A White Noise Approach to the Self Intersection Local Times of a Gaussian Process.
J o u r n a l
OF THE
.,,,toDONESIAN|
ME^^TICAL
^.
fi
'-'^t
yxwvutsrponmlkjihgfedcbaYXWVUTSRQNMLKJIGFECA
W ^
^i ^
'•
'^ 1
S o c i e t y
Vol. 20 No. 2
October 2014
..
'
0,
A White Noise Approach
113
as £ —> 0, where Pe is the heat kernel given by
Pe{x)
•-
1
—f=
zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
(
x^S
exp -zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRPONMLKJIHGFE
—
, X 6 K.
V 27r£
V
2£ /
T h i s a p p r o x i m a t i o n procedure w i l l make the l i m i t i n g object, which we denote by
L x ( T ' ) , more and more singular as the dimension of the process X increases. Hence,
we need to do a renormalization, i.e. cancelation of the divergent terms, to o b t a i n
a well-defined and sufficiently regular object.
Now we describe briefly our m a i n results. Under some conditions on the spat i a l dimension of the b-Gaussian process X and the number of subtracted terms
in the truncated Donsker's delta function, we are able to show the existence of the
renormalized (or truncated) self-intersection local t i m e Lx (T) as a well-defined object i n some w h i t e noise d i s t r i b u t i o n space. Moreover, we derive the chaos decomposition of Lx{T) i n terms of W i c k tensor powers of w h i t e noise. T h i s decomposition
corresponds to t h a t i n terms of multiple W i e n e r - I t o integrals when one works i n
the classical stochastic analysis using Wiener space as the u n d e r l y i n g p r o b a b i l i t y
space and B r o w n i a n m o t i o n as the basic r a n d o m variable. Finally, we also analyze
a regularization corresponding to the Gaussian a p p r o x i m a t i o n described above and
prove a convergence result. The organization of the paper is as follows. I n section
2 we summarize some of the standard facts from the theory of w h i t e noise analysis.
Section 3 contains a detailed exposition of the m a i n results and their proofs.
2.
BASICS
OF W H I T E NOISE
ANALYSIS
I n order to make the paper self-contained, we summarize some fundamental
concepts of w i i i t e noise analysis used throughout this paper. For a more comprehensive explanation including various applications of w h i t e noise theory, see for
example, the books of H i d a et al [4], K u o [6] and Obata [9]. Let ( 5 ^ ( R ) , C , / i ) be
the R'^-valued w h i t e noise space, i.e., 5^(]R) is tfie space of R'^-valued tempered
distributions,yxwvutsrponmlkjihgfedcbaYXWVUTSRQNMLKJIGFECA
C is the Borel cr-algebra generated by weak topology on C is the S-transforrn of a unique Hida
distribution in {S)' if and only if it satisfies the conditions:
(1) F
is ray analytic,
i.e., for every f,g£
Sd{R)
F ^ A / + 5 ^ has an entire extension to X £C,
the mapping R 9 A i->
and
(2) F has growth of second order, i.e., there exist constants K\,K2
continuous seminorm
F{zf)
||-|| on SdfR)
such that for all z eC,
0 and a
/ G >Sd(R)
/ "
There are two i m p o r t a n t consequences of the above characterization theorem. T h e
first one deals w i t h the Bochner integration of a family of Hida distributions which
depend on an additional parameter and the second one concerns tlie convergence
of sequences of H i d a distributions. For details and proofs see [5].
C o r o l l a r y 2.2. [5] Let {U,A,ir)
be a measure space and A
$ A 6e a mapping
from n to [Sy.
If the S-transform O/4>A fulfds the folloviing two conditions:
(1) the mapping A i-> 5 ( $ A ) ( / ) is measurable for all f G «Sd(R), and
C2(A) G L°°{n,A,u)
and a continuous
(2) there exist C i ( A ) G L^{n,A,ir),
seminorm
\\-\\ on Sd{R) such that for all z £C,
5($A)U/)
s. Define a m a p p i n g F A :
5 d ( R ) ^ C by F A : = S (exp {iX{Xt
Fxif)
=
( ( e x p [iX {{;
- X, - c ) ) ) . T h e n we have for any / G Sd{R)
l[o,t)/) -
(•, 1 ( 0 , . ] / ) -
c)) , : exp ( ( • , / " ) )
:))
A White Noise Approach
/
1
= exp
2
117
/ ^ expzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
i-iXc)
exp ( ( w , i A l [ s , t ] / + fj^ dp{Q)
= exp [ - \ | A p
l / M p d n ) exp (zA ( ( / ,
- c)) .
