The small deviations of many-dimensional diffusion processes and
  rarefaction by boundaries
arXiv:0704.0315v1  [math.PR]  3 Apr 2007
THE SMALL DEVIATIONS OF MANY-DIMENSIONAL DIFFUSION
PROCESSES AND RAREFACTION BY BOUNDARIES
Vitalii A. Gasanenko
Abstract. We lead the algorithm of expansion of sojourn probability of many-dimensional
diffusion processes in small domain. The principal member of this expansion defines nor-
malizing coefficient for special limit theorems.
Introduction.
Let ξ(t) be a random process with measurable phase space (X,Σ(X)). Consider the
measurable connected domain D ∈ Σ(X) and small parameter ǫ. The investigations of
asymptotics of sojourn probability (small deviations)
P (ξ(t) ∈ ǫD, t ∈ [0, T ]) (1)
is jointed with many practice and theoretical problems [1-4]. In the literature, it was
researched both rough asymptotics of principal member of (1)(log from it)[5] and exact
asymptotics of diffusion processes of (1)[6-8]. In the works [9,10] was proved of algo-
rithms of expansions of exact asymptotics of small deviation for diffusion and piecewise
deterministic random processes for one-dimensional case.
The purpose this article is to present the algorithm of expansion of small deviation for
many-dimensional diffusion processes and to define all constants of principal member.
In Section 1 our main result is stated and proved. In section 2 we consider the limits
theorems about numbers of unabsorbed diffusion particles by boundaries of small domain.
I. The expansion.
We shall investigate of asymptote of following probability
P (ǫ, x) = P (ξ(t) ∈ ǫD, 0 ≤ t ≤ T ) , ǫ → 0,
where ξ(t) ∈ Rd is solution of the following stochastic differential equation
dξ(t) = a(t, ξ(t))dt+
bi(ξ(t))dwi(t), ξ(0) = x ∈ ǫD. (2)
where functions
1991 Mathematics Subject Classification. 60 J 65.
Key words and phrases. parabolic problem,small domain, algorithm of expansion, number of unab-
sorbed processes.
Typeset by AMS-TEX
http://arxiv.org/abs/0704.0315v1
2 VITALII A. GASANENKO
bi(x), a(t, x) : R
d → Rd and R+ ×R
d → Rd.
are differentiable.
Set σij(x) =
bik(x)b
k(x).
It is known that P (ǫ, x) = uǫ0(T, x). Here u
0(t, x) is solution of the following parabolic
boundary problem at 0 ≤ t ≤ T
∂uǫ0(t, x)
i,j=1
σij(x)
∂2uǫ0(t, x)
∂xi∂xj
ai(T − t, x)
∂uǫ0(t, x)
, x ∈ Dǫ;
u(t, x)|t=0 = 1; x ∈ Dǫ; u(t, x) = 0 x ∈ ∂Dǫ, 0 ≤ t ≤ T. (3)
where Dǫ = ǫD. It is assumed that D is a connected bounded domain from R
m; the
boundary ∂Q is the Lyapunov surface C(1,λ) and 0 ∈ D. We interest of the asymptotic
expansion ǫ → 0 of solution this problem uǫ0(t, x) at ǫ → 0.
We define the differential operator A : 1
1≤i,j≤d
σij(0)
∂xi∂xj
. Let σ be a matrix with
the following property
1≤i,j≤d
σij(0)zizj ≥ µ|~z|
Here µ, there is a fixed positive number, and ~z = (z1, · · · , zd) is an arbitrary real
vector.
This operator acts in the following space
HA = {u : u ∈ L2(D) ∩ Au ∈ L2(D) ∩ u(∂D) = 0}
with inner product (u, v)A = (Au, v). Here (, ) is inner product in L2(Q). The opera-
tor A is a positive operator[11]. It is known that the following eigenvalue problem
Au = −λu, u(∂D) = 0
has infinite set of real eigenvalues λi → ∞ and
0 < λ1 < λ2 < · · · < λs < · · · .
The corresponding eigenfunctions
f11, . . . , f1n1 , · · · , fs1, . . . , fsns , · · ·
form the complete system of functions both in HA and L
2(Q) := {u : u ∈ L2(Q) ∩
u(∂Q) = 0}. Here the number nk is equal to multiplicity of eigenvalue λk.
It is often convenient to present the system of eigenfunctions by one index: {fn(z)}.
