On the Self-Intersection Local Time of Subfractional Brownian Motion

Abstract

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 mean zero Gaussian process 𝑆^𝐻₀={𝑆^𝐻₀(𝑡),𝑡≥0} with the covariance function 𝑅_𝐻(𝑡,𝑠)=𝔼𝑆^𝐻₀(𝑡)𝑆^𝐻₀(𝑠)=𝑠^2𝐻+𝑡^2𝐻−12(𝑠+𝑡)^2𝐻+|𝑡−𝑠|^2𝐻,(1.1) for all 𝑠,𝑡≥0. For 𝐻=1/2, 𝑆^𝐻₀ coincides with the standard Brownian motion. 𝑆^𝐻₀ 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 𝑆^𝐻₀. 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: 2−2^2𝐻−1∧1(𝑡−𝑠)^2𝐻≤𝔼𝑆^𝐻₀(𝑡)−𝑆^𝐻₀(𝑠)²≤2−2^2𝐻−1∨1(𝑡−𝑠)^2𝐻.(1.2)

Let 𝐒^𝐻={𝐒^𝐻(𝑡),𝑡∈ℕ₊} be a 𝑑-dimensional subfractional Brownian motion with multiparameters 𝐻=(𝐻₁,𝐻₂,…,𝐻_𝑑). Suppose that 𝑑≥2, we are interested in, when it exists, the self-intersection local time of subfractional Brownian motion 𝐒^𝐻 which is formally defined as ℓ^𝐻_𝑇=^𝑇₀^𝑡₀𝛿₀𝐒^𝐻(𝑡)−𝐒^𝐻(𝑠)𝑑𝑠𝑑𝑡,(1.3) 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 ℓ^𝐻_𝑇,𝜀=^𝑇₀^𝑡₀𝑝_𝜀𝐒^𝐻(𝑡)−𝐒^𝐻(𝑠)𝑑𝑠𝑑𝑡,(1.4) where 𝑝_𝜀(𝑥)=(2𝜋𝜀)^−𝑑/2−|𝑥|exp²=()2𝜀2𝜋^−𝑑_ℝ𝑑𝑒^{𝑖𝜉⋅𝑥−(𝜀|𝜉|2/2)}𝑑𝜉,𝑥∈ℝ^𝑑.(1.5)

