# On the Self-Intersection Local Time of Subfractional Brownian Motion

Authors: Junfeng Liu; Zhihang Peng; Donglei Tang; Yuquan Cang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We study the problem of self-intersection local time ofd-dimensional subfractional Brownian motion based on the property of chaotic representation and the white noise analysis.


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## Body

## 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 mean zero Gaussian process S0H={S0H(t),t≥0} with the covariance function
(1.1)RH(t,s)=𝔼[S0H(t)S0H(s)]=s2H+t2H-12[(s+t)2H+|t-s|2H],
for all s,t≥0. For H=1/2, S0H coincides with the standard Brownian motion. S0H 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 S0H. The subfractional Brownian motion has properties analogous to those of fractional Brownian motion, such as self-similarity, Hölder continuous paths, and so forth. But its increments are not stationary, because, for s≤t, we have the following estimates:
(1.2)[(2-22H-1)∧1](t-s)2H≤𝔼[(S0H(t)-S0H(s))2]≤[(2-22H-1)∨1](t-s)2H.LetSH={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
(1.3)ℓTH=∫0T∫0tδ0(SH(t)-SH(s))dsdt,
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 whenε tends to zero, of the following sequence of processes
(1.4)ℓT,εH=∫0T∫0tpε(SH(t)-SH(s))dsdt,
where
(1.5)pε(x)=(2πε)-d/2exp(-|x|22ε)=(2π)-d∫ℝd‍eiξ·x-(ε|ξ|2/2)dξ,x∈ℝd.ForH=1/2, the process S0H 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 Brownian motion, Varadhan [6] has proved that ℓT,ε1/2 does not converge in L2 but it can be renormalized so that ℓT,ε1/2-(T/2π)log(1/ε) converges in L2 as ε tends to zero. The limit is 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 proved that for 1/2<H<3/4, ℓT,εH,fBm-CHTε-1+(1/2H) converges in L2 as ε tends to zero, where CH is a constant depending only on H. Hu [8] showed that, under the condition H<min(3/(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 parameter. They also showed that in the case 3/4>H≥3/2d, r(ε)ℓT,εH,fBm converges in distribution to a normal law N(0,Tσ2), as ε tends to zero, and r(ε)=|logε|-1 if H=3/(2d), and r(ε)=εd-3/(2H) if 3/(2d)<H. Wu and Xiao  [10] proved the existence of the intersection 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 (0,1)d. 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 intersection local times in L2 was also derived. For the case of subfractional Brownian motion, Yan and Shen [13] studied the so-called collision local time ℓT=∫0T‍δ(S0H1(t)-S0H2(t))dt of two independent subfractional Brownian motion with respective indices Hi∈(0,1),i=1,2. By an elementary method, they showed that ℓT is smooth in the sense of Meyer-Watanabe if and only if min(H1,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 self-intersection 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.Most of the estimates of this paper contain unspecified constants. An unspecified positive and finite constant will be denoted byC, 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 ofd-dimensional, d≥1, subfractional Brownian motion is the real Gélfand triple
(2.1)𝒮d(ℝ)⊂L2(ℝ,ℝd)⊂𝒮d′(ℝ),
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
(2.2)〈g,f〉=∑i=1d‍∫ℝ‍gi(x)fi(x)dx,
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
(2.3)C(f):=∫𝒮d′(ℝ)‍ei〈ω→,f〉dμ(ω→)=e-(1/2)|f|2,f∈𝒮d(ℝ).
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
(2.4)SHj(t)=〈ωj,KHj〉=∫0t‍KHj(t,s)ωj(s)ds,ωi∈𝒮d′(ℝ).
We recall the explicit formula for the kernel KH(t,s) of a one-dimensional subfractional Brownian motion with parameter H∈(0,1)(2.5)KH(t,s)=-πs1/2-H2HΓ(H+1/2)dds(∫st‍(u2-s2)H-1/2du)1[0,t](s)=πs3/2-H2HΓ(H+1/2)((t2-s2)H-1/2t+∫st‍(u2-s2)H-1/2u2du)1[0,t](s).
Especially, for H>1/2, we have
(2.6)KH(t,s)=21-HπΓ(H-1/2)s3/2-H(∫st‍(u2-s2)H-3/2du)1[0,t](s).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
(2.7)L2(𝒮d′(ℝ),ℬ,μ)≃(⨁k=0∞SymL2(ℝk,k!dkx))⊗2d=𝒢
leading to the chaos expansion of the elements in L2(𝒮d′(ℝ),ℬ,μ)(2.8)F(ω1,ω2,…,ωd)=∑n=(n1,n2,…,nd)∈ℕd‍〈:ω1⊗n1:⊗⋯⊗:ωd⊗nd:,fn〉,
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(2.9)n=(n1,…,nd)∈ℕd,n=∑i=1d‍ni,n!=∏i=1d‍ni!,
which reduces expansion (2.8) to
(2.10)F(ω)=∑n∈ℕd‍〈:ω⊗n:,fn〉,ω∈𝒮d′(ℝ).
The norm of F is given by
(2.11)∥F∥(L2)2=∑n∈ℕd‍n!|fn|2,n2,
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
(2.12):exp〈ω,f〉≔exp(〈ω,f〉-12(f,f))=∑n∈ℕd‍1n!〈:ω⊗n:,f⊗n〉=C(f)e〈ω,f〉,ω∈𝒮d′(ℝ).