The mapping A i-4- % ( / ) is measurable for all / e 5 d ( R ) . Furthermore, letyvutsronmljihgfedcaVSQMJIE
2 GC
and / GyxwvutsrponmlkjihgfedcbaYXWVUTSRQNMLKJIGFECA
SfR),
then
Fxizf)
rt
< exp ( - ^ | A p r\f{u)fdu\exp
([AjUl | ( / , l [ . , q / ) |)
< exp ( - - I
|/(u)p du ) exp
< exp I
|/(n)pdn
- ^ 1
| 2|/?(t-6')^sup|/,(n)
exp
\rjf{uwdu
< e x p ( - l | A | V ( t - 5 ) ) exp
for some positive constants a and f, and || •
defined as
is a continuous seminorm on Sf
d
||/||oo:=$7sup|/,(K)|.
1=1
The first factor is an integrable function of A, and the second factor is constant
with respect to A. Hence, according to the Corollary 2.2 dU {Xt - X^ - c) £ ( 5 ) * .
Now, we integrate F A over Rf to obtain an explicit expression for tfie S-transforrn.
S{bd{Xt-X,-c)){f)
= (y^)
S (exp {iX{Xt - X , - c))) ( / ) riA
2\
V^'AL
271
I exp
271
\ D2
(U{!lfj{u)J{u)du-c,))
2/;|/(iz)|2dn
/
2\
exp
27i/;|/(u)Pdn
/
ffu)J{u)du-c,
\
2/;i/(ii)Prfix^tt
Now we prove our main results on self-intersection local times Lx{T)
I
and their
subtracted counterparts L ' ' Y { T ) I n the sequel we fix tlie following notations:
A := { ( s , t) G R % 0 < s < t < T } and d'^{s, t) is the Lebesgue measure on A . For
118
H . P . SURYAWAN
simplicity, we also take c = 0. Moreover, we define the truncated exponential series
e x p ( ^ ) ( x ) : =yxwvutsrponmlkjihgfedcbaYXWVUTSRQNMLKJIGFECA
Y
and the truncated Donsker's delta function
AM
sT'i^t-xf
via
5(0
: = jjT{Xt-Xf
L'PiT)
b-Gaussian process.
For
such that 2N > d - 2, the Bochner integral
d\s,t)
is a Hida distribution.
PROOF.
From the definition of the truncated Donsker's delta function we see
immediately t h a t S
(Xt - Xs)j
( / ) is a measurable function for every / G
S d { R ) . Furthermore, for everyyvutsronmljihgfedcaVSQMJIE
2 G C and / G 5 d ( R )
{Xt - Xf)
<
{zf)
Y'^
1YV
(
\2nfl\f{uWdu)
<
<
where {t-s)'^
(
v
(
d/2
1
27ra2 )
1
j-^i^
2 \
exp
d^
1
jt-
(xV"
27ra2
2/;|/(n)|2dn'
exp
n
2a2
(t-sO^-'^/^expf^lzP
V 2Q2
f
OO
-I 2 \
oo/
2
"^/^ is integrable w i t h respect to d^(s, t) on A if and only i f N-d/2
- 1 . Therefore we can conclude, using Corollary 2 . 2 , t h a t L { ^ \ T ) G ( 5 ) * .
>
•
Moreover, we are able t o derive the chaos decomposition for the (truncated) selfintersection local times L ' ' Y { T ) .
A White Noise Approach
119
P r o p o s i t i o n 3.3. L e i X = (XtyvutsronmljihgfedcaVSQMJIE
)jgjQ zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
be a d-dimensional b-Gaussian process. For
any pair of integers d>l
F2m of ^Y
and N > 0 such that 2N > d - 2, the kernel functions
{T) are given by
/'
[[i=i\---is,t\j
,2/
( 1 Y "yxwvutsrponmlkjihgfedcbaYXWVUTSRQNMLKJIGFECA
f
n ? r i iVM
( i M / )J (y-ii
^o
V27r,
(/:i/(a)Pd«)
{-\r
L 2 m ( w i , • • • ,'"2m) =
m!
for each m £ NQ such that m>
P R O O F . Let / = ( / i , . . . , ff
follow:
SYV
Y^x^'iT))
N. All other odd kernel functions Fm vanish.
The S-transform of L ) ^ ' ( T ) is obtained as
e SfR).
1
if) = 1^
d/2
(2nJl\f{uydu)
I
exp
X
{N)
dfs,t)
\
2/;i/(
YY
m+d/2
[rjfiuWdu)
2m i
mj,...,md
^
mi+...+md=m
j=l Ms
J
Remember the general form of the chaos decomposition
FTY)=
Y
(••^^'"••:Fm)
and sYx^\T)){f)=
Y
{Fm,
f^""
meN:'0
Hence, we can read ofi' the kernel functions F ^ for
L^YY):
®2m
F,.
=
^
V27r/
m!
