t), t ≥ 0, the particle position at time t. We have that
x(n)(t) = v
ξ(t)∑
i=0
η
(n)
i θi + v η
(n)
ξ(t) (t− τξ(t)). (1)
Basically, Eq. (1) determines the random evolution in the semi-Markov medium ξ(t).
Lemma 1. The probability distribution of the random vector x(n)(t) is determined
by the probability distribution of its projection x(n)(t) = v
∑ξ(t)
i=0
η
(n)
i θi + v η
(n)
ξ(t)(t −
− τξ(t)) on a fixed line, where η(n)
i is the projection of η(n)
i on the line.
Proof. Let us consider the cumulative distribution function (cdf) Fx(n)(t)(y) =
= P (x(n)(t) ≤ y). Then, the characteristic function ϕx(n)(t)(α) of x(n)(t) is given
by
ϕx(n)(t) = E exp
{
i
(
α,x(n)(t)
)}
= E exp
{
i ‖α‖
(
e,x(n)(t)
)}
=
= E exp
{
i ‖α‖x
′(n)(t)
}
=
∞∫
0
exp {i ‖α‖y} dFx(n)(t)(y),
where ‖α‖ =
√
α2
1 + α2
2 + . . .+ α2
n, x
(n)(t) is the projection of x(n)(t) onto the unit
vector e and it has a cdf Fx(n)(t)(y).
Lemma 1 is proved.
It is well known that if f(x1, x2, . . . , xn) ∈ L1(Rn) depends only on ‖x‖ =
=
√
x2
1 + x2
2 + . . .+ x2
n, i.e., f(x1, x2, . . . , xn) = g(r), then the function
ϕ(s1, s2, . . . , sn) =
∫
Rn
f(x) exp
{
−i
n∑
k=1
skxk
}
dx
depends only on s = ‖s‖. Such functions are called radial functions and for these
functions the Fourier transform in several variables goes over into the “Bessel transform”
in one variable as follows:
ϕ(s) =
(2π)n/2
s(n−2)/2
∞∫
0
g(r) rn/2J(n−2)/2(sr)dr,
where Jp(x) denotes the Bessel function, of the first kind, of order p [10].
Since ϕx(α) depends only on α = ‖α‖, meaning that ϕx(α) = ϕ(α) then the pdf
fx(t)(y) corresponding to the distribution
Fx(t)(y) = P
v ξ(t)+1∑
i=0
η
(n)
i θi + v η
(n)
ξ(t) (t− τξ(t)) ≤ y
depends only on r = ‖y‖, that is, fx(t)(y) = h(r) and we have
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 4
574 A. A. POGORUI, R. M. RODRÍGUEZ-DAGNINO
ϕx(t)(α) =
(2π)n/2
α(n−2)/2
∞∫
0
h(r) rn/2J(n−2)/2(αr)dr.
It also follows that if h(r) is continuous on [0,+∞) and
∫ ∞
0
rn−1h(r)dr <∞, and if∫ ∞
0
αn−1ϕ(α)dα <∞, then [10]
fx(t)(y) = h(r) =
1
(2π)n/2 r(n−2)/2
∞∫
0
ϕ(α)αn/2J(n−2)/2(αr)dα.
Now, let us define x̂(n)(t) = v
∑ξ(t)
i=0
η
(n)
i θi and ∆(t) = v η
(n)
ξ(t) (t − τξ(t)), and
we will denote as Fx̂(n)(t)(y) (resp. F∆(t)(y)) the cdf of x̂(n)(t) (resp. ∆(t)). It is
easy to verify that x̂(n)(t) and ∆(t) are independent. Hence, we have Fx(n)(t)(y) =
= Fx̂(n)(t)(y) ∗ F∆(t)(y).
Therefore, by using Lemma 1 we can study the cdf of x(n)(t) but we need to know
the cdf of x̂(n)(t) and ∆(t).
Lemma 2. Let Fn(t) be the cdf of η(n)
i θi and it is of the following form:
Fn(t) =
1
2
+
Γ
(
n
2
)
√
π Γ
(
n−1
2
) 1∫
0
G
(
t
x
)
(1− x2)(n−3)/2dx, if t ≥ 0,
1
2
−
Γ
(
n
2
)
√
π Γ
(
n−1
2
) 1∫
0
G
(
− t
x
)
(1− x2)(n−3)/2dx, if t < 0.