We define the S-transform of a Φ∈(S)* by
(2.13)SΦ(f):=〈〈Φ,:exp(〈·,f〉):〉〉,∀f∈𝒮d(ℝ).
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)
(2.14)SΦ(f):=∑n‍〈Fn,fn⊗n〉,
extends the chaos expansion to Φ∈(S)* with distribution valued kernels Fn such that
(2.15)〈〈Φ,φ〉〉=∑n‍n!〈Fn,φn〉,
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 functionF:𝒮d(ℝ)→ℂ is called a U-functional whenever(1)
for everyf1,f2∈𝒮d(ℝ) and λ∈ℝ, the mapping λ→F(λf1+f2) has an entire extension to λ∈ℂ,(2)
there are constantsK1,K2>0 such that
(2.16)|F(zf)|≤K1exp(K2|z|2∥f∥2),∀z∈ℂ,f∈𝒮d(ℝ),
for some continuous norm ∥·∥ on 𝒮d(ℝ).We are now ready to state the aforementioned characterization result.Theorem 2.2.
TheS-transform defines a bijection between the space (S)* and the space of U-functionals.As a consequence of Theorem2.2, 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 allf∈𝒮d(ℝ), ((SΦn)(f))n∈ℕ is Cauchy sequence in ℂ;(2)
there areK1,K2>0 such that for some continuous norm ∥·∥ on 𝒮d(ℝ) one has
(2.17)|(SΦn)(zf)|≤K1eK2|z|2∥f∥2,z∈ℂ,f∈𝒮d(ℝ),n∈ℕ,
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 mappingλ→(Sϕλ)(f) is measurable for every f∈𝒮d(ℝ),(2)
the(SΦλ)(f) obeys a U-estimate
(2.18)|(SΦλ)(zf)|≤C1(λ)eC2(λ)|z|2∥f∥2,z∈ℂ,f∈𝒮d(ℝ),for some continuous ∥·∥ on 𝒮2d(ℝ) and for C1∈L1(Ω,m), C2∈L∞(Ω,m). Then
(2.19)∫Ω‍dm(λ)Φλ∈(S)′,S(∫Ω‍dm(λ)Φλ)(f)=∫Ω‍dm(λ)SΦλ(f).


## 2.1. Subfractional Brownian Motion

The starting point of white noise analysis for the construction ofd-dimensional, d≥1, subfractional Brownian motion is the real Gélfand triple
(2.1)𝒮d(ℝ)⊂L2(ℝ,ℝd)⊂𝒮d′(ℝ),
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
(2.2)〈g,f〉=∑i=1d‍∫ℝ‍gi(x)fi(x)dx,
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
(2.3)C(f):=∫𝒮d′(ℝ)‍ei〈ω→,f〉dμ(ω→)=e-(1/2)|f|2,f∈𝒮d(ℝ).
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
(2.4)SHj(t)=〈ωj,KHj〉=∫0t‍KHj(t,s)ωj(s)ds,ωi∈𝒮d′(ℝ).
We recall the explicit formula for the kernel KH(t,s) of a one-dimensional subfractional Brownian motion with parameter H∈(0,1)(2.5)KH(t,s)=-πs1/2-H2HΓ(H+1/2)dds(∫st‍(u2-s2)H-1/2du)1[0,t](s)=πs3/2-H2HΓ(H+1/2)((t2-s2)H-1/2t+∫st‍(u2-s2)H-1/2u2du)1[0,t](s).
Especially, for H>1/2, we have
(2.6)KH(t,s)=21-HπΓ(H-1/2)s3/2-H(∫st‍(u2-s2)H-3/2du)1[0,t](s).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
(2.7)L2(𝒮d′(ℝ),ℬ,μ)≃(⨁k=0∞SymL2(ℝk,k!dkx))⊗2d=𝒢
leading to the chaos expansion of the elements in L2(𝒮d′(ℝ),ℬ,μ)(2.8)F(ω1,ω2,…,ωd)=∑n=(n1,n2,…,nd)∈ℕd‍〈:ω1⊗n1:⊗⋯⊗:ωd⊗nd:,fn〉,
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(2.9)n=(n1,…,nd)∈ℕd,n=∑i=1d‍ni,n!=∏i=1d‍ni!,
which reduces expansion (2.8) to
(2.10)F(ω)=∑n∈ℕd‍〈:ω⊗n:,fn〉,ω∈𝒮d′(ℝ).
The norm of F is given by
(2.11)∥F∥(L2)2=∑n∈ℕd‍n!|fn|2,n2,
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
(2.12):exp〈ω,f〉≔exp(〈ω,f〉-12(f,f))=∑n∈ℕd‍1n!〈:ω⊗n:,f⊗n〉=C(f)e〈ω,f〉,ω∈𝒮d′(ℝ).
We define the S-transform of a Φ∈(S)* by
(2.13)SΦ(f):=〈〈Φ,:exp(〈·,f〉):〉〉,∀f∈𝒮d(ℝ).