More precisely, for every m
F2m{u\,. . .,U2m)
G
IA
f
ft
Y'lfiuWdu)
Ng such t h a t
m>
N
and
m+d/2
d^(s,t).
u i , . . . , U2m
=
while ail other odd kernels
m!
V2W
4(j;|/(.)|2,,)
m+d/2
G
R i t holds
dfs,t),
are identically equal to zero. •
Theorem 3.2 asserts t h a t for one-dimensional b-Gaussian process ail self-intersection
local times ^YiT)
are well-defined as Hida distributions. For d > 2, self-intersection
local times only become well-defined after omission of the divergent terms. I n particular, we o b t a i n the generalized expectation of renormalized self-intersection local
H.P.
120
SURYAWAN
times L yvutsronmljihgfedcaVSQMJIE
) 7'yxwvutsrponmlkjihgfedcbaYXWVUTSRQNMLKJIGFECA
{T) which is given by
7)''7AzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRPONMLKJIHG
7 ^ ^ - 4 7 ^
E . ( L 4 m ) = Fo =
It is immediate t h a t the generalized expectation is finite only in dimension one,
and for higher dimension (d > 2) the expectation blows up. Now we extend the
result i n Theorem 3.2 to local times of intersection of higher order m € N . The
basic m o t i v a t i o n for this investigation comes from the situation when we want to
count the amount of time i n which the sample p a t h of a b-Gaussian process spends
intersect itself m-times w i t h i n the time interval [0, T ] . The following theorem gives
a generalization to a result of Mendonca and Streit [8] on Brownian motion.
T h e o r e m 3.4.zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
Let X = {Xt)t^[Qg\
be a b-Gaussian process. For any pair of
integers d > 1 and N > 0 such that 2N > d —2 the (truncated) rn-tuple intersection
local time of X
4 E m
AM
-
(%,-X,J...54(X,,„-X,,„_,)d-t,
where A , „ : = { ( t i , • • • , t,„) e R™ : 0 < G < • • • < t,„ < T }
Lebesgue measure on A „ , , is a Hida distribution.
PROOF.
(4)
and d'^t denotes
Let
m-l
Im:=l[^d''\Xt,,,-Xty
k-fc=l
denote the integrand i n (4). T h e n
m-l
/
/
s{im)u)=n
fc=l
\ \
\ d/2
1
2 7 r/ ;;- i/ ( u)|2dRy
/
X
exp
2\ \
1
(M
f3{u)f{u)du
v
which is a measurable function for every / G 5 d ( R ) . Now, we ciieck the boundedness
condition.
Let 2 G C and / G «Sd(R). Let us also define a continuous
seminorm
Hoo.fc o n 5 ( R ) by l / j U . f c : = sup„g(t^,t,,^,] |/j(u)| and a continuous seminorm |M|oo,fc
on 5rf(R) by ||/||^_, : = E •=! UMo.k- Then, we have
m-\
s{im){zf)\
=
I (
n
fc=i V
\
d/2
1
\2rrl\Y\f{uydxy
2\ \
X
expkM
\f{u)\du
2/;-'|/H|2dn
the
A White Noise Approach
m-l
fc=i
/ m - l
X d/2
exp
27ra2(G.
121
\
{N)
\fc=l
Finally, by defining another continuous seminorm || • ||» on 0. To conclude the proof, we notice t h a t for X — d/2 + 1 > 0 the
coefficient i n the front of the exponential is integrable w i t h respect to d ' " t , t h a t is
/•
7
Y\(f
(r(i + /v-rf/2) r-'
t xN-dn,m,
l[
r(m + l +
"''^
(m-l)(iV-d/2))'
where r ( - ) denotes the usual G a m m a function. Therefore we may apply Gorollary
2.2 t o establish the existence of the Bochner integral asserted i n the theorem.
•
To conclude the section, we present a regularization result corresponding
to the renormalization procedure as described in Theorem 3.2.
We define the
regularized Donsker's delta function of b-Gaussian process as
1
0 and d > 1 the regularized self-intersection
distribution with kernel functions
Fe,2m(Ml, • • • ,M2m)
=
b-Gaussian process.
local time Lx,e{T)
in the chaos decomposition given by
m+d/2
m!
for each m e Ng, and F j . m is identically to zero if m is an odd number.
the (truncated) regularized self-intersection local times
LYAT)
: = JjZ^
{Xt-Xf
dfs,t)
Moreover,
d\s,t)
converges strongly as £ —> 0 in ( 5 ) * to the (truncated) local times L^YiY,
2 X > d - 2 .
For
is a Hida
provided
122
H.P.