The corresponding system of eigenvalues {λn} will be with recurrences. We shall use it
We introduce the spectral function
e(x, y, λ) =
fj(x)fj(y).
We shall need in the following theorem from the monograph [12].
THE SMALL DEVIATION OF MANY-DIMENSIONAL DIFFUSION PROCESSES 3
Theorem 1 ([12].Th.17.5.3). . There exists such constant Cα that
x,y∈D
|Dαx,ye(x, y, λ)| ≤ Cαλ
(n+|α|)/2
Here α is multi-index.
Theorem 2. . If the surface ∂D is Lyapunov surface and
(t,z)∈[0,T ]×D,1≤i,j≤d
∂ai(t, z)
∂bi(z)
∂ai(T − t, z)
then the following relation takes place at ǫ → 0
P (ǫ, zǫ) = exp
µ(t)dt
c1mf1m(z) (1 +O(ǫ)) , at z ∈ D,
where
µ(t) =
σij(0)ai(t, 0)aj(t, 0)− δijai(t, 0)aj(t, 0)
and c1m =
f1m(z)dz.
Proof. Make the change of variables and function
xi = ziǫ, u
1 = u
0 exp
ak(T − t, 0)zk
Now we obtain the following parabolic problem for function uǫ1
∂uǫ1(t, z)
i,j=1
σij(ǫz)
∂2uǫ1(t, z)
∂zi∂zj
ai(T − t, ǫz)−
σij(ǫz)aj(T − t, 0)
∂uǫ1(t, z)
σij(ǫz)ai(T − t, 0)aj(T − t, 0)− δijai(T − t, 0)aj(T − t, ǫz)− ǫ
∂ai(T − t, 0)
uǫ1, z ∈ D;
1(t, z)|t=0 = exp
ak(T, 0)zk
; z ∈ D; uǫ1(t, z) = 0 z ∈ ∂D, 0 ≤ t ≤ T.
We will construct the asymptotic expansion of solution for this initial - boundary
problem in the following form
uǫ1(t, z) =
vk(t, z)ǫ
k. (5)
Note that the famous expansion
4 VITALII A. GASANENKO
ak(T, 0)zk
= 1 + ǫ
ak(T, 0)zk +
ak(T, 0)zk
+ · · · ,
defines the initial conditions for vk, k ≥ 0:
v0(0, z) = 1, v1(0, z) =
ak(T, 0)zk, v2(0, z) =
ak(T, 0)zk
· · · .
Using the first fragment of Taylor series in zero point under conditions of theorem we
can obtain the following representations
σij(ǫz) = σij(0) + ǫσ
ij(z), ai(T − t, ǫz) = ai(T − t, 0) + ǫa
i(T − t, z), 1 ≤ i, j ≤ d (6)
where
z∈D,ǫ∈[0,1],1≤i,j≤d
|σǫij(z)| < ∞, sup
z∈D,t∈[0,T ],ǫ∈[0,1],1≤i≤d
|aǫi(T − t, z)| < ∞
Now, after substitution of (5),(6) to (4) we conclude that the v0 satisfies the problem
i,j=1
σij(0)
∂zi∂zj
 v0 + µ(t)v0 (7)
v0|∂D = 0; v0(0, z) = 1, z ∈ D.
µ(t) =
σij(0)ai(T − t, 0)aj(T − t, 0)− δijai(T − t, 0)aj(T − t, 0)
Further, let us denote by Bǫ(t, z) the operator C
2(D) → C(D), for f ∈ C2(D) it’s
defined as follows:
ǫ(t, z)f =
i,j=1
σǫij(z)
∂zi∂zj
ai(T − t, ǫz)−
σij(ǫz)aj(T − t, 0)
i,j=1
σǫij(z)ai(T − t, 0)aj(T − t, 0)− δijai(T − t, 0)a
j(T − t, z)−
∂ai(T − t, 0)
i,j=1
σǫij(z)
∂zi∂zj
Aǫ1(t, z)f + ǫA
2(t, z).
THE SMALL DEVIATION OF MANY-DIMENSIONAL DIFFUSION PROCESSES 5
Now, formally the functions vk, k ≥ 1 are defined by the following recurrence system
problems
i,j=1
σij(0)
∂zi∂zj
 vk +Bǫ(t, z)vk−1 (8)
v0|∂D = 0; vk(0, z) =
ak(T − t, 0)zk
, z ∈ D.
