Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 414195, 27 pages
doi:10.1155/2012/414195
Research Article
On the Self-Intersection Local Time of
Subfractional Brownian Motion
Junfeng Liu,1 Zhihang Peng,2 Donglei Tang,1 and Yuquan Cang1
1
School of Mathematics and Statistics, Nanjing Audit University, 86 West Yu Shan Road, Pukou,
Nanjing 211815, China
2
Department of Epidemiology and Biostatistics, Nanjing Medical University, 140 Hanzhong Road,
Gulou, Nanjing 210029, China
Correspondence should be addressed to Junfeng Liu, jordanjunfeng@163.com
Received 16 May 2012; Accepted 24 October 2012
Academic Editor: Ahmed El-Sayed
Copyright q 2012 Junfeng Liu et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We study the problem of self-intersection local time of d-dimensional subfractional Brownian
motion based on the property of chaotic representation and the white noise analysis.
1. Introduction
As an extension of Brownian motion, Bojdecki et al. 1 introduced and studied a rather
special class of self-similar Gaussian process. This process arises from occupation time fluctuations of branching particles with Poisson initial condition. It is called the subfractional
Brownian motion. The so-called subfractional Brownian motion with index H ∈ 0, 1 is a
H
mean zero Gaussian process SH
0 {S0 t, t ≥ 0} with the covariance function
1
2H
2H
H
2H
2H
s
,
RH t, s E SH
t
−
−
s|
t
tS
s
s
|t
0
0
2
1.1
H
for all s, t ≥ 0. For H 1/2, SH
0 coincides with the standard Brownian motion. S0 is neither
a semimartingale nor a Markov process unless H 1/2, so many of the powerful techniques
from classical stochastic analysis are not available when dealing with SH
0 . The subfractional
Brownian motion has properties analogous to those of fractional Brownian motion, such as
2
Abstract and Applied Analysis
self-similarity, Hölder continuous paths, and so forth. But its increments are not stationary,
because, for s ≤ t, we have the following estimates:
2 H
≤ 2 − 22H−1 ∨ 1 t − s2H .
−
S
2 − 22H−1 ∧ 1 t − s2H ≤ E SH
t
s
0
0
1.2
Let SH {SH t, t ∈ N } be a d-dimensional subfractional Brownian motion with multiparameters H H1 , H2 , . . . , Hd . Suppose that d ≥ 2, we are interested in, when it
exists, the self-intersection local time of subfractional Brownian motion SH which is formally
defined as
TH
T t
0
δ0 SH t − SH s dsdt,
1.3
0
where δ0 is the Dirac delta function. It measures the amount of time that the processes spend
intersecting itself on the time interval 0, T and has been an important topic of the theory of
stochastic process.
More precisely, we study the existence of the limit when ε tends to zero, of the following sequence of processes
H
T,ε
T t
0
pε SH t − SH s dsdt,
1.4
0
where
pε x 2πε
−d/2
|x|2
exp −
2ε
2π−d
2
Rd
eiξ·x−ε|ξ| /2 dξ,
x ∈ Rd .
1.5
For H 1/2, the process SH
0 is a classical Brownian motion. The self-intersection local
time of the Brownian motion has been studied by many authors such as Albeverio et al. 2,
Calais and Yor 3, He et al. 4, Hu 5, Varadhan 6, and so forth. In the case of planar
1/2
does not converge in L2 but it can be
Brownian motion, Varadhan 6 has proved that T,ε
1/2
− T/2π log1/ε converges in L2 as ε tends to zero. The limit is
renormalized so that T,ε
called the renormalized self-intersection local time of the planar Brownian motion. This result
has been extended by Rosen 7 to the planar fractional Brownian motion, where it is
H,fBm
− CH T ε−11/2H converges in L2 as ε tends to zero,
proved that for 1/2 < H < 3/4, T,ε
where CH is a constant depending only on H. Hu 8 showed that, under the condition H <
min3/2d, 2/d 2, the renormalized self-intersection local time of fractional Brownian
motion is in the Meyer-Watanabe test functional space, that is, the L2 space of “differentiable”
functionals. In 2005, Hu and Nualart 9 proved that the renormalized self-intersection local
time of d-dimensional fractional Brownian motion exists in L2 if and only if H < 3/2d, which
generalizes the Varadhan renormalization theorem to any dimension and with any Hurst
H,fBm
converges in
parameter. They also showed that in the case 3/4 > H ≥ 3/2d, rεT,ε
2
−1
distribution to a normal law N0, T σ , as ε tends to zero, and rε | log ε| if H 3/2d,
and rε εd−3/2H if 3/2d < H. Wu and Xiao 10 proved the existence of the intersection
Abstract and Applied Analysis
3
local times of Ni , d, i 1, 2-fractional Brownian motions, and had a continuous version.
They also established Hölder conditions for the intersection local times and determined
the Hausdorff and packing dimensions of the sets of intersection times and intersection
points. They extended the results of Nualart and Ortiz-Latorre 11, where the existence
of the intersection local times of two independent 1, d-fractional Brownian motions with
the same Hurst index was studied by using a different method. Moreover, Wu and Xiao
10 also showed that anisotropy brings subtle differences into the analytic properties of the
intersection local times as well as rich geometric structures into the sets of intersection times
and intersection points. Oliveira et al. 12 presented expansions of intersection local times of
fractional Brownian motions in Rd , for any dimension d ≥ 1, with arbitrary Hurst coefficients
in 0, 1d . The expansions are in terms of Wick powers of white noises corresponding
to multiple Wiener integrals, being well-defined in the sense of generalized white noise
functionals. As an application of their approach, a sufficient condition on d for the existence of
Brownian motion,
intersection local times in L2 was also derived. For the case of subfractional
T
H2
1
t−S
Yan and Shen 13 studied the so-called collision local time T 0 δSH
0
0 tdt of two
independent subfractional Brownian motion with respective indices Hi ∈ 0, 1, i 1, 2. By
an elementary method, they showed that T is smooth in the sense of Meyer-Watanabe if and
only if minH1 , H2 < 1/3.