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)
(2.14)SΦ(f):=∑n‍〈Fn,fn⊗n〉,
extends the chaos expansion to Φ∈(S)* with distribution valued kernels Fn such that
(2.15)〈〈Φ,φ〉〉=∑n‍n!〈Fn,φn〉,
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 functionF:𝒮d(ℝ)→ℂ is called a U-functional whenever(1)
for everyf1,f2∈𝒮d(ℝ) and λ∈ℝ, the mapping λ→F(λf1+f2) has an entire extension to λ∈ℂ,(2)
there are constantsK1,K2>0 such that
(2.16)|F(zf)|≤K1exp(K2|z|2∥f∥2),∀z∈ℂ,f∈𝒮d(ℝ),
for some continuous norm ∥·∥ on 𝒮d(ℝ).We are now ready to state the aforementioned characterization result.Theorem 2.2.
TheS-transform defines a bijection between the space (S)* and the space of U-functionals.As a consequence of Theorem2.2, 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 allf∈𝒮d(ℝ), ((SΦn)(f))n∈ℕ is Cauchy sequence in ℂ;(2)
there areK1,K2>0 such that for some continuous norm ∥·∥ on 𝒮d(ℝ) one has
(2.17)|(SΦn)(zf)|≤K1eK2|z|2∥f∥2,z∈ℂ,f∈𝒮d(ℝ),n∈ℕ,
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 mappingλ→(Sϕλ)(f) is measurable for every f∈𝒮d(ℝ),(2)
the(SΦλ)(f) obeys a U-estimate
(2.18)|(SΦλ)(zf)|≤C1(λ)eC2(λ)|z|2∥f∥2,z∈ℂ,f∈𝒮d(ℝ),for some continuous ∥·∥ on 𝒮2d(ℝ) and for C1∈L1(Ω,m), C2∈L∞(Ω,m). Then
(2.19)∫Ω‍dm(λ)Φλ∈(S)′,S(∫Ω‍dm(λ)Φλ)(f)=∫Ω‍dm(λ)SΦλ(f).


## 3. Self-Intersection Local Time

Let us now consider thed-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).Proposition 3.1.
For eacht and s strictly positive real numbers, the Bochner integral
(3.1)δ(SH(t)-SH(s)):=(12π)d∫ℝd‍dλeiλ(SH(t)-SH(s))
is a Hida distribution with S-transform given by
(3.2)Sδ(SH(t)-SH(s))(f)=(2π)-(d/2)∏j=1d‍1(δHj(t,s))1/2exp{-〈fj,ΔKj〉22δHj(t,s)}
for all f=(f1,…,fd)∈𝒮d(ℝ) and δHj(t,s)=|ΔKj|22.Proof.
The proof of this result follows from an applications of Corollary2.4 to the S-transform of the integrand function
(3.3)Φ(ω→):=eiλ(SH(t)-SH(s)),ω→=(ω1,…,ωd).
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
(3.4)SΦ(f)=∏j=1d‍S(eiλj〈ΔKj,ωj〉)(fj),S(eiλj〈ΔKj,ωj〉)(fj)=e-(|fj|22/2)∫S′(ℝ)‍dμj(ωj)eiλj〈ΔKj,ωj〉+〈fj,ωj〉=e-(|fj|22/2)∫S′(ℝ)‍dμj(ωj)ei〈λjΔKj-ifj,ωj〉=e-(λj2|ΔKj|22/2)eiλj〈ΔKj,fj〉.
The last equality was obtained by the definition of μ. Then we obtain
(3.5)SΦ(f)=∏j=1d‍e-(λj2/2)|ΔKj|22eiλj〈ΔKj,fj〉,
which clearly fulfills the measurability condition. Moreover, for all z∈ℂ, we find
(3.6)|SΦ(zf)|=∏j=1d‍e-(λj2|ΔKj|22/4)∏j=1d‍|e-(λj2|ΔKj|22/4)+izλj〈ΔKj,fj〉|≤∏j=1d‍e-(λj2|ΔKj|22/4)∏j=1d‍e-(λj2|ΔKj|22/4)+|z||λj||〈ΔKj,fj〉|,
where, for each j=1,…,d, the corresponding term in the second product is bounded by
(3.7)exp(|z|2|ΔKj|22·|〈ΔKj,fj〉|2)
because
(3.8)-λj24|ΔKj|22+|z||λj||〈ΔKj,fj〉|=-(|λj|2|ΔKj|2-|z||ΔKj|2|〈ΔKj,fj〉|)2+|z|2|ΔKj|22|〈ΔKj,fj〉|2.
As a result
(3.9)|SΦ(zf)|≤e-(1/4)∑j=1dλj2|ΔKj|22e|z|2∑j=1d(1/|ΔKj|22)|〈ΔKj,fj〉|2,
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.For the sequence of processesℓT,εH given by (1.4) and (3.5), combining
(3.10)(2π)-d∫ℝddλe-ε(|λ|2/2)S(eiλ(SH(t)-SH(s)))(f)=(2π)-d∫ℝd‍dλ∏j=1d‍e-(λj2/2)(|ΔKj|22+ε)eiλj〈ΔKj,fj〉
and the elementary equality
(3.11)∫ℝ‍e-α(x2/2)+iβxdx=2παe-(β2/2α),
then we have the following.Corollary 3.2.