For 𝐻=1/2, the process 𝑆^𝐻₀ 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 ℓ^1/2_𝑇,𝜀 does not converge in 𝐿² but it can be renormalized so that ℓ^1/2_𝑇,𝜀−(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, ℓ^{𝐻,𝑓𝐵𝑚}_𝑇,𝜀−𝐶_𝐻𝑇𝜀^{−1+(1/2𝐻)} converges in 𝐿² as 𝜀 tends to zero, where 𝐶_𝐻 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)ℓ^{𝐻,𝑓𝐵𝑚}_𝑇,𝜀 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)=𝜀^{𝑑−3/(2𝐻)} if 3/(2𝑑)<𝐻. Wu and Xiao [10] proved the existence of the intersection local times of (𝑁_𝑖,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 ℝ^𝑑, 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 ℓ_𝑇=∫^𝑇₀𝛿(𝑆^𝐻1₀(𝑡)−𝑆^𝐻2₀(𝑡))𝑑𝑡 of two independent subfractional Brownian motion with respective indices 𝐻_𝑖∈(0,1),𝑖=1,2. By an elementary method, they showed that ℓ_𝑇 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 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 𝒮_𝑑(ℝ)⊂𝐿²ℝ,ℝ^𝑑⊂𝒮^′_𝑑(ℝ),(2.1) where 𝐿²(ℝ,ℝ^𝑑) is the real Hilbert space of all vector-valued square integrable functions with respect to Lebesgue measure on ℝ and 𝒮_𝑑(ℝ),𝒮^′_𝑑(ℝ) are the Schwartz spaces of the vectors valued test functions and tempered distributions, respectively. Denote the norm in 𝐿²(ℝ,ℝ^𝑑) by |⋅|_𝑑 or if there is no risk of confusion simply by |⋅| and the dual pairing between 𝒮^′_𝑑 and 𝒮_𝑑(ℝ) by ⟨⋅,⋅⟩, which is defined as the bilinear extension of the inner product on 𝐿²(ℝ,ℝ^𝑑), that is ⟨𝐠,𝐟⟩=^𝑑_𝑖=1ℝ𝑔_𝑖(𝑥)𝑓_𝑖(𝑥)𝑑𝑥,(2.2) for all 𝐠=(𝑔₁,𝑔₂,…𝑔_𝑑)∈𝐿²(ℝ,ℝ^𝑑) and all 𝐟=(𝑓₁,𝑓₂,…,𝑓_𝑑)∈𝒮_𝑑(ℝ). By the Minlos theorem, there is a unique probability measure 𝜇 on the 𝜎-algebra ℬ generated by the cylinder sets on 𝒮^′_𝑑(ℝ) with characteristic function given by 𝐶(𝐟)∶=_{𝒮′𝑑(ℝ)}𝑒^{𝑖⟨→𝜔,𝐟⟩}𝑑𝜇^→𝜔=𝑒^{−(1/2)|𝐟|2},𝐟∈𝒮_𝑑(ℝ).(2.3) In this way, we have defined the white noise measure space (𝒮^′_𝑑(ℝ),ℬ,𝜇). Then a realization of vector of independent subfractional Brownian motion 𝑆^𝐻𝑗,𝑗=1,2,…,𝑑, is given by 𝑆^𝐻𝑗(𝑡)=⟨𝜔_𝑗,𝐾_𝐻𝑗⟩=^𝑡₀𝐾_𝐻𝑗(𝑡,𝑠)𝜔_𝑗(𝑠)𝑑𝑠,𝜔_𝑖∈𝒮^′_𝑑(ℝ).(2.4) We recall the explicit formula for the kernel 𝐾_𝐻(𝑡,𝑠) of a one-dimensional subfractional Brownian motion with parameter 𝐻∈(0,1)𝐾_𝐻()√𝑡,𝑠=−𝜋𝑠^1/2−𝐻2^𝐻Γ()𝑑𝐻+1/2𝑑𝑠^𝑡_𝑠𝑢²−𝑠^2𝐻−1/2𝑑𝑢1_[0,𝑡](𝑠)=√𝜋𝑠^3/2−𝐻2^𝐻Γ()𝑡𝐻+1/2²−𝑠^2𝐻−1/2𝑡+^𝑡_𝑠𝑢²−𝑠^2𝐻−1/2𝑢²𝑑𝑢1_[0,𝑡](𝑠).(2.5) Especially, for 𝐻>1/2, we have 𝐾_𝐻()=2𝑡,𝑠^1−𝐻√𝜋Γ()𝑠𝐻−1/2^{3/2−𝐻𝑡}_𝑠𝑢²−𝑠^2𝐻−3/2𝑑𝑢1_[0,𝑡](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(𝒮^′_𝑑(ℝ),ℬ,𝜇). This space is canonically isomorphic to the symmetric Fock space of symmetric square integrable functions 𝐿²𝒮^′_𝑑(ℝ),ℬ,𝜇≃^∞_𝑘=0Sym𝐿²(ℝ^𝑘,𝑘!𝑑^𝑘𝑥)^⊗2𝑑=𝒢(2.7) leading to the chaos expansion of the elements in 𝐿²(𝒮^′_𝑑(ℝ),ℬ,𝜇)𝐹𝜔₁,𝜔₂,…,𝜔_𝑑=_{𝑛𝐧=1,𝑛2,…,𝑛𝑑∈ℕ𝑑}∶𝜔^⊗𝑛1₁∶⊗⋯⊗∶𝜔^⊗𝑛𝑑_𝑑∶,𝐟_𝐧,(2.8) with kernel functions 𝐟_𝐧 in the Fock space, that is, square integrable functions of the 𝑚 arguments and symmetric in each 𝑛_𝑖-tuple.

