Renormalization constant for the local times of self-intersections of a diffusion process in the plane
The self-intersection local times of a diffusion process in the plane are studied. The main result consists in investigating asymptotic behavior of renormalizing constant for this local time.
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irk-123456789-1647802020-02-11T01:27:21Z Renormalization constant for the local times of self-intersections of a diffusion process in the plane Izyumtseva, O.L. Статті The self-intersection local times of a diffusion process in the plane are studied. The main result consists in investigating asymptotic behavior of renormalizing constant for this local time. Розглянуто локальний час самоперетинів для дифузійного процесу на площині. Головним результатом є дослідження асимптотичної поведінки константи перенормування цього локального часу. 2008 Article Renormalization constant for the local times of self-intersections of a diffusion process in the plane / O.L. Izyumtseva // Український математичний журнал. — 2008. — Т. 60, № 11. — С. 1489–1498. — Бібліогр.: 10 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/164780 519.21 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Izyumtseva, O.L. Renormalization constant for the local times of self-intersections of a diffusion process in the plane Український математичний журнал |
| description |
The self-intersection local times of a diffusion process in the plane are studied. The main result consists
in investigating asymptotic behavior of renormalizing constant for this local time. |
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Article |
| author |
Izyumtseva, O.L. |
| author_facet |
Izyumtseva, O.L. |
| author_sort |
Izyumtseva, O.L. |
| title |
Renormalization constant for the local times of self-intersections of a diffusion process in the plane |
| title_short |
Renormalization constant for the local times of self-intersections of a diffusion process in the plane |
| title_full |
Renormalization constant for the local times of self-intersections of a diffusion process in the plane |
| title_fullStr |
Renormalization constant for the local times of self-intersections of a diffusion process in the plane |
| title_full_unstemmed |
Renormalization constant for the local times of self-intersections of a diffusion process in the plane |
| title_sort |
renormalization constant for the local times of self-intersections of a diffusion process in the plane |
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Інститут математики НАН України |
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2008 |
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Статті |
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http://dspace.nbuv.gov.ua/handle/123456789/164780 |
| citation_txt |
Renormalization constant for the local times of self-intersections of a diffusion process in the plane / O.L. Izyumtseva // Український математичний журнал. — 2008. — Т. 60, № 11. — С. 1489–1498. — Бібліогр.: 10 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT izyumtsevaol renormalizationconstantforthelocaltimesofselfintersectionsofadiffusionprocessintheplane |
| first_indexed |
2025-07-14T17:21:52Z |
| last_indexed |
2025-07-14T17:21:52Z |
| _version_ |
1837643819735056384 |
| fulltext |
UDC 519.21
O. L. Izyumtseva (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
THE CONSTANT OF RENORMALIZATION
FOR SELF-INTERSECTION LOCAL TIME
OF DIFFUSION PROCESS IN THE PLANE
∗∗∗∗
KONSTANTA PERENORMUVANNQ LOKAL|NOHO ÇASU
SAMOPERETYNIV DYFUZIJNOHO PROCESU
NA PLOWYNI
The self-intersection local times of a diffusion process in the plane are studied. The main result consists
in investigating asymptotic behavior of renormalizing constant for this local time.
Rozhlqnuto lokal\nyj ças samoperetyniv dlq dyfuzijnoho procesu na plowyni. Holovnym re-
zul\tatom [ doslidΩennq asymptotyçno] povedinky konstanty perenormuvannq c\oho lokal\no-
ho çasu.
Introduction and the main result. The local time of self-intersections of Brownian
motion arose in the study of Euclidian field theory [1]. Since then, many papers devo-
ted to renormalized self-intersection local time have appeared. For instance Dynkin in
[2] have studied the local time of self-intersections for planar Brownian motion.
Le Gall in [3] defined the Wiener sausage and got some results for it in terms of the
local time of self-intersections of Wiener process.
Rosen in [4, 5, 6] investigated the local time of self-intersections for the stable
process in R2.
