A limit theorem for symmetric Markovian random evolution in R^m
We consider the symmetric Markovian random evolution X(t) performed by a particle that moves with constant finite speed c in the Euclidean space R^m, m >= 2. Its motion is subject to the control of a homogeneous Poisson process of rate λ > 0. We show that, under the Kac condition c → ∞, λ →∞,...
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| Цитувати: | A limit theorem for symmetric Markovian random evolution in R^m / A.D. Kolesnik // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 1. — С. 69–75. — Бібліогр.: 15 назв.— англ. |
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irk-123456789-45372009-11-26T12:00:41Z A limit theorem for symmetric Markovian random evolution in R^m Kolesnik, A.D. We consider the symmetric Markovian random evolution X(t) performed by a particle that moves with constant finite speed c in the Euclidean space R^m, m >= 2. Its motion is subject to the control of a homogeneous Poisson process of rate λ > 0. We show that, under the Kac condition c → ∞, λ →∞, (c^2/λ) → ρ, ρ > 0, the transition density of X(t) converges to the transition density of the homogeneous Wiener process with zero drift and the diffusion coefficient σ^2 = 2ρ/m. 2008 Article A limit theorem for symmetric Markovian random evolution in R^m / A.D. Kolesnik // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 1. — С. 69–75. — Бібліогр.: 15 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4537 519.21 en Інститут математики НАН України |
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We consider the symmetric Markovian random evolution X(t) performed by a particle that moves with constant finite speed c in the Euclidean space R^m, m >= 2. Its motion is subject to the control of a homogeneous Poisson process of rate λ > 0. We show that, under the Kac condition c → ∞, λ →∞, (c^2/λ) → ρ, ρ > 0, the transition density of X(t) converges to the transition density of the homogeneous Wiener process with zero drift and the diffusion coefficient σ^2 = 2ρ/m. |
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Article |
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Kolesnik, A.D. |
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Kolesnik, A.D. A limit theorem for symmetric Markovian random evolution in R^m |
| author_facet |
Kolesnik, A.D. |
| author_sort |
Kolesnik, A.D. |
| title |
A limit theorem for symmetric Markovian random evolution in R^m |
| title_short |
A limit theorem for symmetric Markovian random evolution in R^m |
| title_full |
A limit theorem for symmetric Markovian random evolution in R^m |
| title_fullStr |
A limit theorem for symmetric Markovian random evolution in R^m |
| title_full_unstemmed |
A limit theorem for symmetric Markovian random evolution in R^m |
| title_sort |
limit theorem for symmetric markovian random evolution in r^m |
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Інститут математики НАН України |
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2008 |
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http://dspace.nbuv.gov.ua/handle/123456789/4537 |
| citation_txt |
A limit theorem for symmetric Markovian random evolution in R^m / A.D. Kolesnik // Theory of Stochastic Processes. — 2008. — Т. 14 (30), № 1. — С. 69–75. — Бібліогр.: 15 назв.— англ. |
| work_keys_str_mv |
AT kolesnikad alimittheoremforsymmetricmarkovianrandomevolutioninrm AT kolesnikad limittheoremforsymmetricmarkovianrandomevolutioninrm |
| first_indexed |
2025-07-02T07:45:29Z |
| last_indexed |
2025-07-02T07:45:29Z |
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1836520392781463552 |
| fulltext |
Theory of Stochastic Processes
Vol. 14 (30), no. 1, 2008, pp. 69–75
UDC 519.21
ALEXANDER D. KOLESNIK
A LIMIT THEOREM FOR SYMMETRIC
MARKOVIAN RANDOM EVOLUTION IN Rm
We consider the symmetric Markovian random evolution X(t) performed by a particle
that moves with constant finite speed c in the Euclidean space �m, m ≥ 2. Its motion
is subject to the control of a homogeneous Poisson process of rate λ > 0. We show
that, under the Kac condition c → ∞, λ → ∞, (c2/λ) → ρ, ρ > 0, the transition
density of X(t) converges to the transition density of the homogeneous Wiener process
with zero drift and the diffusion coefficient σ2 = 2ρ/m.