For simplicity, in the sequel, we will use the notations 𝐧=𝑛₁,…,𝑛_𝑑∈ℕ^𝑑,𝑛=^𝑑_𝑖=1𝑛_𝑖,𝐧!=^𝑑_𝑖=1𝑛_𝑖!,(2.9) which reduces expansion (2.8) to 𝐹(𝜔)=_{𝐧∈ℕ𝑑}∶𝜔^⊗𝐧∶,𝐟_𝐧,𝜔∈𝒮^′_𝑑(ℝ).(2.10) The norm of 𝐹 is given by ‖𝐹‖²_(𝐿2)=_{𝐧∈ℕ𝑑}𝐧!𝐟_𝐧²_2,𝑛,(2.11) where |⋅|_2,𝑛 is the norm in 𝐿²(ℝ^𝑛,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∈𝒮_𝑑(ℝ), let us consider the Wick exponential ⟨⟩⟨⟩−1∶exp𝜔,𝐟∶=exp𝜔,𝐟2(𝐟,𝐟)=_{𝐧∈ℕ𝑑}1𝐧!∶𝜔^⊗𝐧∶,𝐟^⊗𝐧=𝐶(𝐟)𝑒^{⟨𝜔,𝐟⟩},𝜔∈𝒮^′_𝑑(ℝ).(2.12) We define the 𝑆-transform of a Φ∈(𝑆)^∗ by 𝑆Φ(𝐟)∶=Φ,∶exp(⟨⋅,𝐟⟩)∶,∀𝐟∈𝒮_𝑑(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) 𝑆Φ(𝐟)∶=_𝐧𝐹_𝐧,𝐟_𝐧^⊗𝐧,(2.14) extends the chaos expansion to Φ∈(𝑆)^∗ with distribution valued kernels 𝐹_𝐧 such that Φ,𝜑=_𝐧𝐧!𝐹_𝐧,𝜑_𝐧,(2.15) for every generalized test function 𝜑∈(𝑆) with kernel function 𝜑_𝐧.

In order to characterize the space (𝑆)^∗ through its 𝑆-transform, we need the following definition:

Definition 2.1. A function 𝐹∶𝒮_𝑑(ℝ)→ℂ is called a 𝑈-functional whenever(1)for every 𝐟₁,𝐟₂∈𝒮_𝑑(ℝ) and 𝜆∈ℝ, the mapping 𝜆→𝐹(𝜆𝐟₁+𝐟₂) has an entire extension to 𝜆∈ℂ,(2)there are constants 𝐾₁,𝐾₂>0 such that |𝐹(𝑧𝐟)|≤𝐾₁exp𝐾₂|𝑧|²‖𝐟‖²,∀𝑧∈ℂ,𝐟∈𝒮_𝑑(ℝ),(2.16) for some continuous norm ‖⋅‖ on 𝒮_𝑑(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 𝑈-functionals.

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 (Φ_𝑛,𝑛∈ℕ) be a sequence in (𝑆)^∗ such that(1)for all 𝐟∈𝒮_𝑑(ℝ), ((𝑆Φ_𝑛)(𝐟))_𝑛∈ℕ is Cauchy sequence in ℂ;(2)there are 𝐾₁,𝐾₂>0 such that for some continuous norm ‖⋅‖ on 𝒮_𝑑(ℝ) one has 𝑆Φ_𝑛(𝑧𝐟)≤𝐾₁𝑒^{𝐾2|𝑧|2‖𝐟‖2},𝑧∈ℂ,𝐟∈𝒮_𝑑(ℝ),𝑛∈ℕ,(2.17)
then (Φ_𝑛,𝑛∈ℕ) converges strongly in (𝑆)^∗ to a unique Hida distribution.

Corollary 2.4. Let (Ω,ℬ,𝑚) be a measure space and 𝜆→Φ_𝜆 be a mapping from Ω to (𝑆)^∗. We assume that the 𝑆-transform of Φ_𝜆 fulfills the following two properties:(1)the mapping 𝜆→(𝑆𝜙_𝜆)(𝐟) is measurable for every 𝐟∈𝒮_𝑑(ℝ),(2)the (𝑆Φ_𝜆)(𝐟) obeys a 𝑈-estimate 𝑆Φ_𝜆(𝑧𝐟)≤𝐶₁(𝜆)𝑒^{𝐶2(𝜆)|𝑧|2‖𝐟‖2},𝑧∈ℂ,𝐟∈𝒮_𝑑(ℝ),(2.18)for some continuous ‖⋅‖ on 𝒮_2𝑑(ℝ) and for 𝐶₁∈𝐿¹(Ω,𝑚), 𝐶₂∈𝐿^∞(Ω,𝑚). Then 𝑑𝑚(_Ω𝜆)Φ_𝜆∈(𝑆)^′,𝑆_Ω𝑑𝑚(𝜆)Φ_𝜆(𝐟)=_Ω𝑑𝑚(𝜆)𝑆Φ_𝜆(𝐟).(2.19)