Bass and Khoshnevisan in[7] gave a new method of constructing intersection local
times for Brownian motion in R2 and R
3 by using stochastic calculus and additive
functionals of Markov process and obtained Tanaka formula for self-intersection local
times of planar Brownian motion.
Chen, Xia in [8] have studied large deviation and law of the iterated logarithm for
self-intersection local times of additive process.
More references can be found in papers mentioned above.
The goal of this paper is to consider the self-intersection local times for diffusion
process in the plane and to study the asymptotic behaviour of renormalizing constant
for it.
In studying double self-intersections of planar Wiener process w t t( ), ≥{ }0 one
naturally tries to use the formal expression
δ0 2 1
0 12
( ( ) ( ))
( , )
w s w s ds−∫
�
∆
, (1)
where ∆2 0 1( , ) = 0 11 2≤ ≤ ≤{ }s s , δ0 is the δ -function concentrated at the point
zero.
Let f xε ε( ), >{ }0 be an approximate sequence for δ0 given by the following
formula:
f xε( ) = 1
2
2
2
πε
εe
x−
. (2)
Consider
∗
Partially supported by the Ministry of Education and Science of Ukraine (project No GP/F13/0095).
© O. L. IZYUMTSEVA, 2008
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11 1489
1490 O. L. IZYUMTSEVA
f w s w s dsε( ( ) ( ))
( , )
2 1
0 12
−∫
�
∆
. (3)
The expectation of (3) blows up as ε → 0 + . Therefore, instead of (3) consider
T2
ε : =
f w s w s ds E f w s w s dsε ε( ( ) ( )) ( ( ) ( ))
( , ) ( , )
2 1
0 1
2 1
0 12 2
− − −∫ ∫
� �
∆ ∆
. (4)
Dynkin in [2] proved that there exists
L
2 0
2−
→ +
lim
ε
εT , where this limit is called the
renormalized self-intersection local time for planar Wiener process.
Let us call the following expression as the renormalizing constant for self-intersec-
tion local time of Wiener process in the plane:
Cw
ε : =
E f w s w s dsε ( ( ) ( ))
( , )
2 1
0 12
−∫
�
∆
. (5)
In [2] it was showed that
Cw
ε ∼ 1
2
1
π ε
ln , ε → 0 + .
Let Y be a diffusion process in R
2 described by the stochastic differential equa-
tion
dY s a Y s ds B Y s dw s
Y x
( ) ( ( )) ( ( )) ( ),
( ) .
= +
=0 0
(6)
Here the coefficients a and B are Lipschitz functions. We suppose that
m I1 < B B∗ < m I2 , (7)
where m1, m2 are some positive constants, and I is identity matrix in R
2.
Note that by using the fact that B is Lipschitz function and taking into account (7),
one can show that det B is Lipschitz function too.
The purpose of the article is to find the asymptotic behavior of renormalizing con-
stant for self-intersection local times of diffusion process given in (6), namely, to find
asymptotic behavior of
CY
ε : =
E f Y s Y s dsε( ( ) ( ))
( , )
2 1
0 12
−∫
�
∆
. (8)
The main result of this paper is the following statement.
Theorem. Suppose that diffusion process Y satisfies (6), fε is given by the
formula (2). Then
CY
ε ∼ E
B Y s
ds1 1
2
1
0
1
det ( ( ))
ln∫ π ε
, ε → 0 + . (9)
Note that there exists another case, when the renormalization can be described pre-
cisely.
By using this statement the following conclusion can be made.