Motivated by all these results, we will study the self-intersection local time of the
so-called subfractional Brownian motion see below for a precise definition, which has
been proposed by Bojdecki et al. 1. Recently, the long-range dependence property has
become an important aspect of stochastic models in various scientific area including
hydrology, telecommunication, turbulence, image processing, and finance. It is well known
that fractional Brownian motion fBm in short is one of the best known and most widely used
processes that exhibits the long-range dependence property, self-similarity, and stationary
increments. It is a suitable generalization of classical Brownian motion. On the other hand,
many authors have proposed to use more general self-similar Gaussian process and random
fields as stochastic models. Such applications have raised many interesting theoretical
questions about self-similar Gaussian processes and fields in general. However, in contrast
to the extensive studies on fractional Brownian motion, there has been little systematic
investigation on other self-similar Gaussian processes. The main reason for this is the complexity of dependence structures for self-similar Gaussian processes which does not have
stationary increments. The subfractional Brownian motion has properties analogous to
those of fractional Brownian motion self-similarity, long-range dependence, Hölder paths,
the variation, and the renormalized variation. However, in comparison with fractional
Brownian motion, the subfractional Brownian motion has nonstationary increments and the
increments over nonoverlapping intervals are more weakly correlated and their covariance
decays polynomially as a higher rate in comparison with fractional Brownian motion for
this reason in Bojdecki et al. 1 is called subfractional Brownian motion. The above
mentioned properties make subfractional Brownian motion a possible candidate for models
which involve long-range dependence, self-similarity, and nonstationary. Therefore, it seems
interesting to study the self-intersection local time of subfractional Brownian motion. And we
need more precise estimates to prove our results because of the nonstationary increments. We
will view the self-intersection local time of subfractional Brownian motion as the generalized
white noise functionals. Furthermore, we discuss the existence and expansions of the selfintersection local times in L2 . We have organized our paper as follows: Section 2 contains the
notations, definitions, and results for Gaussian white noise analysis. In Section 3, we present
the main results and their demonstrations.
4
Abstract and Applied Analysis
Most of the estimates of this paper contain unspecified constants. An unspecified positive and finite constant will be denoted by C, which may not be the same in each occurrence.
Sometimes we will emphasize the dependence of these constants upon parameters.
2. Gaussian White Noise Analysis
In this section, we briefly recall the concepts and results of white noise analysis used through
out this work, and for details, see Kuo 15, Obata 16, and so forth.
2.1. Subfractional Brownian Motion
The starting point of white noise analysis for the construction of d-dimensional, d ≥ 1, subfractional Brownian motion is the real Gélfand triple
Sd R ⊂ L2 R, Rd ⊂ Sd R,
2.1
where L2 R, Rd is the real Hilbert space of all vector-valued square integrable functions with
respect to Lebesgue measure on R and Sd R, Sd R are the Schwartz spaces of the vectors
valued test functions and tempered distributions, respectively. Denote the norm in L2 R, Rd by | · |d or if there is no risk of confusion simply by | · | and the dual pairing between Sd and
Sd R by ·, ·
, which is defined as the bilinear extension of the inner product on L2 R, Rd ,
that is
g, f
d i1
R
2.2
gi xfi xdx,
for all g g1 , g2 , . . . gd ∈ L2 R, Rd and all f f1 , f2 , . . . , fd ∈ Sd R. By the Minlos
theorem, there is a unique probability measure μ on the σ-algebra B generated by the cylinder
sets on Sd R with characteristic function given by
Cf :
eiω,f
dμω
e−1/2|f| ,
2
Sd R
f ∈ Sd R.
2.3
In this way, we have defined the white noise measure space Sd R, B, μ. Then a realization
of vector of independent subfractional Brownian motion SHj , j 1, 2, . . . , d, is given by
S t ωj , KHj Hj
t
0
KHj t, sωj sds,
ωi ∈ Sd R.
2.4
Abstract and Applied Analysis
5
We recall the explicit formula for the kernel KH t, s of a one-dimensional subfractional
Brownian motion with parameter H ∈ 0, 1
t
H−1/2
d
πs1/2−H
2
2
u −s
du 10,t s
KH t, s − H
2 ΓH 1/2 ds
s
⎛
H−1/2 t 2
H−1/2 ⎞
√ 3/2−H
t2 − s2
u − s2
πs
⎝
H
du⎠10,t s.
t
u2
2 ΓH 1/2
s
√
2.5
Especially, for H > 1/2, we have
t
√
H−3/2
21−H π 3/2−H
2
2
KH t, s u −s
s
du 10,t s.
ΓH − 1/2
s
2.6
We refer to Bojdecki et al. 1, 17–19, Dzhaparidze and van Zanten 20, Liu and
Yan 21, Liu et al. 22, Shen and Yan 23, Tudor 24–27, Yan and Shen 13, 14, and the
references therein for a complete description of subfractional Brownian motion.
2.2. Hida Distributions and Characterization Results
Let us now consider the complex Hilbert space L2 : L2 Sd R, B, μ. This space is
canonically isomorphic to the symmetric Fock space of symmetric square integrable functions
L
2
Sd R, B, μ
∞
⊗2d
Sym L R , k!d x
2
k
G
k
2.7
k0
leading to the chaos expansion of the elements in L2 Sd R, B, μ
Fω1 , ω2 , . . . , ωd nn1 ,n2 ,...,nd ∈Nd
: ω1⊗n1 : ⊗ · · · ⊗ : ωd⊗nd :, fn ,
2.8
with kernel functions fn in the Fock space, that is, square integrable functions of the m
arguments and symmetric in each ni -tuple.