SURYAWAN
P R O O F . T h e first part of the proof again follows by an application of Corollary 2.2
w i t h respect to the Lebesgue measure on A . For all / € 5,i(R) we o b t a i n
s{Sd,AXt-xf){f)
zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFE
\
/
d/2
1
2^ ( e + / ; i / ( u ) P d i z )
2\
•Y(
i =U
l
X exp
2(f
+ Jl\f{u)\^du)
f
Mu)f{u)du
^-^^
which is evidently measurable. Hence for allyvutsronmljihgfedcaVSQMJIE
2 6 C we have
S{6d,AXt-Xf){zf)
^d/2
I
(t -
exp
<
sfA
\2(e + /;|/(K)Pdn)
eY\l\!(u)\^du)}
7
\ d/2
We observe t h a t
, A, j K i , j
is bounded on A and ( — 7 — - - ,
|/(")Pdu
, 1.
\^27r(£ + F | / ( u ) P d u ) J
tegrabie on A . Hence, by Corollary 2.2, we have t h a t Lx.fT)
IS i n -
G ( 5 ) * . Moreover,
for e v e r y / G 5 d ( R ) ,
d/2
1
siwAmn-A)
LY VYVV
,
m+d/2
d
p
j A l i i
E
mi,...,md
mi +
...+md=m
Sl\f{uWdu)
[e +
pt
.2mj
fAAAAdu]
dfs,t).
j - i V
I t follows from tlie last expression t i i a t the kernel functions F ^ ^ m appearing i n the
chaos decomposition
LxAT)=
E
{••^^^'"••'Ym)
are of the form
Fe,2m(wi,---,R2m)
=
m!
V2^y
[e +
fAfiuWdu)
m+d/2
dfs,t),
and are identically equal to zero if m is an odd number. F i n a l l y we have to check
the convergence of L ^ l i T ) as e ^ 0. For all 2 G C and all / G Sd{R) we have
S [ L Y A T ) ) (2/) I < y ^ |S
{Xt - Xf) (2/) I dfsA)
A White Noise Approach
d/2
123
. ^ s
2
TA_
exp
<
\
2zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK
E
giving the boundedness condition. Furthermore, using similar calculations as i n the
proof of Theorem 3.2 we obtain t h a t for ail (s, t) e A
s
if)
{Xt - Xf)
d/2
\2ir
<
f
exp
Z\f{u)\^du
1
\
2
(N)
,2/;i/(w)Pdn
d/2
_
f]
27ra2
«lY-d/2
exp
\
/
YT
2
2Q2
\
oo/
The last upper bound is an integrable function w i t h respect to d 2 ( s , t). Finally, we
can apply Lebesgue's dominated convergence theorem to get the other condition
needed for the application of Corollary 2.3. T i l l s finisiies tiie proof. •
4.
CONCLUDING
REMARKS
We have proved under some conditions on tiie number of divergent terms
must be subtracted and the spatial dimension, t h a t self-intersection local times of ddimensionai b-Caussian process as well as their regularizations, after appropriatelly
renormalized, are H i d a distributions. The power of the method of t r u n c a t i o n based
on tiie fact t i i a t tiie kernel functions i n tlie chaos decomposition of decreasing order
are more and more singular i n the sense of Lebesgue integrable function. F x p i i c i t
expressions for the chaos decompositions of the self-intersection local times are also
presented. We also remark t h a t white noise approach provides a general idea on
renormalization procedures. T h i s idea can be further developed using another tools
such as M a i i i a v i n calculus to obtain regularity results.
REFERENCES
[1] d a F a r i a , M . , H i d a , T . , S t r c i t , L . , a n d W a t a n a b e , H . , " I n t e r s e c t i o n local t i m e s as generalized
w h i t e noise f u n c t i o n a l s ' " , Acta
[2] D r u m o n d , C.,
B r o w n i a n motions w i t h H
Quantum
Appl.
Math.
4 6 (1997), 351 - 362.
O l i v c i r a , M . J . , a n d d a S i l v a , J . L . , " I n t e r s e c t i o n local t i m e s of f r a c t i o n a l
Dynamics
G ( 0 , 1 ) as generalized w h i t e noise f u n c t i o n a l s " . Stochastic
of Biomolecular
Systems.
AIP
Conf.
Proc.
[3] E d w a r d s , S. P . , " T h e s t a t i s t i c a l m e c h a n i c s of p o l y m e r s w i t h e x c l u d e d v o l u m e " , Proc.
Soc.
and
( 2 0 0 8 ) , 34 - 45.
Phys.
8 5 ( 1 9 6 5 ) , 613 - 624.
[4] H i d a , T . . K u o , H - H . , Potthoff,
Calculus,
J . , a n d S t r c i t , L . , White
Noise.
An
Infinite
Dimensional
K l u w c r A c a d e m i c P u b l i s h e r s , 1993.