For eacht and s strictly positive real numbers,
(3.12)ℓT,εH=∫0T‍∫0t‍dsdt1(2π)d∫ℝd‍dλeiλ·(SH(t)-SH(s))-(ελ2/2),
is a Hida distribution with S-transform given by
(3.13)SℓT,εH(f)=(2π)-(d/2)∫0T‍∫0t‍dsdt∏j=1d‍1(δHj(t,s)+ε)1/2exp{-〈fj,ΔKj〉22(δHj(t,s)+ε)}
for all f=(f1,…,fd)∈𝒮d(ℝ) and δHj(t,s)=|ΔKj|22.Theorem 3.3.
ForH∈(0,1), every positive integer d≥1, and ε>0, the self-intersection local time ℓT,εH has the following chaos expansion:
(3.14)ℓT,εH=1(2π)d/2∑m=0∞‍1m!(-12)m〈:ω⊗2m:,Gm,ε〉,
where
(3.15)Gm,ε=∫0T‍∫0t‍dsdt∏j=1d‍1(δHj(t,s)+ε)mj+(1/2)ΔK⊗2m.Proof.
For everyf∈𝒮(ℝ), we calculate the S-transform of ℓT,ε as follows:
(3.16)SℓT,ε(f)=(2π)-(d/2)∫0T‍∫0t‍dsdt∏j=1d‍1(δHj(t,s)+ε)1/2exp{-〈fj,ΔKj〉22(δHj(t,s)+ε)}=(2π)-(d/2)∫0T‍∫0t‍dsdt∏j=1d‍∑mj=0∞‍1mj!(-12)mj1(δHj(t,s)+ε)mj+(1/2)〈fj,ΔKj〉2mj=(2π)-(d/2)∫0T‍∫0t‍dsdt∑m=0∞‍1m!(-12)m∏j=1d‍1(δHj(t,s)+ε)mj+(1/2)〈fj⊗2m,ΔKj⊗2m〉.
Comparing with the general form of the chaos expansion, we find that the kernel functions are equal to
(3.17)Gm,ε=∫0T‍∫0t‍dsdt∏j=1d‍1(δHj(t,s)+ε)mj+(1/2)ΔK⊗2m.For simplicity, we assume that the notationF≍G means that there are positive constants C1 and C2 such that
(3.18)C1G(x)≤F(x)≤C2G(x),
in the common domain of definition for F and G.Let us now compute the expectation of the self-intersection local time of the subfractional Brownian motion,𝔼(ℓT,εH), it is just the first chaos. So in view of Theorem 3.3(3.19)𝔼(ℓT,εH)=(2π)-(d/2)∫0T‍∫0t‍dsdt∏j=1d‍1(δHj(t,s)+ε)1/2.
Moreover, for all s≤t, the second moment of increments δHj(t,s)=𝔼(SHj(t)-SHj(s))2 satisfying the following estimate:
(3.20)[βHj∧1](t-s)2Hj≤δHj(t,s)≤[βHj∨1](t-s)2Hj,
with βHj=2-22Hj-1. As the integrand in 𝔼(ℓT,εH) is always positive, we have
(3.21)𝔼(ℓT,εH)≤(2π)-(d/2)∫0T‍∫0t‍dsdt∏j=1d‍1([βHj∧1](t-s)2Hj+ε)1/2=(2π)-(d/2)∫0T‍dsT-s∏j=1d([βHj∧1]s2Hj+ε)1/2.
We use the change of variables s=εd/2H*z:=α(ε)z with H*=∑j=1d‍Hj,
(3.22)𝔼(ℓT,εH)≤α(ε)(2π)d/2∫0T/α(ε)‍dzT-α(ε)z∏j=1d([βHj∧1]εdHj/H*z2Hj+ε)1/2=α(ε)(2πε)d/2∫0T/α(ε)‍dzT-α(ε)z∏j=1d([βHj∧1]ε(dHj/H*)-1z2Hj+1)1/2.
We divide the integral in two parts
(3.23)𝔼(ℓT,εH)≤α(ε)(2πε)d/2{∫01‍dzT-α(ε)z∏j=1d([βHj∧1]ε(dHj/H*)-1z2Hj+1)1/2+∫1T/α(ε)‍dzT-α(ε)z∏j=1d([βHj∧1]ε(dHj/H*)-1z2Hj+1)1/2}.
The first integral in braces is bounded. Set I the second integral in braces, so
(3.24)I≤T∏j=1d[βHj∧1]1/2∫1T/α(ε)‍dz1zH*={T2-H*ε(d/2)-(d/2H*)-1(1-H*)∏j=1d[βHj∧1]1/2,H*≠1T[logT-(d/2H*)logε]∏j=1d[βHj∧1]1/2,H*=1.Proposition 3.4.
Denote byH*=∑j=1d‍Hj. Let T>0 and d≥1, then if H*≥1,
(3.25)limε→0𝔼(ℓT,εH)=∞.
Moreover if H*=1,
(3.26)𝔼(ℓT,εH)≤CT,Hj,d|logε|.
If H*>1(3.27)𝔼(ℓT,εH)≤CT,Hj,dε(d/2)-(d/2H*).
If H*<1, there is no blow up, that is,
(3.28)limε→0𝔼(ℓT,εH)∈[1(2π)d/2∏j=1d([βHj∨1])1/2∫0T‍dsT-ssH*,1(2π)d/2∏j=1d([βHj∧1])1/2∫0T‍dsT-ssH*].In particular, we haveProposition 3.5.