3. Self-Intersection Local Time

Let us now consider the 𝑑-dimensional subfractional Brownian motion 𝐒^𝐻(𝑡) with parameter 𝐻=(𝐻₁,𝐻₂,…,𝐻_𝑑). In view of (2.4), for 𝑗=1,…,𝑑, 𝑆^𝐻𝑗(𝑡)−𝑆^𝐻𝑗(𝑠)=⟨Δ𝐾_𝑗,𝜔_𝑗⟩ with Δ𝐾_𝑗=𝐾_𝐻𝑗(𝑡,𝑢)−𝐾_𝐻𝑗(𝑠,𝑢).

Proposition 3.1. For each 𝑡 and 𝑠 strictly positive real numbers, the Bochner integral 𝛿𝐒^𝐻(𝑡)−𝐒^𝐻(𝑠)1∶=2𝜋^𝑑_ℝ^𝑑𝑑𝜆𝑒^{𝑖𝜆(𝐒𝐻(𝑡)−𝐒𝐻(𝑠))}(3.1) is a Hida distribution with 𝑆-transform given by 𝑆𝛿𝐒^𝐻(𝑡)−𝐒^𝐻(𝑠)(𝐟)=(2𝜋)^{𝑑−(𝑑/2)}_𝑗=11𝛿^𝐻𝑗(𝑡,𝑠)^1/2−𝑓exp^𝑗,Δ𝐾_𝑗²2𝛿^𝐻𝑗(𝑡,𝑠)(3.2) for all 𝐟=(𝑓₁,…,𝑓_𝑑)∈𝒮_𝑑(ℝ) and 𝛿^𝐻𝑗(𝑡,𝑠)=|Δ𝐾_𝑗|²₂.