[5] K o n d r a t i e v , Y . , L e u k e r t , P., Potthoff,
functionals in G a u s s i a n s p a c e s :
J . , Streit, L . , and Westerkamp, W . , " G e n e r a l i z e d
T h e c h a r a c t e r i z a t i o n t h e o r e m r e v i s i t e d " , J. Funct.
1 4 1 ( 1 9 9 6 ) , 301 - 318.
[6] K u o , H - H . , White
Noise
Distribution
Theory,
C R C P r e s s , B o c a R a t o n , 1996.
Anal.
zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
H . P . SURYAWAN
124
zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
to Stochastic
Integration,
S p r i n g e r V c r l a g , H e i d e l b e r g , 2006.
[7] K u o , H - H . , Introduction
[8] M e n d o n c a ,
Infinite
S. a n d S t r c i t , L . , " M u l t i p l e intersection local t i m e s in t e r m s of w h i t e n o i s e " .
Dimensional
[9] O b a t a , N . , White
Analysis,
Noise
Quantum
Calculus
Probability
and Fock Space,
and Related
Topics
4 ( 2 0 0 1 ) , 533 - 543.
Springer Vcrlag, Heidelberg,
[10] S y m a n z i k , K . , " E u c l i d e a n q u a n t u m field t h e o r y " , R . J o s t ( e d ) . Local Quantum
demic Press, New York,
1994,
Theory,
Aca-
1969.
[11] W a t a n a b e , H . , " T h e l o c a l time of sclf-intorscctions of B r o w n i a n m o t i o n s as generalized B r o w n i a n f u n c t i o n a l s " , Lett.
Math.
Phys.
2 3 ( 1 9 9 1 ) , 1 - 9.
OF THE
.,,,toDONESIAN|
ME^^TICAL
^.
fi
'-'^t
yxwvutsrponmlkjihgfedcbaYXWVUTSRQNMLKJIGFECA
W ^
^i ^
'•
'^ 1
S o c i e t y
Vol. 20 No. 2
October 2014
..
'
0,
A White Noise Approach
113
as £ —> 0, where Pe is the heat kernel given by
Pe{x)
•-
1
—f=
zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
(
x^S
exp -zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRPONMLKJIHGFE
—
, X 6 K.
V 27r£
V
2£ /
T h i s a p p r o x i m a t i o n procedure w i l l make the l i m i t i n g object, which we denote by
L x ( T ' ) , more and more singular as the dimension of the process X increases. Hence,
we need to do a renormalization, i.e. cancelation of the divergent terms, to o b t a i n
a well-defined and sufficiently regular object.
Now we describe briefly our m a i n results. Under some conditions on the spat i a l dimension of the b-Gaussian process X and the number of subtracted terms
in the truncated Donsker's delta function, we are able to show the existence of the
renormalized (or truncated) self-intersection local t i m e Lx (T) as a well-defined object i n some w h i t e noise d i s t r i b u t i o n space. Moreover, we derive the chaos decomposition of Lx{T) i n terms of W i c k tensor powers of w h i t e noise. T h i s decomposition
corresponds to t h a t i n terms of multiple W i e n e r - I t o integrals when one works i n
the classical stochastic analysis using Wiener space as the u n d e r l y i n g p r o b a b i l i t y
space and B r o w n i a n m o t i o n as the basic r a n d o m variable. Finally, we also analyze
a regularization corresponding to the Gaussian a p p r o x i m a t i o n described above and
prove a convergence result. The organization of the paper is as follows. I n section
2 we summarize some of the standard facts from the theory of w h i t e noise analysis.
Section 3 contains a detailed exposition of the m a i n results and their proofs.
2.
BASICS
OF W H I T E NOISE
ANALYSIS
I n order to make the paper self-contained, we summarize some fundamental
concepts of w i i i t e noise analysis used throughout this paper. For a more comprehensive explanation including various applications of w h i t e noise theory, see for
example, the books of H i d a et al [4], K u o [6] and Obata [9]. Let ( 5 ^ ( R ) , C , / i ) be
the R'^-valued w h i t e noise space, i.e., 5^(]R) is tfie space of R'^-valued tempered
distributions,yxwvutsrponmlkjihgfedcbaYXWVUTSRQNMLKJIGFECA
C is the Borel cr-algebra generated by weak topology on C is the S-transforrn of a unique Hida
distribution in {S)' if and only if it satisfies the conditions:
(1) F
is ray analytic,
i.e., for every f,g£
Sd{R)
F ^ A / + 5 ^ has an entire extension to X £C,
the mapping R 9 A i->
and
(2) F has growth of second order, i.e., there exist constants K\,K2
continuous seminorm
F{zf)
||-|| on SdfR)
such that for all z eC,
0 and a
/ G >Sd(R)
/ "
There are two i m p o r t a n t consequences of the above characterization theorem. T h e
first one deals w i t h the Bochner integration of a family of Hida distributions which
depend on an additional parameter and the second one concerns tlie convergence
of sequences of H i d a distributions. For details and proofs see [5].