(2)
Proof. Let us denote by fηi(x) the pdf of the projection η(n)
i of the vector η(n)
i onto
a fixed line. It is showed in [8] that fηi(x) is of the following form:
f
η
(n)
i
(x) =
Γ
(
n
2
)
√
π Γ
(
n−1
2
) (1− x2)(n−3)/2, if x ∈ [−1, 1],
0, if x /∈ [−1, 1].
(3)
Since η(n)
i and θi are independent it is easy to verify that the cdf of η(n)
i θi is of the form
(2).
Lemma 2 is proved.
The process γ(t) = t− τξ(t) is a Markov process and it has the following generator
operator A [9]:
Aϕ(s) = ϕ′(s) +
g(s)
1−G(s)
(ϕ(0)− ϕ(s)) , s ≥ 0,
where ϕ ∈ C1(R).
Lemma 3. The cdf F∆(t)(s) = P
(
v η
(n)
ξ(t) (t− τξ(t)) ≤ s
)
is given by
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 4
MULTIDIMENSIONAL RANDOM MOTION WITH UNIFORMLY DISTRIBUTED CHANGES . . . 575
F∆(t)(s) =
1
2
+
Γ
(
n
2
)
√
π Γ
(
n−1
2
) 1∫
0
Fγ(t)
( s
vx
)
(1− x2)(n−3)/2dx, if s ≥ 0,
1
2
−
Γ
(
n
2
)
√
π Γ
(
n−1
2
) 1∫
0
Fγ(t)
(
− s
vx
)
(1− x2)(n−3)/2dx, if s < 0.
Proof. The cdf Fγ(t)(u) = P(γ(t) ≤ u) satisfies the following Markov renewal
equation [9]
Fγ(t)(u) = V (t, u) +
t∫
0
g(s)Fγ(t−s)(u)ds, (4)
where V (t, u) = P(γ(t) ≤ u, τ1 > t) = (1−G(t)) I{t≤u}.
Let us define the function R(t) =
∑∞
k=0
g∗(k)(t), where the symbol ∗(n) denotes
the k-fold convolution of g(t) with itself. Then, Eq. (4) can be rewritten as
Fγ(t)(u) = (V ∗R)(t, u) =
t∫
0
V (t− s, u)dR(s).
Since v η(n)
ξ(t) and γ(t) are independent that concludes the proof.
2. Evolution in odd-dimensional spaces. Now, let us assume that n = 2l + 3, l =
= 0, 1, 2, . . . , and θk has a (n−1)-Erlang distribution, that is g(t) =
λn−1
Γ(n− 1)
tn−2e−λt.
It follows from Lemma 2 that the pdf fn(t) of the random variable η(n)
i θi has the form
fn(t) =
Γ
(
n
2
)
√
π Γ
(
n−1
2
)
Γ(n− 1)
λ
1∫
0
(λt)2l+1
x2l+2
e−λt/x(1− x2)ldx
or equivalently,
fn(t) =
λΓ
(
l + 3
2
)
√
π Γ(l + 1)Γ(2l + 2)
l∑
k=0
(
l
k
)
(−1)k(λt)2k
∞∫
λt
s2(l−k)e−sds,
for t ≥ 0. Furthermore, the following equivalent expression can be found after some
algebraic simplifications
fn(t) =
λe−λt
l! 22l+1
l∑
k=0
(−1)k
(2(l − k)!)
k!(l − k)!
2(l−k)∑
m=0
(λt)2k+m
m!
. (5)
We have fn(t) = fn(−t) for the case when t < 0.
3. Evolution in three dimensions. Let us consider the particular case when n = 3.
Thus, by taking into account Lemma 2, we have that η(3)
i is uniformly distributed on
[−1, 1].
Let random variables θk, k = 0, 1, 2, . . . , be 2-Erlang distributed, i.e., g(t) =
= λ2te−λt, λ > 0, t ≥ 0.