Proof. The proof of this result follows from an applications of Corollary 2.4 to the 𝑆-transform of the integrand function Φ^→𝜔∶=𝑒^{𝑖𝜆(𝐒𝐻(𝑡)−𝐒𝐻(𝑠))},^→𝜔=𝜔₁,…,𝜔_𝑑.(3.3) With respect to the Lebesgue measure on ℝ^𝑑. By symmetry, we assume that 𝑠≤𝑡. For this purpose, we begin by observing that, for 𝐟=(𝑓₁,…,𝑓_𝑑)∈𝒮_𝑑(ℝ), and all 𝜆∈ℂ, one has 𝑆Φ(𝐟)=^𝑑_𝑗=1𝑆𝑒^{𝑖𝜆𝑗⟨Δ𝐾𝑗,𝜔𝑗⟩}𝑓_𝑗,𝑆𝑒^{𝑖𝜆𝑗⟨Δ𝐾𝑗,𝜔𝑗⟩}𝑓_𝑗=𝑒^{−(|𝑓𝑗|22/2)}_𝑆′(ℝ)𝑑𝜇_𝑗𝜔_𝑗𝑒^{𝑖𝜆𝑗⟨Δ𝐾𝑗,𝜔𝑗⟩+⟨𝑓𝑗,𝜔𝑗⟩}=𝑒^{−(|𝑓𝑗|22/2)}_𝑆′(ℝ)𝑑𝜇_𝑗𝜔_𝑗𝑒^{𝑖⟨𝜆𝑗Δ𝐾𝑗−𝑖𝑓𝑗,𝜔𝑗⟩}=𝑒^{−(𝜆2𝑗|Δ𝐾𝑗|22/2)}𝑒^{𝑖𝜆𝑗⟨Δ𝐾𝑗,𝑓𝑗⟩}.(3.4) The last equality was obtained by the definition of 𝜇. Then we obtain 𝑆Φ(𝐟)=^𝑑_𝑗=1𝑒^{−(𝜆2𝑗/2)|Δ𝐾𝑗|22}𝑒^{𝑖𝜆𝑗⟨Δ𝐾𝑗,𝑓𝑗⟩},(3.5) which clearly fulfills the measurability condition. Moreover, for all 𝑧∈ℂ, we find |𝑆Φ(𝑧𝐟)|=^𝑑_𝑗=1𝑒^{−(𝜆2𝑗|Δ𝐾𝑗|22𝑑/4)}_𝑗=1𝑒^{−(𝜆2𝑗|Δ𝐾𝑗|22/4)+𝑖𝑧𝜆𝑗⟨Δ𝐾𝑗,𝑓𝑗⟩}≤^𝑑_𝑗=1𝑒^{−(𝜆2𝑗|Δ𝐾𝑗|22𝑑/4)}_𝑗=1𝑒^{−(𝜆2𝑗|Δ𝐾𝑗|22/4)+|𝑧||𝜆𝑗||⟨Δ𝐾𝑗,𝑓𝑗⟩|},(3.6) where, for each 𝑗=1,…,𝑑, the corresponding term in the second product is bounded by |𝑧|exp²Δ𝐾_𝑗²₂⋅Δ𝐾_𝑗,𝑓_𝑗²(3.7) because −𝜆²_𝑗4Δ𝐾_𝑗²₂+|𝑧|𝜆_𝑗Δ𝐾_𝑗,𝑓_𝑗𝜆=−^𝑗2Δ𝐾_𝑗2−|𝑧|Δ𝐾_𝑗2Δ𝐾_𝑗,𝑓_𝑗²+|𝑧|²Δ𝐾_𝑗²₂Δ𝐾_𝑗,𝑓_𝑗².(3.8) As a result |𝑆Φ(𝑧𝐟)|≤𝑒^{−(1/4)∑𝑑𝑗=1𝜆2𝑗|Δ𝐾𝑗|22}𝑒^{|𝑧|2∑𝑑𝑗=1(1/|Δ𝐾𝑗|22)|⟨Δ𝐾𝑗,𝑓𝑗⟩|2},(3.9) where, as a function of 𝜆, the first exponential is integrable on ℝ^𝑑 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 processes ℓ^𝐻_𝑇,𝜀 given by (1.4) and (3.5), combining (2𝜋)^−𝑑_ℝ𝑑𝑑𝜆𝑒^{−𝜀(|𝜆|2/2)}𝑆𝑒^{𝑖𝜆(𝐒𝐻(𝑡)−𝐒𝐻(𝑠))}(𝐟)=(2𝜋)^−𝑑_ℝ𝑑𝑑𝜆^𝑑_𝑗=1𝑒^{−(𝜆2𝑗/2)(|Δ𝐾𝑗|22+𝜀)}𝑒^{𝑖𝜆𝑗⟨Δ𝐾𝑗,𝑓𝑗⟩}(3.10) and the elementary equality _ℝ𝑒^{−𝛼(𝑥2/2)+𝑖𝛽𝑥}𝑑𝑥=2𝜋𝛼𝑒^{−(𝛽2/2𝛼)},(3.11) then we have the following.

Corollary 3.2. For each 𝑡 and 𝑠 strictly positive real numbers, ℓ^𝐻_𝑇,𝜀=^𝑇₀^𝑡₀1𝑑𝑠𝑑𝑡(2𝜋)^𝑑ℝ𝑑𝑑𝜆𝑒^{i𝜆⋅(𝐒𝐻(𝑡)−𝐒𝐻(𝑠))−(𝜀𝜆2/2)},(3.12) is a Hida distribution with 𝑆-transform given by 𝑆ℓ^𝐻_𝑇,𝜀(𝐟)=(2𝜋)^{−(𝑑/2)𝑇}₀^𝑡₀𝑑𝑠𝑑𝑡^𝑑_𝑗=11𝛿^𝐻𝑗(𝑡,𝑠)+𝜀^1/2−𝑓exp^𝑗,Δ𝐾_𝑗²2𝛿^𝐻𝑗(𝑡,𝑠)+𝜀(3.13) for all 𝐟=(𝑓₁,…,𝑓_𝑑)∈𝒮_𝑑(ℝ) and 𝛿^𝐻𝑗(𝑡,𝑠)=|Δ𝐾_𝑗|²₂.