Let X t( ) be Ito process in terms of [9, Sec. 2] with the following representation:
X t( ) = w t s ds
t
( ) ( )+ ∫
0
α , (10)
where α is measurable bounded random function adapted to the flow of w.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
THE CONSTANT OF RENORMALIZATION FOR SELF-INTERSECTION LOCAL TIME … 1491
By using Girsanov theorem and Dynkin result for Brownian motion, it can be che-
cked that there exists both
lim ( )
ε
ε ε
→ +
−
0
C CX w (11)
and
L f X s X s ds CX
2 0
2 1
0 12
−
→ +
− −
∫lim ( ( ) ( ))
( , )
ε
ε ε
�
∆
. (12)
As a result we have the following equivalence:
CX
ε ∼ 1
2
1
π ε
ln , ε → 0 + . (13)
Let us rewrite (6) as follows:
Y s( ) = x a Y r dr B Y r dw r
s s
0
0 0
+ +∫ ∫( ( )) ( ( )) ( ) . (14)
In case of B = I, by using (9) we get
CY
ε ∼ 1
2
1
π ε
ln , ε → 0 + . (15)
Therefore, the hypothesis can be put forward that the renormalizing constant for the
class of Ito processes of the form
X s( ) = X B r dw r r dr
s s
0
0 0
+ +∫ ∫( ) ( ) ( )α
is equivalent to
E
B s
ds1 1
2
1
0
1
det ( )
ln∫ π ε
, ε → 0 + .
We will need the following estimations for transition density of diffusion process
[10, Sec. 6].
Denote by EY s x s x( , ; , )1 1 2 2 transition density of the process Y, then there exist
C1 > 0, C2 > 0 such that
EY s x s x( , ; , )1 1 2 2 ≥
C
s s
e
x x
C s s1
2 1
1 2
2
1 2 1
−
− −
−( ) , (16)
EY s x s x( , ; , )1 1 2 2 ≤
C
s s
e
x x
C s s2
2 1
1 2
2
2 2 1
−
− −
−( ) . (17)
By using the estimations (16), (17), one can show that
lim
lnε
ε
ε
→ +0 1
CY
≤ C∗, (18)
lim
lnε
ε
ε
→ +0
1
CY
≥ C∗ , (19)
where C∗, C∗ are some positive constants.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
1492 O. L. IZYUMTSEVA
It should not be made any conclusion about the precise asymptotic behaviour of
CY
ε as ε → 0 + . That is why to prove the theorem we use another approach related
not only to estimations (16), (17) but also to the parametrix method consisting of the
following procedure.
Asume that Y satisfies the equation
Y s( ) = Y s a Y r dr B Y r dw r
s
s
s
s
( ) ( ( )) ( ( )) ( )0
0 0
+ +∫ ∫ , (20)
where s0 is some fixed point under Y s( )0 = x0 being fixed. According to the para-
metrix method, the transition density of process Y on the interval [ , ]s0 1 can be ex-
pressed as follows:
EY s x s x( , ; , )1 1 2 2 =
E EX
s
s
Xs x s x s x s x s x s x dx ds( , ; , ) ( , ; , ) ( , ; , )1 1 2 2 1 1 3 3 3 3 2 2 3 3
1
2
2
+ ∫ ∫
�
Φ ,
where the process X has the representation
X s( ) = x a x dr B x dw r
s
s
s
s
0 0 0
0 0
+ +∫ ∫( ) ( ) ( ) (22)
and Φ is some function satisfying the estimation [10, Sec. 4].
There exists C3 > 0 such that
Φ( , ; , )s x s x1 1 3 3 ≤
C
s s
e
x x
C s s3
3 1
3 2
1 3
2
3 3 1
( ) /
( )
−
− −
− . (23)
So, transition density of diffusion process Y on the small time intervals is close to
the transition density of Wiener process.