Suppose that allHj=H, let T>0 and d≥1.(1)
IfH=1/d, then 𝔼ℓT,εH=Cln(1/ε)+o(ε).(2)
If1/d<H<3/(2d), then 𝔼ℓT,εH=Cε(1/2H)-(d/2)+o(ε).From the above results, ifH*<1, the self-intersection local time ℓTH 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 δ(SH(t)-SH(s)), we could obtain an integrable function in factor of the remaining part, then the second 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(3.29)expN(x)=∑n=N∞‍xnn!.
It follows from (3.2) that the S-transform of δ(N) is given by
(3.30)Sδ(N)(SH(t)-SH(s))(f)=(2π)-(d/2)∑n,n≥N‍(-12)n∏j=1d‍〈fj,ΔKj〉2njδHj(t,s)nj+(1/2)·1(nj)!,
so
(3.31)|Sδ(N)(SH(t)-SH(s))(f)|≤(2π)-(d/2)∑n,n≥N‍(12)n∏j=1d‍|fj|∞2nj·|ΔKj|12nj(βHj∧1)nj+(1/2)|t-s|(2nj+1)Hj·1(nj)!.
We need to estimate the L1-norm of ΔKj, |ΔKj|1 for fixed j.We will treat the case when allHj>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)
(3.32)ΔKj=KHj(t,u)-KHj(s,u)=CHju3/2-Hj{∫ut‍(x2-u2)Hj-3/2dx1[0,t](u)-∫us‍(x2-u2)Hj-3/2dx1[0,s](u)}=CHju3/2-Hj{∫ut‍(x2-u2)Hj-3/2dx1[s,t](u)+∫ut‍(x2-u2)Hj-3/2dx1[0,s](u)-∫us‍(x2-u2)Hj-3/2dx1[0,s](u)}=CHju3/2-Hj{∫ut‍(x2-u2)Hj-3/2dx1[s,t](u)+∫st‍(x2-u2)Hj-3/2dx1[0,s](u)}.
We obtain
(3.33)|ΔKj|1=∫0T‍ΔKj(u)du=Chj{∫0T‍u3/2-Hjdu∫st‍(x2-u2)Hj-3/2dx1[0,s](u)+∫0T‍u3/2-Hjdu∫ut‍(x2-u2)Hj-3/2dx1[s,t](u)}:=CHj(I1+I2).
For I1, we obtain
(3.34)I1=∫0T‍u3/2-Hjdu∫st‍(x2-u2)Hj-3/2dx1[0,s](u)≤∫0T‍u3/2-Hjdu∫st‍(x-u)Hj-3/2xHj-3/2dx1[0,s](u)≤∫st‍xHj-3/2dx∫0x‍(x-u)Hj-3/2u3/2-Hj1[0,s](u)du≤B(5/2-Hj,Hj-1/2)Hj+1/2|t-s|Hj+1/2.
For I2, we obtain
(3.35)I2=∫0T‍u3/2-Hjdu∫ut‍(x2-u2)Hj-3/2dx1[s,t](u)≤∫0T‍u3/2-Hjdu∫ut‍(x-u)Hj-3/2xHj-3/2dx1[s,t](u)≤∫st‍xHj-3/2dx∫0x‍(x-u)Hj-3/2u3/2-Hjdu≤B(5/2-Hj,Hj-1/2)Hj+1/2|t-s|Hj+1/2.
So
(3.36)|ΔKj|1≤CHj|t-s|Hj+1/2.
Suppose |t-s| is small enough, we get
(3.37)|Sδ(N)(SH(t)-SH(s))(f)|≤(2π)-(d/2)∑n,n≥N‍(12)n∏j=1d‍CHj|fj|∞2nj(βHj∧1)nj+(1/2)|t-s|nj(Hj-1/2)+Hj·1(nj)!≤CH∑n,n≥N‍∏j=1d‍1|t-s|nj(Hj-1/2)+Hj12n∏j=1d‍|fj|∞2nj·1(nj)!≤CH1|t-s|N(Hmax-1/2)+H*exp{12∑j=1d‍|fj|∞2},
with H*=∑j=1d‍Hj. Then we have.Theorem 3.6.
LetT>0 and n∈ℕ, suppose that
(3.38)H*+N(Hmax-12)<1,
then
(3.39)ℓTH,(N)=∫0T‍∫0t‍δ(N)(SH(t)-SH(s))dsdt,
is well defined as an element of (S)* and
(3.40)limε→0ℓT,εH,(N)=ℓTH,(N),in(S)*.Theorem 3.7.
ForH∈(0,1), every positive integer d≥1, and ε>0, suppose that  (3.38) holds, then the truncated self-intersection local time ℓTH,(N) has the following chaos expansion:
(3.41)ℓTH,(N)=1(2π)d/2∑m=0∞‍1m!(-12)m〈:ω⊗2m:,Gm〉,
where
(3.42)Gm=∫0T‍∫0t‍dsdt∏j=1d‍1(δHj(t,s))mj+(1/2)ΔK⊗2m,
for each m∈ℕd and m≥N. All other kernel functions Gm are identically equal to zero.Proof.
By Corollary2.4, 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
(3.43)(SℓTH,(N))(f)=(2π)-(d/2)∫0T‍∫0t‍dsdt∑n,n≥N‍(-12)n∏j=1d‍〈fj,ΔKj〉2njδHj(t,s)nj+(1/2)·1(nj)!.