For simplicity, in the sequel, we will use the notations
n n1 , . . . , nd ∈ Nd ,
n
d
ni ,
n! i1
d
ni !,
2.9
i1
which reduces expansion 2.8 to
Fω n∈Nd
: ω⊗n :, fn ,
ω ∈ Sd R.
2.10
6
Abstract and Applied Analysis
The norm of F is given by
F2L2 n!|fn |22,n ,
2.11
n∈Nd
where | · |2,n is the norm in L2 Rn , dt.
To proceed further, we have to consider a Gelfand triple around the space L2 . We
will use the space S∗ of Hida distributions or generalized Brownian functionals and
the corresponding Gelfand triple S ⊂ L2 ⊂ S∗ . Here S is the space of white noise
test functions such that its dual space with respect to L2 is the space S∗ . Instead of
reproducing the explicit construction of S∗ in Theorem 2.2 below we characterize this space
through its S-transform. We recall that given a f ∈ Sd R, let us consider the Wick exponential
1
: expω, f
: exp ω, f
− f, f
2
1
: ω⊗n :, f⊗n Cfeω,f
,
n!
n∈Nd
ω∈
Sd R.
2.12
We define the S-transform of a Φ ∈ S∗ by
SΦf :
Φ, : exp·, f
: ,
∀f ∈ Sd R.
2.13
Here ·, ·
denotes the dual pairing between S∗ and S which is defined as the bilinear
extension of the sesquilinear inner product on L2 . We observe that the multilinear expansion
of 2.13
SΦf :
Fn , fn ⊗n ,
n
2.14
extends the chaos expansion to Φ ∈ S∗ with distribution valued kernels Fn such that
Φ, ϕ n! Fn , ϕn ,
n
2.15
for every generalized test function ϕ ∈ S with kernel function ϕn .
In order to characterize the space S∗ through its S-transform, we need the following
definition:
Definition 2.1. A function F : Sd R → C is called a U-functional whenever
1 for every f1 , f2 ∈ Sd R and λ ∈ R, the mapping λ → Fλf1 f2 has an entire
extension to λ ∈ C,
Abstract and Applied Analysis
7
2 there are constants K1 , K2 > 0 such that
|Fzf| ≤ K1 exp K2 |z|2 f2 ,
∀z ∈ C, f ∈ Sd R,
2.16
for some continuous norm · on Sd R.
We are now ready to state the aforementioned characterization result.
Theorem 2.2. The S-transform defines a bijection between the space S∗ and the space of Ufunctionals.
As a consequence of Theorem 2.2, one may derive the next two statements. The first
one concerns the convergence of sequences of Hida distributions and the second one the
Bochner integration of families of distributions of the same type.
Corollary 2.3. Let Φn , n ∈ N be a sequence in S∗ such that
1 for all f ∈ Sd R, SΦn fn∈N is Cauchy sequence in C;
2 there are K1 , K2 > 0 such that for some continuous norm · on Sd R one has
|SΦn zf| ≤ K1 eK2 |z|
2
f2
z ∈ C, f ∈ Sd R, n ∈ N,
,
2.17
then Φn , n ∈ N converges strongly in S∗ to a unique Hida distribution.
Corollary 2.4. Let Ω, B, m be a measure space and λ → Φλ be a mapping from Ω to S∗ . We
assume that the S-transform of Φλ fulfills the following two properties:
1 the mapping λ → Sφλ f is measurable for every f ∈ Sd R,
2 the SΦλ f obeys a U-estimate
|SΦλ zf| ≤ C1 λeC2 λ|z| f ,
2
2
z ∈ C, f ∈ Sd R,
2.18
for some continuous · on S2d R and for C1 ∈ L1 Ω, m, C2 ∈ L∞ Ω, m. Then
S
Ω
Ω
dmλΦλ ∈ S ,
dmλΦλ f dmλSΦλ f.
2.19
Ω
3. Self-Intersection Local Time
Let us now consider the d-dimensional subfractional Brownian motion SH t with parameter
H H1 , H2 , . . . , Hd . In view of 2.4, for j 1, . . . , d, SHj t − SHj s ΔKj , ωj with
ΔKj KHj t, u − KHj s, u.
8
Abstract and Applied Analysis
Proposition 3.1. For each t and s strictly positive real numbers, the Bochner integral
δ S t − S s :
H
H
1
2π
d H
Rd
dλeiλS
t−SH s
3.1
is a Hida distribution with S-transform given by
Sδ S t − S s f 2π
H
H
−d/2
d
j1
1
δHj t, s
1/2
2 fj , ΔKj
exp − H
2δ j t, s
3.2
for all f f1 , . . . , fd ∈ Sd R and δHj t, s |ΔKj |22 .
Proof. The proof of this result follows from an applications of Corollary 2.4 to the S-transform
of the integrand function
H
Φω
: eiλS
t−SH s
,
ω
ω1 , . . . , ωd .
3.3
With respect to the Lebesgue measure on Rd . By symmetry, we assume that s ≤ t. For this
purpose, we begin by observing that, for f f1 , . . . , fd ∈ Sd R, and all λ ∈ C, one has
SΦf d
S eiλj ΔKj ,ωj fj ,
j1
S eiλj ΔKj ,ωj fj
e
−|fj |22 /2
e
−|fj |22 /2
S R
S R
dμj ωj e
3.4
iλj ΔKj ,ωj fj ,ωj 2
2
dμj ωj eiλj ΔKj −ifj ,ωj e−λj |ΔKj |2 /2 eiλj ΔKj ,fj .