Theorem 3.3. For 𝐻∈(0,1), every positive integer 𝑑≥1, and 𝜀>0, the self-intersection local time ℓ^𝐻_𝑇,𝜀 has the following chaos expansion: ℓ^𝐻_𝑇,𝜀=1(2𝜋)^∞𝑑/2_𝑚=01−1𝐦!2^𝐦∶𝜔^⊗2𝐦∶,𝐆_𝐦,𝜀,(3.14) where 𝐆_𝐦,𝜀=^𝑇₀^𝑡₀𝑑𝑠𝑑𝑡^𝑑_𝑗=11𝛿^𝐻𝑗(𝑡,𝑠)+𝜀^{𝑚𝑗+(1/2)}Δ𝐾^⊗2𝐦.(3.15)

Proof. For every 𝐟∈𝒮(ℝ), we calculate the 𝑆-transform of ℓ_𝑇,𝜀 as follows: 𝑆ℓ_𝑇,𝜀(𝐟)=(2𝜋)^{−(𝑑/2)𝑇}₀^𝑡₀𝑑𝑠𝑑𝑡^𝑑_𝑗=11𝛿^𝐻𝑗(𝑡,𝑠)+𝜀^1/2−𝑓exp^𝑗,Δ𝐾_𝑗²2𝛿^𝐻𝑗()=()𝑡,𝑠+𝜀2𝜋^{−(𝑑/2)𝑇}₀^𝑡₀𝑑𝑠𝑑𝑡^𝑑∞_{𝑗=1𝑚𝑗=0}1𝑚_𝑗!−12^𝑚𝑗1𝛿^𝐻𝑗(𝑡,𝑠)+𝜀^{𝑚𝑗+(1/2)}𝑓_𝑗,Δ𝐾_𝑗^2𝑚𝑗=(2𝜋)^{−(𝑑/2)𝑇}₀^𝑡₀𝑑𝑠𝑑𝑡^∞_𝐦=01−1𝐦!2^𝐦𝑑_𝑗=11𝛿^𝐻𝑗(𝑡,𝑠)+𝜀^{𝑚𝑗+(1/2)}𝑓_𝑗^⊗2𝐦,Δ𝐾_𝑗^⊗2𝐦.(3.16) Comparing with the general form of the chaos expansion, we find that the kernel functions are equal to 𝐆_𝐦,𝜀=^𝑇₀^𝑡₀𝑑𝑠𝑑𝑡^𝑑_𝑗=11𝛿^𝐻𝑗(𝑡,𝑠)+𝜀^{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 𝐶₁𝐺(𝑥)≤𝐹(𝑥)≤𝐶₂𝐺(𝑥),(3.18) 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(ℓ^𝐻_𝑇,𝜀), it is just the first chaos. So in view of Theorem 3.3𝔼ℓ^𝐻_𝑇,𝜀=(2𝜋)^{−(𝑑/2)𝑇}₀^𝑡₀𝑑𝑠𝑑𝑡^𝑑_𝑗=11𝛿^𝐻𝑗(𝑡,𝑠)+𝜀^1/2.(3.19) Moreover, for all 𝑠≤𝑡, the second moment of increments 𝛿^𝐻𝑗(𝑡,𝑠)=𝔼(𝑆^𝐻𝑗(𝑡)−𝑆^𝐻𝑗(𝑠))² satisfying the following estimate: 𝛽_𝐻𝑗∧1(𝑡−𝑠)^2𝐻𝑗≤𝛿^𝐻𝑗(𝑡,𝑠)≤𝛽_𝐻𝑗∨1(𝑡−𝑠)^2𝐻𝑗,(3.20) with 