Comparing with the general form of chaos expansion, the result is proved.Next we will estimate theL2-norm of the chaos of the self-intersection local time of subfractional Brownian motion. Now we state the result.Theorem 3.8.
Suppose allHj=H, let T>0, n≠0 and d≥2. (1)
IfHd=1, then
(3.44)limε→01|logε||G2n,ε|2,2n
exists.(2)
If1<dH<3/2, then
(3.45)limε→0|G2n,ε|2,2n
exists.(3)
IfH>3/2, then limε→0εd/2-(3/4H)|G2n,ε|2,2n exists.Proof.
Forn∈ℕd, the 2nth chaos is given by
(3.46)ℓT,2n,ε=(2π)-d/21n!12n〈:ω⊗2n:,G2n,ε〉,
where
(3.47)G2n,ε=∫0T‍dt∫0t‍ds∏j=1d‍ΔKj⊗2n(δHj(t,s)+ε)nj+1/2.
So
(3.48)𝔼(ℓT,2n,ε)2=(2n)!(2π)d(n!)222n|G2n,ε|L2(ℝn)2.
In view of  (3.48), we need to estimate |G2n,ε|L2(ℝn)2=|G2n,ε|2,2n2, where
(3.49)|G2n,ε|2,2n2=∫ℝ2n‍d2nu∫0T‍∫0t‍∫0T‍∫0t'‍dtdsdt'ds'∏j=1d‍ΔKj(t,s)⊗2njΔKj(t′,s′)⊗2nj((δHj(t,s)+ε)(δHj(t′,s′)+ε))nj+1/2.
By using Fubini theorem, we first get
(3.50)∫ℝ2n‍d2nu∏j=1d‍ΔKj(t,s)⊗2njΔKj(t′,s′)⊗2nj=∏j=1d‍∏i=12nj‍∫ℝ‍duijΔKj(t,s)(t,s)ΔKj(t,s)(t′,s′)=∏j=1d‍∏i=12nj‍𝔼(SHj(t)-SHj(s))(SHj(t′)-SHj(s′))=∏j=1d‍(Ct,s,t's'Hj)2nj,
with Ct,s,t's'Hj=𝔼(SHj(t)-SHj(s))(SHj(t')-SHj(s')). Moreover denote by Rt,s,t's'Hj=𝔼(BHj(t)-BHj(s))(BHj(t')-BHj(s')) with BH a fractional Brownian motion with Hurst index H∈(0,1). Then, from Bojdecki et al. [1], we know that
(3.51)0<Ct,s,t's'Hj<Rt,s,t's'Hj,H>12,Rt,s,t's'Hj<Ct,s,t's'Hj<0,H<12.
So
(3.52)|G2n,ε|2,2n2≤∫0T‍∫0t‍∫0T‍∫0t'‍dtdsdt'ds'∏j=1d‍(Rt,s,t's'Hj)2nj((δHj(t,s)+ε)(δHj(t′,s′)+ε))nj+1/2.
In view of the symmetry of the domain and integrand function, it suffices to integrate only on
(3.53)𝒯=𝒯1∪𝒯2∪𝒯3,
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' and x+y+z=t-s'<t, where t is considered as a parameter. Set |G2n,ε(i)|2,2n2 the integral over 𝒯i,i=1,2,3.
Step  1. For |G2n,ε(1)|2,2n2, we obtain
(3.54)|G2n,ε(1)|2,2n2≤CH∫0T‍dt∫0≤x+y+z≤t‍dxdydz·∏j=1d‍[(x+z)2Hj+(y+z)2Hj-(x+y+z)2Hj-z2Hj]2nj[(x2Hj+ε)(y2Hj+ε)]nj+(1/2).
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 byθt(ε)=ε-(1/2H)t and make the following change of variables (x,y,z)=ε-(1/2H)(x',y',z'), we get
(3.55)|G2n,ε(1)|2,2n2≤CHε(3/2H)-d∫0T‍dt∫0≤x+y+z≤θt(ε)‍dxdydz·[(x+z)2H+(y+z)2H-(x+y+z)2H-z2H]2n[(x2H+1)(y2H+1)]n+(d/2)=2CHε(3/2H)-d∫0T‍dt∫0≤x+y+z≤θt(ε),0≤x≤y≤θt(ε)‍dxdydz·[(x+z)2H+(y+z)2H-(x+y+z)2H-z2H]2n[(x2H+1)(y2H+1)]n+(d/2).
Denote by
(3.56)fH(x,y,z)=(x+z)2H+(y+z)2H-(x+y+z)2H-z2H,
and by
(3.57)f(t,H,d,n,ε)=∫0≤x+y+z≤θt(ε),0≤x≤y≤θt(ε)‍dxdydz[fH(x,y,z)]2n[(x2H+1)(y2H+1)]n+(d/2).
First note that |fH(x,y,z)|≤x2H and for dH>1,
(3.58)∫0≤x≤y‍dxdyx4Hn[(x2H+1)(y2H+1)]n+(d/2)<∞.
This implies that for every x∈ℝ+,
(3.59)∫0≤x≤y‍dxdyfH(x,y,z)2n[(x2H+1)(y2H+1)]n+(d/2),
is finite. On the other hand, for α∈(1,2-2H), zαfH(x,y,z)2n decreases to zero when z tends to zero, then
(3.60)f(H,d,n)=∫ℝ+‍dz∫0≤x≤y‍dxdyfH(x,y,z)2n[(x2H+1)(y2H+1)]n+(d/2)<∞,
and so that
(3.61)limε→0f(t,H,d,n,ε)=f(H,d,n).