Proof of theorem. Let n be fixed. Consider the partition of ∆2 0 1( , ) such that
∆2 0 1( , ) =
∆2
0
1
2
0
2
1k
n
k
n
R
k
n
k
k
n
, +
=
−
=
−
∪ ∪∪ , (24)
where
∆2
1k
n
k
n
, +
= k
n
s s k
n
≤ ≤ ≤ +{ }1 2
1 , (25)
Rk
2 = k
n
s k
n
k
n
s≤ ≤ + + ≤ ≤{ }1 2
1 1 1, . (26)
Then
CY
ε = I In n
1 2, ,ε ε+ ,
where
In
1,ε =
E f Y s Y s ds
k
n
k
n
k
n
=
−
+
∑ ∫ −
0
1
2 1
1
2
ε( ( ) ( ))
,
�
∆
, (27)
In
2,ε =
E f Y s Y s ds
k
n
Rk=
−
∑ ∫ −
0
2
2 1
2
ε( ( ) ( ))
�
. (28)
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
THE CONSTANT OF RENORMALIZATION FOR SELF-INTERSECTION LOCAL TIME … 1493
Let us consider In
2,ε . By using (17) and usual calculations, In
2,ε can be estimated
as follows:
In
2,ε ≤
k
n
R
c s s
ds
k=
−
∑ ∫ −0
2
2 2 1
1
2
( )
�
≤ c n( ), (29)
where c n( ) is some positive constant which does not depend on ε .
In
1,ε can be rewritten as
In
1,ε =
k
n
k
nk
n
k
n
EE f Y s Y s F ds
=
−
+
∑ ∫ −
0
1
2 1
1
2
ε( ( ) ( ))
,
�
∆
=
=
E f x x k
n
Y k
n
s x s x s x dx ds
k
n
k
n
k
n
Y Y
=
−
+
∑ ∫ ∫
×
−
0
1
1
1 2 1 1 1 1 2 2
2
2 2
∆ ,
( ) , ; , ( , ; , )
R
ε E E
� �
, (30)
where as usual Ft = σ( ( ), )Y s s t≤ .
According to the parametrix method, we get the following representation for In
1,ε:
In
1,ε = E f x x
k
n
Y
k
n
s x
k
n
k
n
k
n
X
=
−
+
∑ ∫ ∫
×
−
0
1
1
1 2 1 1
2
2 2
∆ ,
( ) , ; ,
R
Eε +
+
k
n
s
X
k
n
Y
k
n
x s x s x s dx ds
1
2
3 3 3 3 1 1 3 3∫ ∫
R
EΦ , ; , ( , ; , ) ×
× E E˜ ˜( , ; , ) ( , ; , ) ( , ; , )
X
s
s
X
s x s x s x s x s x s x dx ds dx ds1 1 2 2 1 1 3 3 3 3 2 2 3 3
1
2
2
+
∫ ∫
R
Φ
� �
, (31)
The process X on the interval k n s/ , 1[ ] and X̃ on the interval s s1 2,[ ] have the fol-
lowing representations:
dX s a Y
k
n
ds B Y
k
n
dw s
X
k
n
Y
k
n
( ) ( ),
,
=
+
=
(32)
dX s a X s ds B X s dw s
X s X s
˜ ( ) ( ( )) ( ( )) ( ),
˜ ( ) ( ).