𝛽_𝐻𝑗=2−2^{2𝐻𝑗−1}. As the integrand in 𝔼(ℓ^𝐻_𝑇,𝜀) is always positive, we have 𝔼ℓ^𝐻_𝑇,𝜀≤(2𝜋)^{−(𝑑/2)𝑇}₀^𝑡₀𝑑𝑠𝑑𝑡^𝑑_𝑗=11𝛽_𝐻𝑗∧1(𝑡−𝑠)^2𝐻𝑗+𝜀^1/2=(2𝜋)^{−(𝑑/2)𝑇}₀𝑑𝑠𝑇−𝑠∏^𝑑_𝑗=1𝛽_𝐻𝑗∧1𝑠^2𝐻𝑗+𝜀^1/2.(3.21) We use the change of variables 𝑠=𝜀^{𝑑/2𝐻∗}𝑧∶=𝛼(𝜀)𝑧 with 𝐻^∗=∑^𝑑_𝑗=1𝐻_𝑗, 𝔼ℓ^𝐻_𝑇,𝜀≤𝛼(𝜀)(2𝜋)^{𝑑/20𝑇/𝛼(𝜀)}(𝜀)𝑧𝑑𝑧𝑇−𝛼∏^𝑑_𝑗=1𝛽_𝐻𝑗∧1𝜀^{𝑑𝐻𝑗/𝐻∗}𝑧^2𝐻𝑗+𝜀^1/2=𝛼(𝜀)(2𝜋𝜀)^{𝑑/20𝑇/𝛼(𝜀)}(𝜀)𝑧𝑑𝑧𝑇−𝛼∏^𝑑_𝑗=1𝛽_𝐻𝑗∧1𝜀^{(𝑑𝐻𝑗/𝐻∗)−1}𝑧^2𝐻𝑗+1^1/2.(3.22) We divide the integral in two parts 𝔼ℓ^𝐻_𝑇,𝜀≤𝛼(𝜀)(2𝜋𝜀)^𝑑/210(𝜀)𝑧𝑑𝑧𝑇−𝛼∏^𝑑_𝑗=1𝛽_𝐻𝑗∧1𝜀^{(𝑑𝐻𝑗/𝐻∗)−1}𝑧^2𝐻𝑗+1^1/2+₁^{𝑇/𝛼(𝜀)}(𝜀)𝑧𝑑𝑧𝑇−𝛼∏^𝑑_𝑗=1𝛽_𝐻𝑗∧1𝜀^{(𝑑𝐻𝑗/𝐻∗)−1}𝑧^2𝐻𝑗+1^1/2.(3.23) The first integral in braces is bounded. Set 𝐼 the second integral in braces, so 𝑇𝐼≤∏^𝑑_𝑗=1𝛽_𝐻𝑗∧1^{1/21𝑇/𝛼(𝜀)}1𝑑𝑧𝑧^𝐻∗=𝑇^2−𝐻∗𝜀^{(𝑑/2)−(𝑑/2𝐻∗)}−1(1−𝐻^∗)∏^𝑑_𝑗=1𝛽_𝐻𝑗∧1^1/2,𝐻^∗𝑇≠1log𝑇−𝑑/2𝐻^∗log𝜀∏^𝑑_𝑗=1𝛽_𝐻𝑗∧1^1/2,𝐻^∗=1.(3.24)

Proposition 3.4. Denote by 𝐻^∗=∑^𝑑_𝑗=1𝐻_𝑗. Let 𝑇>0 and 𝑑≥1, then if 𝐻^∗≥1, lim_𝜀→0𝔼ℓ^𝐻_𝑇,𝜀=∞.(3.25) Moreover if 𝐻^∗=1, 𝔼ℓ^𝐻_𝑇,𝜀≤𝐶_{𝑇,𝐻𝑗,𝑑}log𝜀.(3.26) If 𝐻^∗>1𝔼ℓ^𝐻_𝑇,𝜀≤𝐶_{𝑇,𝐻𝑗,𝑑}𝜀^{(𝑑/2)−(𝑑/2𝐻∗)}.(3.27) If 𝐻^∗<1, there is no blow up, that is, lim_𝜀→0𝔼ℓ^𝐻_𝑇,𝜀∈1(2𝜋)^𝑑/2∏^𝑑_𝑗=1𝛽_𝐻𝑗∨1^1/2𝑇0𝑑𝑠𝑇−𝑠𝑠^𝐻∗,1(2𝜋)^𝑑/2∏^𝑑_𝑗=1𝛽_𝐻𝑗∧1^1/2𝑇0𝑑𝑠𝑇−𝑠𝑠^𝐻∗.(3.28)