The last equality was obtained by the definition of μ. Then we obtain
d
2
2
e−λj /2|ΔKj |2 eiλj ΔKj ,fj ,
SΦf 3.5
j1
which clearly fulfills the measurability condition. Moreover, for all z ∈ C, we find
|SΦzf| d
e−λj |ΔKj |2 /4
2
2
j1
≤
d
j1
d −λ2j |ΔKj |22 /4izλj ΔKj ,fj e
j1
e
−λ2j |ΔKj |22 /4
d
2
2
e−λj |ΔKj |2 /4|z||λj ||ΔKj ,fj | ,
j1
3.6
Abstract and Applied Analysis
9
where, for each j 1, . . . , d, the corresponding term in the second product is bounded by
⎞
|z|
2
exp⎝ · ΔKj , fj ⎠
ΔKj 2
2
⎛
2
3.7
because
−
λ2j ΔKj 2 |z|λj ΔKj , fj 2
4
2
λj 2
|z| |z|2 ΔKj , fj
−
ΔKj 2 − 2 ΔKj , fj .
2
ΔKj ΔKj 2
2
3.8
As a result
|SΦzf| ≤ e
−1/4
d
j1
λ2j |ΔKj |22 |z|2
e
d
2
2
j1 1/|ΔKj |2 |ΔKj ,fj |
,
3.9
where, as a function of λ, the first exponential is integrable on Rd and the second exponential
is constant.
An application of the result mentioned above completes the proof. In particular, it
yields 3.2 by integrating 3.5 over λ.
H
For the sequence of processes T,ε
given by 1.4 and 3.5, combining
−d
2π
Rd
2
H
H
dλe−ε|λ| /2 S eiλS t−S s f
2π
−d
Rd
3.10
d
2
2
dλ e−λj /2|ΔKj |2 ε eiλj ΔKj ,fj j1
and the elementary equality
R
e
−αx2 /2iβx
dx 2π −β2 /2α
e
,
α
3.11
then we have the following.
Corollary 3.2. For each t and s strictly positive real numbers,
H
T,ε
T t
dsdt
0
0
1
2π
H
d
Rd
dλeiλ·S
t−SH s−ελ2 /2
,
3.12
10
Abstract and Applied Analysis
is a Hida distribution with S-transform given by
H
ST,
ε f
2π
−d/2
T t
0
0
d
dsdt
j1
1
δHj t, s ε
2
fj , ΔKj
1/2 exp − Hj
2 δ t, s ε
3.13
for all f f1 , . . . , fd ∈ Sd R and δHj t, s |ΔKj |22 .
Theorem 3.3. For H ∈ 0, 1, every positive integer d ≥ 1, and ε > 0, the self-intersection local time
H
has the following chaos expansion:
T,ε
H
T,ε
m ∞
1
1
⊗2m
:
ω
,
−
:,
G
m,ε
2
2πd/2 m0 m!
1
3.14
where
Gm,ε T t
0
0
d
dsdt
1
δ t, s ε
j1
Hj
mj 1/2 ΔK
⊗2m
3.15
.
Proof. For every f ∈ SR, we calculate the S-transform of T,ε as follows:
ST,ε f 2π
−d/2
T t
0
2π
−d/2
j1
1
δHj t, s ε
fj , ΔKj
2
1/2 exp − Hj
2 δ t, s ε
T t
0
2π−d/2
0
d
dsdt
mj
d ∞
2mj
1
1
1
−
dsdt
fj , ΔKj
m
1/2
j
m!
2
0
δHj t, s ε
j1 mj 0 j
T t
m d
∞
1
1
1
⊗2m
⊗2m
f
.
,
ΔK
−
mj 1/2 j
H
j
m!
2
m0
j1 δ j t, s ε
dsdt
0
0
3.16
Comparing with the general form of the chaos expansion, we find that the kernel functions
are equal to
Gm,ε T t
0
0
d
dsdt
j1
1
δ t, s ε
Hj
mj 1/2 ΔK
⊗2m
.
3.17
For simplicity, we assume that the notation F G means that there are positive
constants C1 and C2 such that
C1 Gx ≤ Fx ≤ C2 Gx,
in the common domain of definition for F and G.
3.18
Abstract and Applied Analysis
11
Let us now compute the expectation of the self-intersection local time of the subfracH
, it is just the first chaos. So in view of Theorem 3.3
tional Brownian motion, ET,ε
T t
d
−d/2
H
E T,ε 2π
dsdt
0
0
j1
1
δHj t, s ε
3.19
1/2 .
2
Moreover, for all s ≤ t, the second moment of increments δHj t, s ESHj t − SHj s
satisfying the following estimate:
βHj ∧ 1 t − s2Hj ≤ δHj t, s ≤ βHj ∨ 1 t − s2Hj ,
3.20
H
with βHj 2 − 22Hj −1 . As the integrand in ET,ε
is always positive, we have
T t
d
1
H
≤ 2π−d/2
E T,ε
dsdt
1/2
0 0
j1
βHj ∧ 1 t − s2Hj ε
2π−d/2
T
ds
0
∗
We use the change of variables s εd/2H z : αεz with H ∗ H
≤
E T,ε
T/αε
αε
2π
dz
d/2
0
dz
2πεd/2
d
j1
0
d
j1
Hj ,
T − αεz
1/2
∗
d
βHj ∧ 1 εdHj /H z2Hj ε
j1
T/αε
αε
3.21
T −s
1/2 .
d
2Hj
β
∧
1
s
ε
Hj
j1
βHj
T − αεz
1/2 .
∗
∧ 1 εdHj /H −1 z2Hj 1
3.22
We divide the integral in two parts
H
≤
E T,ε
αε
2πε
⎧
⎪
⎨ 1
d/2 ⎪
⎩
dz
d
j1
0
βHj
T − αεz
1/2
∗
∧ 1 εdHj /H −1 z2Hj 1
⎫
⎪
⎬
T/αε
dz
1
d
j1
βHj
T − αεz
1/2 ⎪.
∗
⎭
∧ 1 εdHj /H −1 z2Hj 1
3.23
12
Abstract and Applied Analysis
The first integral in braces is bounded. Set I the second integral in braces, so
I≤
T
1/2
d
j1 βHj ∧ 1
⎧
∗
∗
⎪
T 2−H εd/2−d/2H − 1
⎪
∗
⎪
1
⎪
1/2 , H /
⎪
⎪
d
∗
⎪
T/αε
β
∧
1
−
H
⎨
1
j1 Hj
1
dz H ∗ (
)
⎪
z
T log T − d/2H ∗ log ε
⎪
1
⎪
⎪
, H ∗ 1.