C o r o l l a r y 2.2. [5] Let {U,A,ir)
be a measure space and A
$ A 6e a mapping
from n to [Sy.
If the S-transform O/4>A fulfds the folloviing two conditions:
(1) the mapping A i-> 5 ( $ A ) ( / ) is measurable for all f G «Sd(R), and
C2(A) G L°°{n,A,u)
and a continuous
(2) there exist C i ( A ) G L^{n,A,ir),
seminorm
\\-\\ on Sd{R) such that for all z £C,
5($A)U/)
s. Define a m a p p i n g F A :
5 d ( R ) ^ C by F A : = S (exp {iX{Xt
Fxif)
=
( ( e x p [iX {{;
- X, - c ) ) ) . T h e n we have for any / G Sd{R)
l[o,t)/) -
(•, 1 ( 0 , . ] / ) -
c)) , : exp ( ( • , / " ) )
:))
A White Noise Approach
/
1
= exp
2
117
/ ^ expzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
i-iXc)
exp ( ( w , i A l [ s , t ] / + fj^ dp{Q)
= exp [ - \ | A p
l / M p d n ) exp (zA ( ( / ,
- c)) .
The mapping A i-4- % ( / ) is measurable for all / e 5 d ( R ) . Furthermore, letyvutsronmljihgfedcaVSQMJIE
2 GC
and / GyxwvutsrponmlkjihgfedcbaYXWVUTSRQNMLKJIGFECA
SfR),
then
Fxizf)
rt
< exp ( - ^ | A p r\f{u)fdu\exp
([AjUl | ( / , l [ . , q / ) |)
< exp ( - - I
|/(u)p du ) exp
< exp I
|/(n)pdn
- ^ 1
| 2|/?(t-6')^sup|/,(n)
exp
\rjf{uwdu
< e x p ( - l | A | V ( t - 5 ) ) exp
for some positive constants a and f, and || •
defined as
is a continuous seminorm on Sf
d
||/||oo:=$7sup|/,(K)|.
1=1
The first factor is an integrable function of A, and the second factor is constant
with respect to A. Hence, according to the Corollary 2.2 dU {Xt - X^ - c) £ ( 5 ) * .
Now, we integrate F A over Rf to obtain an explicit expression for tfie S-transforrn.
S{bd{Xt-X,-c)){f)
= (y^)
S (exp {iX{Xt - X , - c))) ( / ) riA
2\
V^'AL
271
I exp
271
\ D2
(U{!lfj{u)J{u)du-c,))
2/;|/(iz)|2dn
/
2\
exp
27i/;|/(u)Pdn
/
ffu)J{u)du-c,
\
2/;i/(ii)Prfix^tt
Now we prove our main results on self-intersection local times Lx{T)
I
and their
subtracted counterparts L ' ' Y { T ) I n the sequel we fix tlie following notations:
A := { ( s , t) G R % 0 < s < t < T } and d'^{s, t) is the Lebesgue measure on A . For
118
H . P . SURYAWAN
simplicity, we also take c = 0. Moreover, we define the truncated exponential series
e x p ( ^ ) ( x ) : =yxwvutsrponmlkjihgfedcbaYXWVUTSRQNMLKJIGFECA
Y
and the truncated Donsker's delta function
AM
sT'i^t-xf
via
5(0
: = jjT{Xt-Xf
L'PiT)
b-Gaussian process.
For
such that 2N > d - 2, the Bochner integral
d\s,t)
is a Hida distribution.
PROOF.
From the definition of the truncated Donsker's delta function we see
immediately t h a t S
(Xt - Xs)j
( / ) is a measurable function for every / G
S d { R ) . Furthermore, for everyyvutsronmljihgfedcaVSQMJIE
2 G C and / G 5 d ( R )
{Xt - Xf)
<
{zf)
Y'^
1YV
(
\2nfl\f{uWdu)
<
<
where {t-s)'^
(
v
(
d/2
1
27ra2 )
1
j-^i^
2 \
exp
d^
1
jt-
(xV"
27ra2
2/;|/(n)|2dn'
exp
n
2a2
(t-sO^-'^/^expf^lzP
V 2Q2
f
OO
-I 2 \
oo/
2
"^/^ is integrable w i t h respect to d^(s, t) on A if and only i f N-d/2
- 1 . Therefore we can conclude, using Corollary 2 . 2 , t h a t L { ^ \ T ) G ( 5 ) * .
>
•
Moreover, we are able t o derive the chaos decomposition for the (truncated) selfintersection local times L ' ' Y { T ) .