The subject of our interest is the following stochastic motion. A particle starts from
the origin x1 = · · · = xm = 0 of the space Rm, m ≥ 2, at time t = 0. It moves with
constant, finite speed c (i.e. c is treated as the constant norm of the velocity). The
initial direction is a random m-dimensional vector with uniform distribution (Riemann-
Lebesgue probability measure) on the unit sphere
Sm
1 =
{
x = (x1, . . . , xm) ∈ R
m : x2
1 + · · ·+ x2
m = 1
}
.
The particle changes its directions at random instants which form a homogeneous Poisson
process of rate λ > 0. At these moments, it instantaneously takes on the new directions
with uniform distribution on Sm
1 , independently of its previous motion.
Let X(t) = (X1(t), . . . , Xm(t)) denote the particle’s position at an arbitrary time
t > 0. At any time t > 0, the particle is located with probability 1 in the m-dimensional
ball of radius ct:
Bm
ct =
{
x = (x1, . . . , xm) ∈ R
m : x2
1 + · · ·+ x2
m ≤ c2t2
}
.
Consider the distribution Pr {X(t) ∈ dx} , x ∈ Bm
ct , t ≥ 0,where dx is an infinitesimal
volume in the space Rm. This distribution consists of two components. The singular
component corresponds to the case where no Poisson event occurs in the interval (0, t)
and is concentrated on the sphere
Sm
ct = ∂Bm
ct =
{
x = (x1, . . . , xm) ∈ R
m : x2
1 + · · ·+ x2
m = c2t2
}
.
In this case, the particle is located on the sphere Sm
ct , and the probability of this event is
Pr {X(t) ∈ Sm
ct} = e−λt.
If one or more than one Poisson events occur, the particle is located strictly inside the
ball Bm
ct , and the probability of this event is
Pr {X(t) ∈ Int Bm
ct} = 1− e−λt.
2000 AMS Mathematics Subject Classification. Primary 82C70; Secondary 82B41, 60K35, 60K37,
70L05.
Key words and phrases. Random motion, finite speed, random evolution, uniformly distributed di-
rections, multidimensional Wiener process.
69
70 ALEXANDER D. KOLESNIK
The part of the distribution Pr {X(t) ∈ dx} corresponding to this case is concentrated
in the interior
Int Bm
ct =
{
x = (x1, . . . , xm) ∈ R
m : x2
1 + · · ·+ x2
m < c2t2
}
,
and forms its absolutely continuous component. Therefore, there exists the density
p(x, t) = p(x1, . . . , xm, t), x ∈ Int Bm
ct , t > 0, of the absolutely continuous compo-
nent of the distribution Pr {X(t) ∈ dx}.
Our goal is to study the limiting behaviour of X(t) as both the speed c and the
intensity of switches λ tend to infinity.
Consider the characteristic function of the process X(t) given by
H(t) = E {exp (i(α,X(t)))} ,
where α = (α1, . . . , αm) ∈ Rm is the real m-dimensional vector of inversion parameters
and (α,X(t)) denotes the inner product of the vectors α and X(t).
It was shown in [3], [4], [6] that the Laplace transform L of the characteristic function
H(t) has the explicit form
L [H(t)] (s) =
F
(
1
2 ,
m−2
2 ; m
2 ; (c‖α‖)2
(s+λ)2+(c‖α‖)2
)
√
(s+ λ)2 + (c‖α‖)2 − λ F
(
1
2 ,
m−2
2 ; m
2 ; (c‖α‖)2
(s+λ)2+(c‖α‖)2
) , Re s > 0,
(1)
where ‖α‖ =
√
α2
1 + · · ·+ α2
m,
F (ξ, η; ζ; z) = 2F1(ξ, η; ζ; z) =
∞∑
k=0
(ξ)k(η)k
(ζ)k
zk
k!
is the Gauss hypergeometric function, and
(a)k = a(a+ 1) . . . (a+ k − 1) =
Γ(a+ k)
Γ(a)
is the Pochhammer symbol.