⎪
1/2
⎪
⎪
d
⎩
j1 βHj ∧ 1
Proposition 3.4. Denote by H ∗ d
j1
3.24
Hj . Let T > 0 and d ≥ 1, then if H ∗ ≥ 1,
H
∞.
lim E T,ε
3.25
H
≤ CT,Hj ,d log ε.
E T,ε
3.26
∗
H
≤ CT,Hj ,d εd/2−d/2H .
E T,ε
3.27
ε→0
Moreover if H ∗ 1,
If H ∗ > 1
If H ∗ < 1, there is no blow up, that is,
⎡
⎢
H
∈⎣
lim E T,ε
ε→0
2πd/2
1
1/2
d
βHj ∨ 1
j1
T
ds
0
T −s
,
sH ∗
2πd/2
1
1/2
d
βHj ∧ 1
j1
⎤
T
ds
0
T −s ⎥
⎦.
sH ∗
3.28
In particular, we have
Proposition 3.5. Suppose that all Hj H, let T > 0 and d ≥ 1.
H
C ln1/ε oε.
1 If H 1/d, then ET,ε
H
2 If 1/d < H < 3/2d, then ET,ε
Cε1/2H−d/2 oε.
From the above results, if H ∗ < 1, the self-intersection local time TH is well defined in
S . This is same as the case of fractional Brownian motion mainly because the covariance
structure and the property 1.2 of the increments of the subfractional Brownian motion.
Suppose now that H ∗ ≥ 1. The idea is that if we subtract some of the first term in the
expansion of the exponential function in the expression of the S-transform of δSH t−SH s,
we could obtain an integrable function in factor of the remaining part, then the second
∗
Abstract and Applied Analysis
13
condition of Corollary 2.4 will be satisfied. And so we could define a renormalization of the
self-intersection local time in S∗ .
Let us denote the truncated exponential series by
expN x ∞
xn
nN
n!
3.29
.
It follows from 3.2 that the S-transform of δN is given by
Sδ
N
S t − S s f 2π
H
H
−d/2
2n
d
1 n fj , ΔKj j
1
· ,
−
n
1/2
Hj
j
2
nj !
n,n≥N
j1 δ t, s
3.30
so
N H
S t − SH s f
Sδ
≤ 2π
−d/2
d
1 n 2
n,n≥N
j1
βHj
2nj fj · ΔKj 2nj
1
1
∞
· .
nj 1/2
nj !
∧1
|t − s|2nj 1Hj
3.31
We need to estimate the L1 -norm of ΔKj , |ΔKj |1 for fixed j.
We will treat the case when all Hj > 1/2. For the case all Hj < 1/2, we do not have a
good estimation of |Kj |1 and so we do not have a result. Let Hj > 1/2, in view of 2.8
ΔKj KHj t, u − KHj s, u
t
s Hj −3/2
Hj −3/2
3/2−Hj
2
2
2
2
x −u
x −u
dx10,t u −
dx10,s u
CHj u
u
CHj u3/2−Hj
t
u
x 2 − u2
Hj −3/2
dx1s,t u
u
t x −u
2
2
Hj −3/2
dx10,s u −
s x −u
u
CHj u
3/2−Hj
t
u
2
2
Hj −3/2
dx10,s u
u
x −u
2
2
Hj −3/2
dx1s,t u t x −u
2
2
Hj −3/2
dx10,s u .
s
3.32
14
Abstract and Applied Analysis
We obtain
ΔKj 1
T
0
Chj
ΔKj udu
T
t x2 − u2
u3/2−Hj du
0
Hj −3/2
dx10,s u
s
T
u3/2−Hj du
0
t x 2 − u2
Hj −3/2
3.33
dx1s,t u
u
: CHj I1 I2 .
For I1 , we obtain
I1 T
u3/2−Hj du
0
≤
x2 − u2
Hj −3/2
dx10,s u
s
T
u
3/2−Hj
t
du
0
≤
t x − uHj −3/2 xHj −3/2 dx10,s u
s
t
x
Hj −3/2
x
dx
s
3.34
Hj −3/2 3/2−Hj
x − u
u
10,s udu
0
B 5/2 − Hj , Hj − 1/2
≤
|t − s|Hj 1/2 .
Hj 1/2
For I2 , we obtain
I2 T
u
3/2−Hj
du
0
≤
T
u3/2−Hj du
Hj −3/2
dx1s,t u
t
x − uHj −3/2 xHj −3/2 dx1s,t u
u
t
x
s
x 2 − u2
u
0
≤
t Hj −3/2
x
dx
3.35
x − u
Hj −3/2 3/2−Hj
u
du
0
B 5/2 − Hj , Hj − 1/2
≤
|t − s|Hj 1/2 .
Hj 1/2
So
ΔKj ≤ CH |t − s|Hj 1/2 .
j
1
3.36
Abstract and Applied Analysis
15
Suppose |t − s| is small enough, we get
N H
S t − SH s f
Sδ
≤ 2π
−d/2
2nj
fj d
1 n CHj 2
n,n≥N
j1
1
∞
· nj 1/2
nj !
∧1
|t − s|nj Hj −1/2Hj
βHj
≤ CH
d
3.37
d 1
fj 2nj · 1 ∞
n
2 j1
nj !
1
|t − s|nj Hj −1/2Hj
⎧
⎫
d ⎬
⎨1
1
f 2 ,
exp
≤ CH
∗
⎩ 2 j1 j ∞ ⎭
|t − s|NHmax −1/2H
n,n≥N j1
with H ∗ d
j1
Hj . Then we have.