We shall solve the problems of (7),(8) by method of separation of variables. According
to this method the solutions are defined in the form
vk(t, z) =
qk,n(t)fn(z). (9)
For definition of principal number it suffices to construct of the v0. If we substitute
(9) at k = 0 to (7) then we obtain
−q̇0,n(t)−
q0,n(t) + µ(t)q0,n(t)
fn(z) = 0.
Set c0,n =
fn(z)dz (coefficients of expansion of indicator of set D). The initial
condition of v0 has the following stating
v0(0, z) =
q0,n(0)fn(z) =
c0,nfn(z) =
c0,lmflm(z), z ∈ D.
By definition of system of functions {fn(z)}, now we have the system of ordinary
differential equations
q̇0,n(t) +
− µ(t)
q0,n(t) = 0, q0,n(0) = c0,n.
From the latter one we have
q0,n(t) = c0,n exp
µ(s)ds
A0 = sup
ǫ≤1,z∈D;i,j
|σǫij(z)|, L0 =
l≥1,1≤m≤nl
(c0,ml)
A1 = sup
0≤ǫ≤1,z∈D,t∈[0,T ];i,j
ai(T − t, ǫz)−
σij(ǫz)aj(T − t, 0)
= sup
0≤ǫ≤1,z∈D,t∈[0,T ];i,j
ij(z)ai(T − t, 0)aj(T − t, 0)− δijai(T − t, 0)a
j(T − t, z)−
∂ai(T − t, 0)
We have the following relations for eigenvalues λl
6 VITALII A. GASANENKO
2/d ≤ λl ≤ k2l
2/d, max(k1, k2) < ∞
Applying Cauchy-Bunyakovskii inequality, Theorem 1 and the latter one, we get
aǫi,j(z)
∂zi∂zj
−λltǫ
µ(s)ds
c0,ml
aǫi,j(z)
∂2fml(z)
∂zi∂zj
≤ A0d
−λltǫ
µ(s)ds
(c0,ml)
∂2fml(z)
∂zi∂zj
≤ A0dC2,2L0
µ(s)ds
l ≤ exp
K0. (10)
Here K0 < ∞.
Reasoning similarly we convince ourselves that for other parts of Bǫ(t, z)v0 the fol-
lowing estimations take place
|Aǫ1(t, z)v0| ≤ A1dC1,1L0
µ(s)ds
l ≤ exp
K0,1; (11)
|Aǫ2(t, z)v0| ≤ A2dC0,0L0
µ(s)ds
l ≤ exp
K0,2, (12)
where max{K0,1,K0,2} < ∞.
Now let us estimate the coefficients βǫn(t) of expansion of B
ǫ(t, z)v0 by system {fn}n≥1.
Applying (10)-(12) and Cauchy-Bunyakovskii inequality, we get
|βǫn(t)| = |
Bǫ(t, z)v0(t, z)fn(z)dz| ≤
(Bǫ(t, z)v0)
n(z)dz
≤ exp(−λ1tǫ
K0 +K0,1
+ ǫK0,2
The latter one now gives
βǫn(s)ds| ≤ ǫγǫ(t), (13)
THE SMALL DEVIATION OF MANY-DIMENSIONAL DIFFUSION PROCESSES 7
where
0≤ǫ≤1,t∈[0,T ]
γǫ(t) < ∞.
Finally, let us estimate the difference rǫ(t, z) = uǫ1(t, z)−v0(t, z). By definition, r
ǫ(t, z)
is solution of the following problem
i,j=1
σij(0)
∂zi∂zj
 rǫ +Bǫ(T − t, z)v0 z ∈ D; (14)
rǫ(t, z)|t=0 = exp
ak(T, 0)zk
− 1; z ∈ D; rǫ(t, z) = 0 z ∈ ∂D, 0 ≤ t ≤ T.
It is clear that rǫ(0, z) we can present as ǫrǫ1(0, z), where r
1(0, z) is uniform bounded
function of variables ǫ ∈ [0, 1] and z ∈ D. So, the coefficients of expansion this function
by system {fn(z)} have the following forms
rǫ(0, z)fn(z)dz = ǫµ
n, where sup
0≤ǫ≤1
(µǫn)
= M < ∞. (13)
Now we have the solution of (14) in the following form
rǫ(t, z) = ǫ
µǫn exp{−λntǫ
βǫn(s)ds}fn(z)
Applying latter one ,(13),(15), Theorem 1 and Cauchy-Bunyakovskii inequality we get
at t > 0
|rǫ(t, z)ǫ−1| ≤
(µǫn)
exp{−λntǫ
βǫn(s)ds}λ
n } ≤
≤ MC0,0 exp{−λ1tǫ
−2}K0,3, where K0,3 < ∞.