For this particular case, the Laplace transform of R(t), say R̂(s), is of the form
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 4
576 A. A. POGORUI, R. M. RODRÍGUEZ-DAGNINO
R̂(s) =
∞∫
0
R(t)e−stdt =
∞∑
k=0
∞∫
0
g∗(k)(t)e−stdt =
∞∑
k=0
(
λ
λ+ s
)2k
=
(λ+ s)2
s2 + 2λs
,
and the Laplace transform V̂ (s, u) of V (t, u) can be written as
V̂ (s, u) =
2λ+ s− (λus+ λ2us+ s+ 2)e−(λ+s)u
(λ+ s)2
.
Therefore, the Laplace transform F̂γ(s, u) of Fγ(t)(u) is given by
F̂γ(s, u) = R̂(s)V̂ (s, u) =
2λ+ s− (λus+ λ2us+ s+ 2)e−(λ+s)u
s(s+ 2λ)
. (6)
After applying the inverse Laplace transform to F̂γ(s, u), we obtain for t > u > 0
Fγ(t)(u) = e−2λt−(2+λu)e−λt sinh (λ(t− u))+2e−λt sinh(λt)−(λu+1)e−λ(2t−u).
Thus, we have the limit result
lim
t→+∞
Fγ(t)(u) = 1− e−λu − λu
2
e−λu.
Taking into account Lemma 3 we can obtain the corresponding expression for
F∆(t)(s).
It follows from Eq. (5) that ηiθi has the Laplace distribution with pdf f3(t) =
=
1
2
λe−λ|t|.
Therefore, the Fourier transform of P
(
v
∑k
i=0
ηiθi ≤ y
)
is given by
∞∫
−∞
e−iλydP
(
v
k∑
i=0
ηiθi ≤ y
)
=
(
λ2
λ2 + v2α2
)k
.
On the other hand, since
Fx̂(3)(t)(y) = P
v ξ(t)∑
i=0
η
(3)
i θi ≤ y
=
∞∑
k=0
P
(
v
k∑
i=0
η
(3)
i θi ≤ y
)
P(ξ(t) = k)
then the characteristic function of x̂(3)(t), say,
ϕx̂(3)(t)(α) = E[e−iαx̂
(3)(t)] =
∞∫
−∞
e−iαydFx̂(3)(t)(y)
can be calculated as follows
ϕx̂(3)(t)(α) =
∞∑
k=0
(
λ2
λ2 + v2α2
)k
P(ξ(t) = k) =
= e−λt
∞∑
k=0
(
λ2
λ2 + v2α2
)k (
(λt)2k
2k!
+
(λt)2k+1
(2k + 1)!
)
.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 4
Let us define Φ =
λ2
√
λ2 + v2α2
, then
ϕx̂(3)(t)(α) = e−λt
[
cosh Φt+
λ2 + v2α2
λ2
sinh Φt
]
.
Therefore, by using the inverse Fourier transform, we can obtain Fx̂(3)(t)(y).
1. Di Crescenzo A. On random motions with velocities alternating at Erlang-distributed random times //
Adv. Appl. Probab. – 2001. – 61. – P. 690 – 701.
2. Pogorui A. A., Rodriguez-Dagnino R. M. One-dimensional semi-Markov evolutions with general Erlang
sojourn times // Random Operators and Stochast. Equat. – 2005. – 13. – P. 1720 – 1724.
3. Pinsky M. Isotropic transport process on a Riemann manifold // Trans. Amer. Math. Soc. – 1976. – 218.
– P. 353 – 360.
4. Orsingher E., De Gregorio A. Random flights in higher spaces // J. Theor. Probab. – 2007. – 20. –
P. 769 – 806.
5. Kolesnik A. D. Random motions at finite speed in higher dimensions // J. Statist. Phys. – 2008. – 131. –
P. 1039 – 1065.
6. Stadje W. Exact solution for non-correlated random walk models // Ibid. – 1989. – 56. – P. 415 – 435.
7. Le Caer G. A Pearson – Dirichlet random walk // Ibid. – 2010. – 40. – P. 728 – 751.
8. Pogorui A. A. Fading evolution in multidimensional spaces // Ukr. Math. J. – 2010. – 62, № 11. –
P. 1577 – 1582.
9. Korolyuk V. S., Limnios N. Stochastic systems in merging phase space. – World Sci. Publ., 2005.
10. Bochner S., Chandrasekharan K. Fourier transforms // Ann. Math. Stud. – 1949. – № 19.
Received 04.11.10
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