Theorem 3.6. Let T > 0 and n ∈ N, suppose that
1
< 1,
H ∗ N Hmax −
2
3.38
then
H,N
T
T t
0
δN SH t − SH s dsdt,
3.39
0
is well defined as an element of S∗ and
H,N
lim ε → 0 T,ε
H,N
T
in S∗ .
,
3.40
Theorem 3.7. For H ∈ 0, 1, every positive integer d ≥ 1, and ε > 0, suppose that 3.38 holds,
H,N
then the truncated self-intersection local time T
has the following chaos expansion:
H,N
T
m ∞
1
1
⊗2m
:
ω
,
:,
G
−
m
2
2πd/2 m0 m!
1
3.41
where
Gm T t
0
0
d
dsdt
j1
1
δ t, s
Hj
mj 1/2 ΔK
⊗2m
,
for each m ∈ Nd and m ≥ N. All other kernel functions Gm are identically equal to zero.
3.42
16
Abstract and Applied Analysis
Proof. By Corollary 2.4, the S-transform of the truncated self-intersection local time is given
as an integral over 3.30. Then given f f1 , . . . , fd ∈ Sd R, we have
H,N
ST
f 2π
−d/2
2n
d
1 n fj , ΔKj j
1
dsdt
· .
−
n
1/2
Hj
j
2
nj !
0
n,n≥N
j1 δ t, s
T t
0
3.43
Comparing with the general form of chaos expansion, the result is proved.
Next we will estimate the L2 -norm of the chaos of the self-intersection local time of
subfractional Brownian motion. Now we state the result.
0 and d ≥ 2.
Theorem 3.8. Suppose all Hj H, let T > 0, n /
1 If Hd 1, then
1
lim 0
|G2n,ε |2,2n
ε→0
log ε
3.44
lim |G2n,ε |2,2n
3.45
exists.
2 If 1 < dH < 3/2, then
ε→0
exists.
3 If H > 3/2, then limε → 0 εd/2−3/4H |G2n,ε |2,2n exists.
Proof. For n ∈ Nd , the 2nth chaos is given by
T,2n,ε 2π−d/2
1 1 ⊗2n
:ω
:, G2n,ε ,
n
n! 2
3.46
where
G2n,ε T
t
dt
0
0
ΔKj⊗2n
d
ds
j1
δHj t, s ε
nj 1/2 .
3.47
So
ET,2n,ε 2 2n!
2πd n!2 22n
|G2n,ε |2L2 Rn .
3.48
Abstract and Applied Analysis
17
In view of 3.48, we need to estimate |G2n,ε |2L2 Rn |G2n,ε |22,2n , where
|G2n,ε |22,2n
T t T t
2n
R2n
d u
dtdsdt ds
0
0
0
0
d
j1
ΔKj t, s⊗2nj ΔKj t , s ⊗2nj
n 1/2 .
H
δ j t, s ε δHj t , s ε j
3.49
By using Fubini theorem, we first get
R2n
d2n u
d
⊗2nj
ΔKj t, s⊗2nj ΔKj t , s
j1
2nj d j1 i1
R
j
dui ΔKj t, st, sΔKj t, s t , s
3.50
2nj d d 2nj
Hj
Ct,s,t s
E SHj t − SHj s SHj t − SHj s ,
j1 i1
j1
Hj
Hj
with Ct,s,t s ESHj t − SHj sSHj t − SHj s . Moreover denote by Rt,s,t s EBHj t −
BHj sBHj t − BHj s with BH a fractional Brownian motion with Hurst index H ∈ 0, 1.
Then, from Bojdecki et al. 1, we know that
Hj
Hj
1
,
2
0 < Ct,s,t s < Rt,s,t s ,
H>
Hj
Rt,s,t s
1
H< .
2
<
Hj
Ct,s,t s
< 0,
3.51
So
|G2n,ε |22,2n ≤
T t T t
0
0
0
0
dtdsdt ds
d
j1
2nj
Hj
Rt,s,t s
n 1/2 .
H
δ j t, s ε δHj t , s ε j
3.52
In view of the symmetry of the domain and integrand function, it suffices to integrate only
on
T T1 ∪ T2 ∪ T3 ,
3.53
where T1 {0 < s < t < s < t}, T2 {0 < s < s < t < t}, T3 {0 < s < s < t < t}. Let us
first integrate over T1 . We make the following change of variables x t − s, t t − s , z s − t
i
and x y z t − s < t, where t is considered as a parameter. Set |G2n,ε |22,2n the integral over
Ti , i 1, 2, 3.
18
Abstract and Applied Analysis
1
Step 1. For |G2n,ε |22,2n , we obtain
1 2
G2n,ε 2,2n
≤ CH
T
dt
0
dxdydz
0≤xyz≤t
2nj
2H 2H
d
x z2Hj y z j − x y z j − z2Hj
·
.
)n 1/2
( 2H
x j ε y2Hj ε j
j1
3.54
It is almost possible to compute the integral when all Hj are different, so let us suppose that
all Hj are equal to some H.
Denote by θt ε ε−1/2H t and make the following change of variables x, y, z ε−1/2H x , y , z , we get
1 2
G2n,ε 2,2n
≤ CH ε3/2H−d
T
dt
0
0≤xyz≤θt ε
dxdydz
2n
2H 2H
− xyz
− z2H
x z2H y z
·
)nd/2
( 2H
x 1 y2H 1
T 3/2H−d
2CH ε
dt
dxdydz
0
3.55
0≤xyz≤θt ε,0≤x≤y≤θt ε
2n
2H 2H
− xyz
− z2H
x z2H y z
·
.
)nd/2
( 2H
x 1 y2H 1
Denote by
2H 2H
fH x, y, z x z2H y z
− xyz
− z2H ,
3.56
and by
)2n
( fH x, y, z
ft, H, d, n, ε dxdydz (
)nd/2 .
0≤xyz≤θt ε,0≤x≤y≤θt ε
x2H 1 y2H 1
3.57
First note that |fH x, y, z| ≤ x2H and for dH > 1,
0≤x≤y
dxdy (
x2H
x4Hn
2H
)nd/2 < ∞.