= +
=
1 1
1 1
(33)
After some transformation, we obtain
In
1,ε = E f x x
k
n
k
n
k
n
=
−
+
∑ ∫ ∫
×
−
0
1
1
1 2
2
2 2
∆ ,
( )
R
ε ×
× E EX X
k
n
Y k
n
s x s x s x dx ds, ; , ( , ; , )˜
1 1 1 1 2 2
� �
+
+ T T T2 3 4, , ,ε ε ε
n n n+ + , (34)
where
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
1494 O. L. IZYUMTSEVA
T2,ε
n =
E f x x s x s x
k
n
k
n
k
n
X
=
−
+
∑ ∫ ∫
×
−
0
1
1
1 2 1 1 2 2
2
2 2
∆ ,
˜( ) ( , ; , )
R
ε E ×
×
k
n
s
X
k
n
Y k
n
x s s x s x dx ds dx ds
1
2
3 3 3 3 1 1 3 3∫ ∫
R
Φ , ; , ( , ; , )E
� �
, (35)
T3,ε
n =
E f x x k
n
Y k
n
s x
k
n
k
n
k
n
X
=
−
+
∑ ∫ ∫
×
−
0
1
1
1 2 1 1
2
2 2
∆ ,
( ) , ; ,
R
ε E ×
×
s
s
X
s x s x s x s x dx ds dx ds
1
2
2
1 1 3 3 3 3 2 2 3 3∫ ∫
R
Φ( , ; , ) ( , ; , )˜E
� �
, (36)
T4,ε
n = E f x x k
n
Y k
n
x s
k
n
k
n
k
n
s
s
k
n
s
=
−
+
∑ ∫ ∫ ∫ ∫ ∫
× ×
−
0
1
1
1 2 3 3
2
2 2
1
2 1
2 2
∆
Φ
,
( ) , ; ,
R R
ε ×
× Φ( , ; , ) ( , ; , ) ( , ; , )˜s x s x s x s x s x s x dx dx ds ds dx dsX X1 1 4 4 3 3 1 1 4 4 2 2 3 4 3 4E E
� �
. (37)
A few remarks about notation should be given. From now, we will denote
B Y s( ( )) =
b Y s b Y s
b Y s b Y s
11 12
21 22
( ( )) ( ( ))
( ( )) ( ( ))
, (38)
�
a Y s( ( )) =
a Y s
a Y s
1
2
( ( ))
( ( ))
. (39)
Let us consider
E f x x k
n
Y k
n
s x s x s x dx ds
k
n
k
n
k
n
X X
=
−
+
∑ ∫ ∫
×
−
0
1
1
1 2 1 1 1 1 2 2
2
2 2
∆ ,
˜( ) , ; , ( , ; , )
R
ε E E
� �
. (40)
After some calculations (40) can be rewritten as
E k
n
Y k
n
s x
k
n
k
n
k
n
X
=
−
+
∑ ∫ ∫
0
1
1
1 1
2
2
1
2
∆ ,
, ; ,
R
E
π
×
×
1
1
2
2 1
2 2
11
2
2 1
2
2
(det ( )) ( ) ( )( )
,
( , , , )
B x s s b x s s
e dx ds
iji j
a B s
− + − +=
−
∑ε ε
ε
� �
� �
, (41)
where
e
a B s− ( , , , )
� �
ε
2 =
= exp
( ) ( ) ( ) ( ) ( )
(det ( )) ( ) ( )( )
,
− −( ) −
− + − +
=∑
1
2
21 1 1 1 11 1 2 1
2
2 1
3
1
2
2 1
2 2
11
2
2 1
2
b x a x b x a x s s
B x s s b x s siji j
ε ε
×
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
THE CONSTANT OF RENORMALIZATION FOR SELF-INTERSECTION LOCAL TIME … 1495
× exp
( ) ( ) ( )
(det ( )) ( ) ( )( )
,
−
+( ) −
− + − +
=∑
ε
ε ε2
1
2
1 2
2
1 2 1
2
1
2
2 1
2 2
11
2
2 1
2
a x a x s s
B x s s b x s siji j
×
× exp
( ) ( ) ( ) ( ) ( )
(det ( )) ( ) ( )( )
,
− −( ) −
− + − +
=∑
1
2
22 1 1 1 12 1 2 1
2
2 3
3
1
2
2 1
2 2
11
2
2 1
2
b x a x b x a x s s
B x s s b x s siji j
ε ε
.
For small ε > 0 the following inequality holds:
e
a B s− ( , , , )
� �
ε
2 ≤ 1. (42)
By using (42) we can write that (41) is less or equal to
E k
n
Y k
n
s x
k
n
k
n
k
n
X
=
−
+
∑ ∫ ∫
0
1
1
1 1
2
2
1
2
∆ ,
, ; ,
R
E
π
×
× 1
1
2
2 1
2 2
11
2
2 1
2 1
(det ( )) ( ) ( )( )
,
B x s s b x s s
dx ds
iji j
− + − +=∑ε ε
� �
=
= E k
n
Y k
n
s x
k
n
k
n
k
n
X
=
−
+
∑ ∫ ∫
0
1
1
1 1
2
2
1
2
∆ ,
, ; ,
R
E
π
1
1det ( )B x
×
× 1
2 1
2
1
2 2
11
2
2 1
2
1
2 1
( ) (det ( )) ( )( ) (det ( ))
,
s s B x b x s s B x
dx ds
iji j
− + − +−
=
−∑ε ε
�
.