A White Noise Approach
119
P r o p o s i t i o n 3.3. L e i X = (XtyvutsronmljihgfedcaVSQMJIE
)jgjQ zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
be a d-dimensional b-Gaussian process. For
any pair of integers d>l
F2m of ^Y
and N > 0 such that 2N > d - 2, the kernel functions
{T) are given by
/'
[[i=i\---is,t\j
,2/
( 1 Y "yxwvutsrponmlkjihgfedcbaYXWVUTSRQNMLKJIGFECA
f
n ? r i iVM
( i M / )J (y-ii
^o
V27r,
(/:i/(a)Pd«)
{-\r
L 2 m ( w i , • • • ,'"2m) =
m!
for each m £ NQ such that m>
P R O O F . Let / = ( / i , . . . , ff
follow:
SYV
Y^x^'iT))
N. All other odd kernel functions Fm vanish.
The S-transform of L ) ^ ' ( T ) is obtained as
e SfR).
1
if) = 1^
d/2
(2nJl\f{uydu)
I
exp
X
{N)
dfs,t)
\
2/;i/(
YY
m+d/2
[rjfiuWdu)
2m i
mj,...,md
^
mi+...+md=m
j=l Ms
J
Remember the general form of the chaos decomposition
FTY)=
Y
(••^^'"••:Fm)
and sYx^\T)){f)=
Y
{Fm,
f^""
meN:'0
Hence, we can read ofi' the kernel functions F ^ for
L^YY):
®2m
F,.
=
^
V27r/
m!
More precisely, for every m
F2m{u\,. . .,U2m)
G
IA
f
ft
Y'lfiuWdu)
Ng such t h a t
m>
N
and
m+d/2
d^(s,t).
u i , . . . , U2m
=
while ail other odd kernels
m!
V2W
4(j;|/(.)|2,,)
m+d/2
G
R i t holds
dfs,t),
are identically equal to zero. •
Theorem 3.2 asserts t h a t for one-dimensional b-Gaussian process ail self-intersection
local times ^YiT)
are well-defined as Hida distributions. For d > 2, self-intersection
local times only become well-defined after omission of the divergent terms. I n particular, we o b t a i n the generalized expectation of renormalized self-intersection local
H.P.
120
SURYAWAN
times L yvutsronmljihgfedcaVSQMJIE
) 7'yxwvutsrponmlkjihgfedcbaYXWVUTSRQNMLKJIGFECA
{T) which is given by
7)''7AzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRPONMLKJIHG
7 ^ ^ - 4 7 ^
E . ( L 4 m ) = Fo =
It is immediate t h a t the generalized expectation is finite only in dimension one,
and for higher dimension (d > 2) the expectation blows up. Now we extend the
result i n Theorem 3.2 to local times of intersection of higher order m € N . The
basic m o t i v a t i o n for this investigation comes from the situation when we want to
count the amount of time i n which the sample p a t h of a b-Gaussian process spends
intersect itself m-times w i t h i n the time interval [0, T ] . The following theorem gives
a generalization to a result of Mendonca and Streit [8] on Brownian motion.
T h e o r e m 3.4.zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
Let X = {Xt)t^[Qg\
be a b-Gaussian process. For any pair of
integers d > 1 and N > 0 such that 2N > d —2 the (truncated) rn-tuple intersection
local time of X
4 E m
AM
-
(%,-X,J...54(X,,„-X,,„_,)d-t,
where A , „ : = { ( t i , • • • , t,„) e R™ : 0 < G < • • • < t,„ < T }
Lebesgue measure on A „ , , is a Hida distribution.
PROOF.
(4)
and d'^t denotes
Let
m-l
Im:=l[^d''\Xt,,,-Xty
k-fc=l
denote the integrand i n (4). T h e n
m-l
/
/
s{im)u)=n
fc=l
\ \
\ d/2
1
2 7 r/ ;;- i/ ( u)|2dRy
/
X
exp
2\ \
1
(M
f3{u)f{u)du
v
which is a measurable function for every / G 5 d ( R ) . Now, we ciieck the boundedness
condition.
Let 2 G C and / G «Sd(R). Let us also define a continuous
seminorm
Hoo.fc o n 5 ( R ) by l / j U . f c : = sup„g(t^,t,,^,] |/j(u)| and a continuous seminorm |M|oo,fc
on 5rf(R) by ||/||^_, : = E •=! UMo.k- Then, we have
m-\
s{im){zf)\
=
I (
n
fc=i V
\
d/2
1
\2rrl\Y\f{uydxy
2\ \
X
expkM
\f{u)\du
2/;-'|/H|2dn
the
A White Noise Approach
m-l
fc=i
/ m - l
X d/2
exp
27ra2(G.