1 y 1
3.58
Abstract and Applied Analysis
19
This implies that for every x ∈ R ,
0≤x≤y
dxdy (
x2H
2n
fH x, y, z
)nd/2 ,
1 y2H 1
3.59
is finite. On the other hand, for α ∈ 1, 2 − 2H, zα fH x, y, z2n decreases to zero when z tends
to zero, then
fH, d, n R
dz
0≤x≤y
dxdy (
x2H
2n
fH x, y, z
)nd/2 < ∞,
1 y2H 1
3.60
and so that
lim ft, H, d, n, ε fH, d, n.
ε→0
3.61
Therefore, we obtain, if Hd ∈ 1, 3/2,
1 2
lim G2n,ε ε→0
2,2n
3.62
0,
and if Hd ≥ 3/2,
1 2
lim εd−3/2H G2n,ε ε→0
2,2n
2T fd, H, n.
3.63
ft, H, d, n, εdt,
3.64
Suppose now Hd 1,
1 2
G2n,ε 2,2n
2ε1/2H
T
0
with
ft, H, d, n, ε θt ε
0≤x≤y≤θt ε
dxdy (
x2H
x4Hn
2H
)nd/2 ,
1 y 1
3.65
and then
ft, H, d, n, ε
aH, d, n
bH, d, n ,
≤ θt ε
log ε
log ε
3.66
20
Abstract and Applied Analysis
where
aH, d, n 0≤x≤y,x≤1
1
bH, d, n sup sup 0≤t≤T ε>0 log ε
dxdy (
x2H
x4Hn
2H
)nd/2 ,
1 y 1
1≤x≤y≤θt ε
dxdy (
x4Hn
2H
)nd/2 .
1 y 1
x2H
3.67
So we obtain
1 1 2
2 aH, d, n
bH, d, n .
≤T
log ε GT,2n,ε 2,2n
log ε
3.68
2
Step 2. Let us now treat |G2n,ε |22,2n .
2 2
G2n,ε 2,2n
2ε3/2H−d
T
dtgt, H, d, n, ε,
3.69
0
where
)2n
gH x, y, z
)nd/2 ,
1 y2H 1
(
gt, H, d, n, ε ≤ CH
0≤xy−z≤θt ε,0≤z≤x≤y≤θt ε
dxdydz (
x2H
3.70
where
2H 2H
− xy−z
− z2H .
gH x, y, z x − z2H y z
3.71
We have that |gH x, y, z| ≤ 2x2H and so
)2n
gH x, y, z
)nd/2
1 y2H 1
)2n
( gH x, y, z
(
z −→
0≤z≤x≤y
dxdy (
z4Hn2
1≤x≤y
x2H
3.72
dxdy (
)nd/2 ,
z2H x2H 1z2H y2H 1
is well defined on R and for Hd ≥ 3/2, one can choose α ∈ 1, 2 − 2H such that when
z → ∞
)2n
( gH x, y, z
dxdy (
)nd/2 −→ 0.
0≤z≤x≤y
x2H 1 y2H 1
α
z
3.73
Abstract and Applied Analysis
21
And then
R
)2n
gH x, y, z
)nd/2 gH, d, n < ∞.
1 y2H 1
(
0≤z≤x≤y
dxdy (
x2H
3.74
So that
lim gt, H, d, n, ε gH, d, n.
ε→0
3.75
1
Suppose Hd 1, the same computations as in the case of |G2n,ε |22,2n leads to
1 2 2
2n 2 aH, d, n
bH, d, n .
≤2 T
log ε G2n,ε 2,2n
log ε
3.76
Suppose now Hd ∈ 1, 3/2, we have
gt, H, d, n, ε ≤ 22n
θt ε
dzz2−2Hd
0
1≤x≤y
x4Hn
2n
H2nd : 2 CH, d, n.
xy
dxdy 3.77
Then
2 2
G2n,ε 2,2n
≤ CH
22nCH,d,n
T 4−2Hd .
3 − 2Hd2 − Hd
1
3.78
2
We know that limε → 0 |G2n,ε |22,2n 0, then limε → 0 |G2n,ε |22,2n exists. Let us suppose that Hd 3/2,
2 2
G2n,ε 2,2n
2
T
dtht, H, d, n, ε
3.79
0
with
)2n
gH x, y, z
ht, H, d, n, ε ≤
dz
dxdy (
)nd/2
0
z≤x≤y≤θt ε
x2H 1 y2H 1
)2n
(
θt ε
gH x, y, z
dz
dxdy (
)nd/2 .
1
z≤x≤y≤θt ε
x2H 1 y2H 1
1
(
3.80
The first term is bounded by
2n
2
0≤x≤y
dxdy (
x2H
x4Hn
2n
2H
)nd/2 2 eH, d, n.
1 y 1
3.81
22
Abstract and Applied Analysis
The second term is bounded by
θt ε
1
1
dz 22n
z
1≤x≤y
dxdy xy
x4Hn
2n
H2nd/2 2 CH, d, n log|θt ε|.
3.82
So
22n1
1 2 2
eH, d, nT
≤ log ε G2n,ε 2,2n log ε
22n
22n1
CH, d, nT H
log ε
3.83
T
log tdt.
0
Therefore
1 2 2
log ε G2n,ε 2,2n
3.84
1 2 2
22n
G2n,ε CH, d, nT.
≤
log ε
2,2n
H
3.85
has a finite nontrivial limit and
3
Step 3. Finally, let us treat |G2n,ε |22,2n . Let x t − t , y t − s , z s − s. We know
3 2
G2n,ε 2,2n
≤ 2ε
3/2H−d
T
dtjt, H, d, n, ε,
3.86
0
where
jt, H, d, n, ε 0≤xyz≤θt ε
dxdydz (
x2H
2n
kH x, y, z
)nd/2 ,
1 y2H 1
3.87
with
2H
2H
kH x, y, z x y
− x2H y z
− z2H .