The proof of theorem is completed.
Remark 1. According to the above system of problems for definition of the functions
vk, k ≥ 1, we outline the construction of coefficients qk,n)(t) for the series (8):
q̇k,n(t) = +
+ µǫk−1,n(t)
qk,n(t),
qk,n(0) =
vk(0, z)fn(z)dz =
am(T, 0)zm
fn(z)dz
Here µǫk−1,n(t) =
fn(z)B
ǫ(t, z)vk−1(t, z)dz.
8 VITALII A. GASANENKO
Remark 2. Theorem 2 is coordinated with results of works [6-8] where the principal
member of small deviations in ball are investigated for more simple SDE.
II. The rarefaction of set of diffusion processes by boundaries of small
domains.
The following problem was investigated in works[13,14]. Let a set identical diffusion
random processes start at the initial time from the different points of domain D. These
processes are diffusion processes with absorbtion on the boundary ∂D. We are interested
in distribution of the number yet absorbed at the moment T . The initial number and
initial position of diffusion processes are defined either a random Poisson measure[14]
or deterministic measure [13]. The proved limits theorems described the situation when
T → ∞ and initial number of diffusion processes depended on T and it increased at the
rise of T . The role of normalizing function played principal member of asymptote of
solution of according parabolic problem at T → ∞.
Henceforth we shall assume that considered diffusion processes satisfy of the SDE (2)
with different initial points.
Now we consider the situation when initial number of absorbing diffusion processes
in small domain ǫD depends on ǫ → 0 and it increase under the condition of decrease
of ǫ. It is not hard to show, that now normalizing function is the principal member of
parabolic problem (3) at ǫ → 0.
The proofs of stated below theorems repeat the proofs of according theorems from
[13,14] almost word for word.
We will denote by η(ǫ, T ) the number of remaining processes in the region ǫD at the
moment T .
We will also assume that σ-additive measure ν is given on the Σν- algebra sets from
D, ν(D) < ∞. All eigenfunctions fij : D → R
1 are (Σν ,ΣY ) measurable. Here ΣY is
system of Borel sets from R1. Let ⇒ denote the weak convergence of random values or
measures.
At the beginning we assume that initial number and position of diffusion processes are
defined by deterministic measure N(ǫB, ǫ), B ∈ D. Thus, N(ǫB, ǫ) is equal to number of
starting points in the set ǫB.
Let us denote by νǫ(·) the measure
νǫ(ǫB) = exp
N(ǫB, ǫ).
where B ∈ Σν .
By definition of measure νǫ(·), we have
dνǫ(x) =
, if x = xk, k = 1, · · · , N(ǫD, ǫ)
0, otherwise.
Theorem 3. Under the assumptions of the Theorem 2 let the N(ǫ·, ǫ) satisfies the con-
dition
νǫ(ǫ ·) ⇒
ν(·).
THE SMALL DEVIATION OF MANY-DIMENSIONAL DIFFUSION PROCESSES 9
Then η(ǫ, T ) ⇒ η(T ) if ǫ → 0 where η(T ) has Poisson distribution function with
parameter
a(T ) = exp
µ(s)ds
F (z)dν(z),
where F (z) =
f1i(z)c1i, c1i =
f1i(z)dz
and µ(t) is the function from Theorem 2.
Now we consider the case when the initial number and positions of processes are
defined by the random Poisson measure µ(·, ǫ) in ǫD:
P (µ(ǫA, ǫ) = k) =
mk(ǫA, ǫ)
−m(ǫA,ǫ)
where m(ǫ ·, ǫ) is finitely additive positive measure on ǫD for fixed ǫ.
We assign
g(ǫ) = exp
Theorem 4. Under the assumptions of the Theorem 2 we suppose that m(ǫ·, ǫ) holds
the condition
m(ǫB, ǫ)g(ǫ) = ν(B), B ∈ Σν .
Then η(ǫ, T ) ⇒ η(T ) if ǫ → 0 where η(T ) has the Poisson distribution function with
the parameter a(T ) from Theorem 3.
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Institute of Mathematics, National Academy of Science of Ukraine, Tereshchenkivska
3, 252601, Kiev, Ukraine
E-mail address: gs@imath.kiev.ua or gsn@ckc.com.ua