We now outline the derivation of formula (1). By using the Markov property and the
classical arguments of renewal theory, one can show that the characteristic function H(t)
satisfies the Volterra integral equation of the second kind
H(t) = e−λtϕ(t) + λ
∫ t
0
e−λ(t−τ)ϕ(t− τ)H(τ) dτ, t ≥ 0,
where ϕ(t) is the so-called normalized Bessel function
ϕ(t) = 2(m−2)/2 Γ
(m
2
) J(m−2)/2(ct‖α‖)
(ct‖α‖)(m−2)/2
, m ≥ 2.
Note that ϕ(t) is the characteristic function (Fourier transform) of the uniform distribu-
tion on the surface of the sphere Sm
ct of radius ct. This Volterra integral equation can be
rewritten in the convolutional form
H(t) = e−λtϕ(t) + λ
[(
e−λtϕ(t)
) ∗H(t)
]
, t ≥ 0.
By applying the Laplace transformation to this equation, we immediately obtain the
general form of the Laplace transform of the characteristic function H(t):
L [H(t)] (s) =
L [ϕ(t)] (s+ λ)
1− λ L [ϕ(t)] (s+ λ)
, Re s > 0.
A LIMIT THEOREM FOR RANDOM EVOLUTION IN �
m 71
By using [1], Table 5.19, formula 6, or [2], formula 6.621(1), we can see that
L [ϕ(t)] (s) =
1√
s2 + (c‖α‖)2 F
(
1
2
,
m− 2
2
;
m
2
;
(c‖α‖)2
s2 + (c‖α‖)2
)
, Re s > 0.
Substituting this into the previous expression, we obtain (1).
Formula (1) produces the already known results in lower dimensions. In particular, in
the planar case (m = 2), formula (1) takes the form
L [H(t)] (s) =
1√
(s+ λ)2 + (c‖α‖)2 − λ,
and this coincides with [9], formula (12) therein. The similar result for the unit speed
c = 1 was given in [13].
In the three-dimensional case (m = 3), in view of the equality
F
(
1
2
,
1
2
;
3
2
;
(c‖α‖)2
s2 + (c‖α‖)2
)
=
√
s2 + (c‖α‖)2
c‖α‖ arctg
(
c‖α‖
s
)
which can be easily checked by means of [2], formula 9.121(26), relation (1) immediately
yields
L [H(t)] (s) =
arctg
(
c‖α‖
s+λ
)
c‖α‖ − λ arctg
(
c‖α‖
s+λ
) .
This exactly coincides with [9], formula (45) therein and, for c = 1, with [14], formulae
(1.6) and (5.8) therein.
Finally, in the four-dimensional case (m = 4), in view of the equality
F
(
1
2
, 1; 2;
(c‖α‖)2
s2 + (c‖α‖)2
)
=
2
√
s2 + (c‖α‖)2
s+
√
s2 + (c‖α‖)2
which can be easily checked by means of [2], formula 9.121(24), we obtain from (1) that
L [H(t)] (s) =
2
s+
√
(s+ λ)2 + (c‖α‖)2 − λ.
Our principal result is given by the following theorem.
Limit Theorem. Under the Kac condition
c→∞, λ→∞, c2
λ
→ ρ, ρ > 0, (2)
the transition density of the process X(t) converges to the transition density of the ho-
mogeneous Wiener process with zero drift and the diffusion coefficient σ2 = 2ρ/m, that
is,
lim
c, λ→∞
(c2/λ)→ρ
p(x, t) =
(
m
4ρπt
)m/2
exp
(
− m
4ρt
‖x‖2
)
, m ≥ 2, (3)
where ‖x‖2 = x2
1 + · · ·+ x2
m.
Proof. Our proof consists of three successive steps. First, we will compute the limit
of function (1) under the Kac condition (2). Then, by evaluating the inverse Laplace
transform, we will find the characteristic function of the limiting process. In the last
step, we will compute the inverse Fourier transform of this characteristic function and
show that the transition density of the limiting process has the form (3).
72 ALEXANDER D. KOLESNIK
It’s easy to see that, under the Kac condition (2), we have
lim
c, λ→∞
(c2/λ)→ρ
(c‖α‖)2
(s+ λ)2 + (c‖α‖)2 = 0
and therefore
lim
c, λ→∞
(c2/λ)→ρ
F
(
1
2
,
m− 2
2
;
m
2
;
(c‖α‖)2
(s+ λ)2 + (c‖α‖)2
)
= 1.