3.88
It is obvious that |kH x, y, z| ≤ 2y2H . So
jt, H, d, n, ε ≤ 2
2n
2y2H
)nd/2 .
1 y2H 1
0≤xyz≤θt ε,0≤x≤y≤θt ε
dxdydz (
x2H
3.89
Abstract and Applied Analysis
23
For Hd > 1,
0≤x≤y
(
x2H
2n
2y2H
)nd/2 dxdy < ∞,
1 y2H 1
3.90
2n
kH x, y, z
)nd/2 dxdy < ∞.
1 y2H 1
3.91
this implies that for every z ∈ R ,
0≤x≤y
(
x2H
On the other hand, for α ∈ 1, 2 − 2H, zα kH x, y, z2n decreases to zero when z tends to
infinity, then
gH, d, n R
dz
0≤x≤y
dxdy (
x2H
2n
kH x, y, z
)nd/2 < ∞,
1 y2H 1
3.92
and so that
lim gt, H, d, n, ε gH, d, n.
ε→0
3.93
Therefore, if Hd ∈ 1.3/2, we obtain
3 2
lim G2n,ε ε→0
2,2n
3.94
0.
If Hd ≥ 3/2,
3 2
lim εd−3/2H G2n,ε ε→0
2,2n
2T gH, d, n.
3.95
dtgt, H, d, n, ε,
3.96
Suppose now Hd 1,
3 2
G2n,ε 2,2n
2ε1/2H
T
0
where
gt, H, d, n, ε ≤ θt ε
2n
2y2H
)nd/2 ,
1 y2H 1
0≤x≤y≤θt ε
dxdy (
x2H
3.97
and then
gt, H, d, n, ε
aH, d, n
bH, d, n ,
≤ θt ε
log ε
log ε
3.98
24
Abstract and Applied Analysis
where
2n
2y2H
)nd/2 ,
1 y2H 1
aH, d, n 0≤x≤y,x≤1
1
bH, d, n sup sup 0≤t≤T ε>0 log ε
dxdy (
x2H
2n
2y2H
)nd/2 .
1 y2H 1
1≤x≤y≤θt ε
dxdy (
x2H
3.99
So we obtain
1 3 2
2 aH, d, n
bH, d, n .
≤T
log ε G2n,ε 2,2n
log ε
3.100
Theorem 3.9. Assume that Hi H, i 1, . . . , d, then for H ∈ 0, 1 and d ≥ 1 satisfying Hd < 1,
in L2 ,
H
−→ TH ,
T,ε
3.101
as ε tends to zero.
H
Proof. By Theorems 3.3 and 3.7, we only need to consider chaos expansion of T,ε
and TH .
Similar techniques in Oliveira et al. 12 allow us to write
2
H
I E T,2n
2n!
2π
d
n!2 22n
|G2n,ε |2L2 Rn .
3.102
In view of 3.102, we need to estimate |G2n |2L2 Rn |G2n |22,2n , where
|G2n |22,2n
m
1
m!
2π
d 2m
1
1
2
m!2
·
2n
R2n
1
2π
d u
T
d T
dtdsdt ds
λt,s λt ,s − μ2
d
ΔKj t, s⊗2nj ΔKj t , s ⊗2nj
n 1/2
H
δ j t, s δHj t , s j
j1
−d/2
dtdsdt ds ,
3.103
Abstract and Applied Analysis
25
where
λt,s |t s|2H |t − s|2H − 22H−1 t2H s2H ,
2H 2H
2H 2H
,
λt ,s t s t − s − 22H−1 t
s
μ
2H 2H 2H 2H
1 s − t s − t − t − t − s − s 2
2H 2H 2H 2H .
s − t s t − t t − s s 3.104
In order to show |I| < ∞, we need some preliminaries. Recall
T
1
2
s, t, s , t : 0 < s < t < T, 0 < s < t < T .
3.105
Without loss of generality, one can assume t < t . For any s, t, s , t ∈ T, we denote T1 {0 ≤
s < s < t < t ≤ T }, T2 {0 ≤ s < s < t < t ≤ T }, T3 {0 ≤ s < t < s < t ≤ T }. From Lemma
2.1 in Yan and Shen 13, we know that there exists a constant κ > 0 such that the following
three statements hold:
1 for s, t, s , t ∈ T1 ,
2H 2H 2H ;
t − s
λt,s λt ,s − μ2 ≥ κ t − s2H t − t
s −s
3.106
2 for s, t, s , t ∈ T2 ,
2H ;
λt,s λt ,s − μ2 ≥ κ t − s2H t − s
3.107
2H .
λt,s λt ,s − μ2 ≥ κ t − s2H t − s
3.108
3 for s, t, s , t ∈ T3 ,
Then we can easily check that, if Hd < 1, |I| < ∞. Moreover, T,ε converges to TH in L2 as ε
tends to zero.
Remark 3.10. In this work, we only study the existence of self-intersection of subfractional
Brownian motion in L2 under mild conditions. The asymptotic behavior and the central limit
H
H
− ET,ε
in L2 as ε tends to zero will be discussed in future works.
theorem of the difference T,ε
Acknowledgments
The authors want to thank the Academic Editor and anonymous referees whose remarks
and suggestions greatly improved the presentation of this paper. The Project is sponsored by
the Mathematical Tianyuan Foundation of China Grant No. 11226198, NSFC 11171062,
26
Abstract and Applied Analysis
NSFC 11201232, NSFC 81001288, NSFC 11271020, NSRC 10023, Innovation Program
of Shanghai Municipal Education Commission 12ZZ063, Priority Academic Program
Development of Jiangsu Higher Education Institutions and Major Program of Key Research
Center in Financial Risk Management of Jiangsu Universities Philosophy Social Sciences No:
2012JDXM009.
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