(43)
Let us note that
1
1det ( )B x
≤ 1
1det ( ( ))/B Y k n
L x Y k
n
+ −
(44)
with some positive constant L .
By using (7) and (44) we can write that (43) is less or equal to
E k
n
Y k
n
s x
B Y k nk
n
k
n
k
n
X
=
−
+
∑ ∫ ∫
0
1
1
1 1
2
2
1
2
1
∆ ,
, ; ,
det ( ( ))/
R
E
π
×
×
1
2 1
2
1 2 1
2
2
1
( ) ( )s s d s s d
dx ds
− + − +ε ε
�
+
+ LE k
n
Y k
n
s x x Y k
nk
n
k
n
k
n
X
=
−
+
∑ ∫ ∫
−
0
1
1
1 1 1
2
2
1
2
∆ ,
, ; ,
R
E
π
×
×
1
2 1
2
1 2 1
2
2
1
( ) ( )s s d s s d
dx ds
− + − +ε ε
�
, (45)
where d1 = d m m1 1 2( , ), d2 = d m2 2( ) are some positive constants.
Denote by Sn
1,ε and Sn
2,ε the first and the second terms of (45). After a usual
calculations we get
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
1496 O. L. IZYUMTSEVA
Sn
1,ε ≤ 1
2
1 1 1
1 2
0
1
ln ( , )
det ( ( ))/ε
+
=
−
∑c d d E
B Y k n nk
n
(46)
with some positive constant c d d( , )1 2 .
For arbitrary r > 0, Sn
2,ε can be rewritten as
LE k
n
Y k
n
s x x Y k
nk
n
k
n
k
n
B Y k
n
r
X
=
−
+
∑ ∫ ∫
−
0
1
1
1 1 1
2
1
2
∆ , ,
, ; ,E
π
×
×
1
2 1
2
1 2 1
2
2
1
( ) ( )s s d s s d
dx ds
− + − +ε ε
�
+
+
LE k
n
Y k
n
s x x Y k
nk
n
k
n
k
n
B Y k
n
r
X
=
−
+
∑ ∫ ∫
−
0
1
1
1 1 1
2
2
1
2
∆ , ,\
, ; ,
R
E
π
×
×
1
2 1
2
1 2 1
2
2
1
( ) ( )s s d s s d
dx ds
− + − +ε ε
�
. (47)
where B Y k
n
r
, = x x Y k
n
r1 1: −
≤{ } .
The first and the second terms of (47) will be denoted by Sn r
2,
,
ε and Sn r
2,
,
ε .
Let us consider Sn r
2,
,
ε . It can be estimated as follows:
Sn r
2,
,
ε ≤
LrE
s s d s s d
ds
k
n
k
n
k
n
=
−
+
∑ ∫ − + − +0
1
1 2 1
2
1 2 1
2
2
2
1
2
1
∆ ,
( ) ( )π ε ε
�
=
= 1
2
1 1
1 2π ε
Lr c d d
n
n
ln ( , )+
− .