Therefore, we obtain, if Hd∈(1,3/2),
(3.62)limε→0|G2n,ε(1)|2,2n2=0,
and if Hd≥3/2,
(3.63)limε→0εd-(3/2H)|G2n,ε(1)|2,2n2=2Tf(d,H,n).
Suppose now Hd=1,
(3.64)|G2n,ε(1)|2,2n2=2ε1/2H∫0T‍f(t,H,d,n,ε)dt,
with
(3.65)f(t,H,d,n,ε)=θt(ε)∫0≤x≤y≤θt(ε)‍dxdyx4Hn[(x2H+1)(y2H+1)]n+(d/2),
and then
(3.66)f(t,H,d,n,ε)|logε|≤θt(ε){a(H,d,n)|logε|+b(H,d,n)},
where
(3.67)a(H,d,n)=∫0≤x≤y,x≤1‍dxdyx4Hn[(x2H+1)(y2H+1)]n+(d/2),b(H,d,n)=sup0≤t≤Tsupε>01|logε|∫1≤x≤y≤θt(ε)‍dxdyx4Hn[(x2H+1)(y2H+1)]n+(d/2).
So we obtain
(3.68)1|logε||GT,2n,ε(1)|2,2n2≤T2{a(H,d,n)|logε|+b(H,d,n)}.
Step  2. Let us now treat |G2n,ε(2)|2,2n2.
(3.69)|G2n,ε(2)|2,2n2=2ε(3/2H)-d∫0T‍dtg(t,H,d,n,ε),
where
(3.70)g(t,H,d,n,ε)≤CH∫0≤x+y-z≤θt(ε),0≤z≤x≤y≤θt(ε)‍dxdydz[gH(x,y,z)]2n[(x2H+1)(y2H+1)]n+(d/2),
where
(3.71)gH(x,y,z)=(x-z)2H+(y+z)2H-(x+y-z)2H-z2H.
We have that |gH(x,y,z)|≤2x2H and so
(3.72)z→∫0≤z≤x≤y‍dxdy[gH(x,y,z)]2n[(x2H+1)(y2H+1)]n+(d/2)=z4Hn+2∫1≤x≤y‍dxdy[gH(x,y,z)]2n[(z2Hx2H+1)(z2Hy2H+1)]n+(d/2),
is well defined on ℝ+ and for Hd≥3/2, one can choose α∈(1,2-2H) such that when z→+∞(3.73)zα∫0≤z≤x≤y‍dxdy[gH(x,y,z)]2n[(x2H+1)(y2H+1)]n+(d/2)→0.
And then
(3.74)∫ℝ+‍∫0≤z≤x≤y‍dxdy[gH(x,y,z)]2n[(x2H+1)(y2H+1)]n+(d/2)=g(H,d,n)<∞.
So that
(3.75)limε→0g(t,H,d,n,ε)=g(H,d,n).
Suppose Hd=1, the same computations as in the case of |G2n,ε(1)|2,2n2 leads to
(3.76)1|logε||G2n,ε(2)|2,2n2≤22nT2{a(H,d,n)|logε|+b(H,d,n)}.
Suppose now Hd∈(1,3/2), we have
(3.77)g(t,H,d,n,ε)≤22n∫0θt(ε)‍dzz2-2Hd∫1≤x≤y‍dxdyx4Hn(xy)H(2n+d):=22nC(H,d,n).
Then
(3.78)|G2n,ε(2)|2,2n2≤CH22nC(H,d,n)(3-2Hd)(2-Hd)T4-2Hd.
We know that limε→0|G2n,ε(1)|2,2n2=0, then limε→0|G2n,ε(2)|2,2n2 exists. Let us suppose that Hd=3/2,
(3.79)|G2n,ε(2)|2,2n2=2∫0T‍dth(t,H,d,n,ε)
with
(3.80)h(t,H,d,n,ε)≤∫01‍dz∫z≤x≤y≤θt(ε)‍dxdy[gH(x,y,z)]2n[(x2H+1)(y2H+1)]n+(d/2)+∫1θt(ε)‍dz∫z≤x≤y≤θt(ε)‍dxdy[gH(x,y,z)]2n[(x2H+1)(y2H+1)]n+(d/2).
The first term is bounded by
(3.81)22n∫0≤x≤y‍dxdyx4Hn[(x2H+1)(y2H+1)]n+(d/2)=22ne(H,d,n).
The second term is bounded by
(3.82)∫1θt(ε)‍dz1z22n∫1≤x≤y‍dxdyx4Hn(xy)H(2n+(d/2))=22nC(H,d,n)log|θt(ε)|.
So
(3.83)1|logε||G2n,ε(2)|2,2n2≤22n+1|logε|e(H,d,n)T+22nHC(H,d,n)T+22n+1|logε|∫0T‍logtdt.
Therefore
(3.84)1|logε||G2n,ε(2)|2,2n2
has a finite nontrivial limit and
(3.85)1|logε||G2n,ε(2)|2,2n2≤22nHC(H,d,n)T.