Then by passing to the limit in (1) under the Kac condition (2), we obtain
lim
c, λ→∞
(c2/λ)→ρ
L [H(t)] (s)
= lim
c, λ→∞
(c2/λ)→ρ
[√
(s+ λ)2 + (c‖α‖)2 − λ F
(
1
2
,
m− 2
2
;
m
2
;
(c‖α‖)2
(s+ λ)2 + (c‖α‖)2
)]−1
= lim
c, λ→∞
(c2/λ)→ρ
⎡⎣(s+ λ)
√
1 +
(
c‖α‖
s+ λ
)2
−λ
∞∑
k=0
(
1
2
)
k
(
m−2
2
)
k(
m
2
)
k
1
k!
(
(c‖α‖)2
(s+ λ)2 + (c‖α‖)2
)k
]−1
(4)
It follows from the Kac condition (2) that, for sufficiently large c and λ, the inequalities∣∣∣∣ c‖α‖s+ λ
∣∣∣∣ < 1,
∣∣∣∣ (c‖α‖)2
(s+ λ)2 + (c‖α‖)2
∣∣∣∣ < 1 (5)
hold for any s and ‖α‖. Therefore, the radical in (4) can be represented in the form of
the absolutely and uniformly converging series√
1 +
(
c‖α‖
s+ λ
)2
= 1 +
1
2
(
c‖α‖
s+ λ
)2
− 1 · 1
2 · 4
(
c‖α‖
s+ λ
)4
+
1 · 1 · 3
2 · 4 · 6
(
c‖α‖
s+ λ
)6
− . . . (6)
The uniform convergence of series (6) follows from the fact that, under condition (5), it
is dominated by the converging numerical series
1 +
1
2
+
1 · 1
2 · 4 +
1 · 1 · 3
2 · 4 · 6 + · · · <∞
(see [2], point 1.110 for details).
Similarly, the uniform convergence of the hypergeometric series in (4) follows from the
second inequality in (5) and the fact that it is dominated by the converging numerical
series
∞∑
k=0
(
1
2
)
k
(
m−2
2
)
k(
m
2
)
k
1
k!
<∞.
A LIMIT THEOREM FOR RANDOM EVOLUTION IN �
m 73
Substituting (6) into (4), we can rewrite it as follows:
lim
c, λ→∞
(c2/λ)→ρ
L [H(t)] (s)
= lim
c, λ→∞
(c2/λ)→ρ
[
(s+ λ)
(
1 +
1
2
(
c‖α‖
s+ λ
)2
− 1 · 1
2 · 4
(
c‖α‖
s+ λ
)4
+ . . .
)
− λ
(
1 +
(
1
2
)
1
(
m−2
2
)
1(
m
2
)
1
(c‖α‖)2
(s+ λ)2 + (c‖α‖)2
+
1
2!
(
1
2
)
2
(
m−2
2
)
2(
m
2
)
2
(
(c‖α‖)2
(s+ λ)2 + (c‖α‖)2
)2
+ . . .
)]−1
= lim
c, λ→∞
(c2/λ)→ρ
[
s+ λ+
1
2
(c‖α‖)2
s+ λ
− 1 · 1
2 · 4
(c‖α‖)4
(s+ λ)3
+ . . .
− λ−
(
1
2
)
1
(
m−2
2
)
1(
m
2
)
1
λ(c‖α‖)2
(s+ λ)2 + (c‖α‖)2
− 1
2!
(
1
2
)
2
(
m−2
2
)
2(
m
2
)
2
λ(c‖α‖)4
((s+ λ)2 + (c‖α‖)2)2 − . . .
]−1
= lim
c, λ→∞
(c2/λ)→ρ
[
s+
1
2
c2
λ ‖α‖2
s
λ + 1
− 1 · 1
2 · 4
c4
λ3 ‖α‖4(
s
λ + 1
)3 + . . .