121
\
{N)
\fc=l
Finally, by defining another continuous seminorm || • ||» on 0. To conclude the proof, we notice t h a t for X — d/2 + 1 > 0 the
coefficient i n the front of the exponential is integrable w i t h respect to d ' " t , t h a t is
/•
7
Y\(f
(r(i + /v-rf/2) r-'
t xN-dn,m,
l[
r(m + l +
"''^
(m-l)(iV-d/2))'
where r ( - ) denotes the usual G a m m a function. Therefore we may apply Gorollary
2.2 t o establish the existence of the Bochner integral asserted i n the theorem.
•
To conclude the section, we present a regularization result corresponding
to the renormalization procedure as described in Theorem 3.2.
We define the
regularized Donsker's delta function of b-Gaussian process as
1
0 and d > 1 the regularized self-intersection
distribution with kernel functions
Fe,2m(Ml, • • • ,M2m)
=
b-Gaussian process.
local time Lx,e{T)
in the chaos decomposition given by
m+d/2
m!
for each m e Ng, and F j . m is identically to zero if m is an odd number.
the (truncated) regularized self-intersection local times
LYAT)
: = JjZ^
{Xt-Xf
dfs,t)
Moreover,
d\s,t)
converges strongly as £ —> 0 in ( 5 ) * to the (truncated) local times L^YiY,
2 X > d - 2 .
For
is a Hida
provided
122
H.P.
SURYAWAN
P R O O F . T h e first part of the proof again follows by an application of Corollary 2.2
w i t h respect to the Lebesgue measure on A . For all / € 5,i(R) we o b t a i n
s{Sd,AXt-xf){f)
zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFE
\
/
d/2
1
2^ ( e + / ; i / ( u ) P d i z )
2\
•Y(
i =U
l
X exp
2(f
+ Jl\f{u)\^du)
f
Mu)f{u)du
^-^^
which is evidently measurable. Hence for allyvutsronmljihgfedcaVSQMJIE
2 6 C we have
S{6d,AXt-Xf){zf)
^d/2
I
(t -
exp
<
sfA
\2(e + /;|/(K)Pdn)
eY\l\!(u)\^du)}
7
\ d/2
We observe t h a t
, A, j K i , j
is bounded on A and ( — 7 — - - ,
|/(")Pdu
, 1.
\^27r(£ + F | / ( u ) P d u ) J
tegrabie on A . Hence, by Corollary 2.2, we have t h a t Lx.fT)
IS i n -
G ( 5 ) * . Moreover,
for e v e r y / G 5 d ( R ) ,
d/2
1
siwAmn-A)
LY VYVV
,
m+d/2
d
p
j A l i i
E
mi,...,md
mi +
...+md=m
Sl\f{uWdu)
[e +
pt
.2mj
fAAAAdu]
dfs,t).
j - i V
I t follows from tlie last expression t i i a t the kernel functions F ^ ^ m appearing i n the
chaos decomposition
LxAT)=
E
{••^^^'"••'Ym)
are of the form
Fe,2m(wi,---,R2m)
=
m!
V2^y
[e +
fAfiuWdu)
m+d/2
dfs,t),
and are identically equal to zero if m is an odd number. F i n a l l y we have to check
the convergence of L ^ l i T ) as e ^ 0. For all 2 G C and all / G Sd{R) we have
S [ L Y A T ) ) (2/) I < y ^ |S
{Xt - Xf) (2/) I dfsA)
A White Noise Approach
d/2
123
. ^ s
2
TA_
exp
<
\
2zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK
E
giving the boundedness condition. Furthermore, using similar calculations as i n the
proof of Theorem 3.2 we obtain t h a t for ail (s, t) e A
s
if)
{Xt - Xf)
d/2
\2ir
<
f
exp
Z\f{u)\^du
1
\
2
(N)
,2/;i/(w)Pdn
d/2
_
f]
27ra2
«lY-d/2
exp
\
/
YT
2
2Q2
\
oo/
The last upper bound is an integrable function w i t h respect to d 2 ( s , t). Finally, we
can apply Lebesgue's dominated convergence theorem to get the other condition
needed for the application of Corollary 2.3. T i l l s finisiies tiie proof. •
4.
CONCLUDING
REMARKS
We have proved under some conditions on tiie number of divergent terms
must be subtracted and the spatial dimension, t h a t self-intersection local times of ddimensionai b-Caussian process as well as their regularizations, after appropriatelly
renormalized, are H i d a distributions. The power of the method of t r u n c a t i o n based
on tiie fact t i i a t tiie kernel functions i n tlie chaos decomposition of decreasing order
are more and more singular i n the sense of Lebesgue integrable function. F x p i i c i t
expressions for the chaos decompositions of the self-intersection local times are also
presented. We also remark t h a t white noise approach provides a general idea on
renormalization procedures. T h i s idea can be further developed using another tools
such as M a i i i a v i n calculus to obtain regularity results.
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w h i t e noise f u n c t i o n a l s ' " , Acta
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