Step  3. Finally, let us treat |G2n,ε(3)|2,2n2. Let x=t-t',y=t'-s',z=s'-s. We know
(3.86)|G2n,ε(3)|2,2n2≤2ε(3/2H)-d∫0T‍dtj(t,H,d,n,ε),
where
(3.87)j(t,H,d,n,ε)=∫0≤x+y+z≤θt(ε)‍dxdydzkH(x,y,z)2n[(x2H+1)(y2H+1)]n+(d/2),
with
(3.88)kH(x,y,z)=(x+y)2H-x2H+(y+z)2H-z2H.
It is obvious that |kH(x,y,z)|≤2y2H. So
(3.89)j(t,H,d,n,ε)≤2∫0≤x+y+z≤θt(ε),0≤x≤y≤θt(ε)‍dxdydz(2y2H)2n[(x2H+1)(y2H+1)]n+(d/2).
For Hd>1,
(3.90)∫0≤x≤y‍(2y2H)2n[(x2H+1)(y2H+1)]n+(d/2)dxdy<∞,
this implies that for every z∈ℝ+,
(3.91)∫0≤x≤y‍kH(x,y,z)2n[(x2H+1)(y2H+1)]n+(d/2)dxdy<∞.
On the other hand, for α∈(1,2-2H), zαkH(x,y,z)2n decreases to zero when z tends to infinity, then
(3.92)g(H,d,n)=∫ℝ+‍dz∫0≤x≤y‍dxdykH(x,y,z)2n[(x2H+1)(y2H+1)]n+(d/2)<∞,
and so that
(3.93)limε→0g(t,H,d,n,ε)=g(H,d,n).
Therefore, if Hd∈(1.3/2), we obtain
(3.94)limε→0|G2n,ε(3)|2,2n2=0.
If Hd≥3/2,
(3.95)limε→0εd-(3/2H)|G2n,ε(3)|2,2n2=2Tg(H,d,n).
Suppose now Hd=1,
(3.96)|G2n,ε(3)|2,2n2=2ε1/2H∫0T‍dtg(t,H,d,n,ε),
where
(3.97)g(t,H,d,n,ε)≤θt(ε)∫0≤x≤y≤θt(ε)‍dxdy(2y2H)2n[(x2H+1)(y2H+1)]n+(d/2),
and then
(3.98)g(t,H,d,n,ε)|logε|≤θt(ε){a(H,d,n)|logε|+b(H,d,n)},
where
(3.99)a(H,d,n)=∫0≤x≤y,x≤1‍dxdy(2y2H)2n[(x2H+1)(y2H+1)]n+(d/2),b(H,d,n)=sup0≤t≤Tsupε>01|logε|∫1≤x≤y≤θt(ε)‍dxdy(2y2H)2n[(x2H+1)(y2H+1)]n+(d/2).
So we obtain
(3.100)1|logε||G2n,ε(3)|2,2n2≤T2{a(H,d,n)|logε|+b(H,d,n)}.Theorem 3.9.
Assume thatHi=H,i=1,…,d, then for H∈(0,1) and d≥1 satisfying Hd<1,
(3.101)ℓT,εH→ℓTH,in(L2),
as ε tends to zero.Proof.
By Theorems3.3 and 3.7, we only need to consider chaos expansion of ℓT,εH and ℓTH. Similar techniques in Oliveira et al. [12] allow us to write
(3.102)I=𝔼(ℓT,2nH)2=(2n)!(2π)d(n!)222n|G2n,ε|L2(ℝn)2.
In view of  (3.102), we need to estimate |G2n|L2(ℝn)2=|G2n|2,2n2, where
(3.103)|G2n|2,2n2=∑m‍m!(12π)d(12)2m1(m!)2·∫ℝ2n‍d2nu∫𝒯‍dtdsdt'ds'∏j=1d‍ΔKj(t,s)⊗2njΔKj(t′,s′)⊗2nj((δHj(t,s))(δHj(t′,s′)))nj+1/2=(12π)d∫𝒯‍[λt,sλt′,s′-μ2]-(d/2)dtdsdt'ds',
where
(3.104)λt,s=|t+s|2H+|t-s|2H-22H-1(t2H+s2H),λt',s'=|t′+s′|2H+|t′-s′|2H-22H-1((t′)2H+(s′)2H),μ=12(|s-t′|2H+|s′-t|2H-|t-t′|2H-|s-s′|2H+|s-t′|2H+|s′+t|2H-|t+t′|2H-|s+s′|2H).
In order to show |I|<∞, we need some preliminaries. Recall
(3.105)𝕋={(s,t,s′,t′):0<s<t<T,0<s′<t′<T}.
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,
(3.106)λt,sλt',s'-μ2≥κ[(t-s)2H(t′-t)2H+(t′-s′)2H(s′-s)2H];(2)
for(s,t,s',t')∈𝕋2,
(3.107)λt,sλt',s'-μ2≥κ[(t-s)2H(t′-s′)2H];(3)
for(s,t,s',t')∈𝕋3,
(3.108)λt,sλt',s'-μ2≥κ[(t-s)2H(t′-s′)2H]. Then we can easily check that, if Hd<1, |I|<∞. Moreover, ℓT,ε converges to ℓTH 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 inL2 under mild conditions. The asymptotic behavior and the central limit theorem of the difference ℓT,εH-𝔼ℓT,εH in L2 as ε tends to zero will be discussed in future works.



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*Source: 414195-2012-12-13.xml*