−
(
1
2
)
1
(
m−2
2
)
1(
m
2
)
1
c2
λ ‖α‖2(
s
λ + 1
)2
+ c2
λ2 ‖α‖2
− 1
2!
(
1
2
)
2
(
m−2
2
)
2(
m
2
)
2
c4
λ3 ‖α‖4((
s
λ + 1
)2 + c2
λ2 ‖α‖2
)2 − . . .
⎤⎥⎦
−1
.
Taking into account both the fact that, under the Kac condition (2), (cn/λn−1) → 0
for any n ≥ 3 (see also [8], formula (4.4) therein), and the uniform convergence of the
series, we obtain
lim
c, λ→∞
(c2/λ)→ρ
L [H(t)] (s) =
[
s+
1
2
ρ‖α‖2 −
(
1
2
)
1
(
m−2
2
)
1(
m
2
)
1
ρ‖α‖2
]−1
.
It’s easy to check that (
1
2
)
1
(
m−2
2
)
1(
m
2
)
1
=
m− 2
2m
, m ≥ 2.
Thus, we finally obtain
lim
c, λ→∞
(c2/λ)→ρ
L [H(t)] (s) =
(
s+
ρ‖α‖2
m
)−1
. (7)
74 ALEXANDER D. KOLESNIK
The inverse Laplace transformation of the function on the right-hand side of (7) yields
L−1
[(
s+
ρ‖α‖2
m
)−1
]
(t) = exp
(
−ρ‖α‖
2
m
t
)
, (8)
where we have used [1], Table 5.2, formula 1. The function on the right-hand side of
(8) is the characteristic function of the m-dimensional homogeneous Wiener process with
zero drift and the diffusion coefficient σ2 = 2ρ/m.
It remains to compute the inverse Fourier transform F−1 of the function on the right-
hand side of (8). By applying the Hankel inversion formula (see [15], Chapter 5, Section
23, page 359, formula (43)), we obtain
w(x, t) := F−1
[
e−(ρ‖α‖2t)/m
]
=
1
(2π)m/2‖x‖(m−2)/2
∫ ∞
0
e−(ρt/m)ξ2
ξm/2 J(m−2)/2(‖x‖ξ) dξ
=
1
(2π)m/2‖x‖(m−2)/2
‖x‖(m−2)/2(
2ρt
m
)m/2
exp
(
−‖x‖
2
4ρt
m
)
=
(
m
4ρπt
)m/2
exp
(
−m‖x‖
2
4ρt
)
,
proving (3). In the last step, we have used [2], formula 6.631(4), and J(m−2)/2(x) is the
Bessel function of the order of (m− 2)/2 with real argument. The theorem is completely
proved.
Function (3) is exactly the transition density of the m-dimensional homogeneous
Wiener process with zero drift and the diffusion coefficient σ2 = 2ρ/m. This entirely
accords with some previous limiting results for random evolutions (see, for instance, [10],
p. 353, Theorem; [11], Proposition 4.8; [12], p. 102, Theorem 4.2.5).
Remark 1. It’s easy to see that if m = 2 and ρ = 1, density (3) turns into the transition
density of the two-dimensional standard Wiener process with zero drift and the diffusion
coefficient σ2 = 1 (see [7], p.1181)). For m = 4 and ρ = 1, density (3) turns into the
transition density of the four-dimensional Wiener process with zero drift and the diffusion
coefficient σ2 = 1/2 (see [5], formula (21) therein). It’s interesting to note also that if
we set ρ = m/2, the limiting process becomes the m-dimensional standard homogeneous
Wiener process with zero drift and the diffusion coefficient σ2 = 1.
Remark 2. Following [15], Chapter 3, Section 11, Subsection 6, we can easily check that
density (3) is the fundamental solution to the m-dimensional heat equation
∂w(x, t)
∂t
=
ρ
m
Δw(x, t), (9)
where Δ denotes them-dimensional Laplacian. For ρ = 1, the differential operator on the
right-hand side of (9) exactly coincides with the generator obtained in [11], Proposition
4.8.
Acknowledgement. I wish to thank an anonymous referee for his insightful com-
ments and remarks that led to improvements to the first draft of the paper.
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