Consider Sn r
2,
,
ε . By using (32) we can write
EX
k
n
Y k
n
s x, ; ,
1 1 ≤ 1
2 1
2
1
2
1
π s k n
e
x Y k n
s k n
−( )
− −
−( )
/
( / )
/ . (48)
By using (48) and polar coordinates transformation we get
Sn r
2,
,
ε ≤ LE
s k n
e x Y k
nk
n
k
n
k
n
B Y k
n
r
x Y k n
s k n
=
−
+
− −
−( )∑ ∫ ∫ −( ) −
0
1
1 1
2
1
2
2
1
2
11
2
∆ , ,
( / )
/
\
/
R
π
×
×
1
2 1
2
1 2 1
2
2
1
( ) ( )s s d s s d
dx ds
− + − +ε ε
�
=
=
LE
s k n
e s k
n
d ds
k
n
k
n
k
n
r=
−
+
+∞
−∑ ∫ ∫−
−
0
1
1 1
1
2
2
21 2
∆ ,
/
ρ ρ ρ
�
.
Put
ρ
s k n1 − /
= v, then
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
THE CONSTANT OF RENORMALIZATION FOR SELF-INTERSECTION LOCAL TIME … 1497
Sn r
2,
,
ε ≤ LE
s k n
s k
n
s k
nk
n
k
n
k
n
=
−
+
∑ ∫ −
−
−
0
1
1 1
1 1
2
1
∆ ,
/
×
×
r s k n
d
e
s s d s s d
ds
1
2
2
2
2 1
2
1 2 1
2
2
1
−
+∞ −
∫ − + − +/ ( ) ( )
v
v v
ε ε
�
≤
≤ 1 1 1
1 2n
L
n
n
c d d∗ − +
ln ( , )
ε
with some positive constant L∗ . By using (17), (23), one can check that
T2,ε
n ≤ c n2
1( ) ln
ε
,
T3,ε
n ≤ c n3
1( ) ln
ε
,
T4,ε
n ≤ c n4( ) ,
where c n2( ) , c n3( ) c n4( ) are some positive constants such that
c n2 0( ) → as n → + ∞ ,
c n3 0( ) → as n → + ∞ .
Therefore, the following inequality holds:
CY
3 ≤ 1
2
1 1 1
1 2
0
1
π ε
ln ( , )
det ( ( ))/
+
=
−
∑c d d E
B Y k n nk
n
+
+ 1
2
1 1 1 1 1
1 2 1 2π ε ε
Lr c d d
n
n n
L
n
n
c d dln ( , ) ln ( , )+
− + − +
∗ +
+ c n c n c n c n2 3 4
1( ) ( ) ln ( ) ( )+( ) + +
ε
.
This inequality implies that
lim
lnε
ε
ε
→ +0 1
CY
≤ lim lim lim
( , )
ln( ) det ( ( ))/ /r n k
nc d d
E
B Y k n n→ + →∞ → + =
−
+
∑
0 0
1 2
0
1
1
2
1
1
1 1
ε π ε
+
+ 1
2
1
1
1 1 1 1
1
1 2 1 2
π ε ε
Lr
c d d n
n n
L
n
n
c d d+
− + − +
∗( , )
ln( )
( , )
ln( )/ /
+
+ ( ( ) ( ))
( )
ln( )
( )
ln( )/ /
c n c n
c n c n
2 3
4
1 1
+ + +
ε ε
= E
B Y s
ds1 1
2
0
1
det ( ( ))∫ π
. (49)
By using the same arguments, one can show that
lim
lnε
ε
ε
→ +0
1
CY
≥ 1
2
1
0
1
π
E
B Y s
ds
det ( ( ))∫ . (50)
The inequalities (49), (50) imply that
CY
ε ∼ E
B Y s
ds1 1
2
1
0
1
det ( ( ))
ln∫ π ε
, ε → 0 + .
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
1498 O. L. IZYUMTSEVA
Theorem is proved.
Remark. In this paper, we have investigated the asymptotic behavior of the con-
stant of renormalization for self-intersection local times of planar diffusion process.
The questions devoted to the existence of double self-intersections and self-intersecti-
ons of order k, the asymptotical behavior of the constant of renormalization for self-
intersection local times of diffusion process in k dimensions will be considered in fur-
ther papers.
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Received 25.05.07
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 11
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