In particular, we have

Proposition 3.5. Suppose that all 𝐻_𝑗=𝐻, let 𝑇>0 and 𝑑≥1.(1)If 𝐻=1/𝑑, then 𝔼ℓ^𝐻_𝑇,𝜀=𝐶ln(1/𝜀)+𝑜(𝜀).(2)If 1/𝑑<𝐻<3/(2𝑑), then 𝔼ℓ^𝐻_𝑇,𝜀=𝐶𝜀^{(1/2𝐻)−(𝑑/2)}+𝑜(𝜀).

From the above results, if 𝐻^∗<1, the self-intersection local time ℓ^𝐻_𝑇 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(𝐒^𝐻(𝑡)−𝐒^𝐻(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 exp_𝑁(𝑥)=^∞_𝑛=𝑁𝑥^𝑛.𝑛!(3.29) It follows from (3.2) that the 𝑆-transform of 𝛿^(𝑁) is given by 𝑆𝛿^(𝑁)𝐒^𝐻(𝑡)−𝐒^𝐻(𝑠)(𝐟)=(2𝜋)^−(𝑑/2)_{𝐧,𝑛≥𝑁}−12^𝐧𝑑_𝑗=1𝑓_𝑗,Δ𝐾_𝑗^2𝑛𝑗𝛿^𝐻𝑗(𝑡,𝑠)^{𝑛𝑗+(1/2)}⋅1𝑛_𝑗!,(3.30) so 𝑆𝛿^(𝑁)𝐒^𝐻(𝑡)−𝐒^𝐻(𝑠)𝐟≤()()2𝜋^−(𝑑/2)_{𝐧,𝑛≥𝑁}12^𝐧𝑑_𝑗=1𝑓_𝑗^2𝑛𝑗_∞⋅Δ𝐾_𝑗^2𝑛𝑗₁𝛽_𝐻𝑗∧1^{𝑛𝑗+(1/2)}|𝑡−𝑠|^{(2𝑛𝑗+1)𝐻𝑗}⋅1𝑛_𝑗!.(3.31) We need to estimate the 𝐿¹-norm of Δ𝐾_𝑗, |Δ𝐾_𝑗|₁ for fixed 𝑗.

We will treat the case when all 𝐻_𝑗>1/2. For the case all 𝐻_𝑗<1/2, we do not have a good estimation of |𝐾_𝑗|₁ and so we do not have a result. Let 𝐻_𝑗>1/2, in view of (2.8) Δ𝐾_𝑗=𝐾_𝐻𝑗(𝑡,𝑢)−𝐾_𝐻𝑗(𝑠,𝑢)=𝐶_𝐻𝑗𝑢^{3/2−𝐻𝑗𝑡}_𝑢𝑥²−𝑢^{2𝐻𝑗−3/2}𝑑𝑥1_[0,𝑡](𝑢)−^𝑠_𝑢𝑥²−𝑢^{2𝐻𝑗−3/2}𝑑𝑥1_[0,𝑠](𝑢)=𝐶_𝐻𝑗𝑢^{3/2−𝐻𝑗𝑡}_𝑢𝑥²−𝑢^{2𝐻𝑗−3/2}𝑑𝑥1_[𝑠,𝑡](𝑢)+^𝑡_𝑢𝑥²−𝑢^{2𝐻𝑗−3/2}𝑑𝑥1_[0,𝑠](𝑢)−^𝑠_𝑢𝑥²−𝑢^{2𝐻𝑗−3/2}𝑑𝑥1_[0,𝑠](𝑢)=𝐶_𝐻𝑗𝑢^{3/2−𝐻𝑗𝑡}_𝑢𝑥²−𝑢^{2𝐻𝑗−3/2}𝑑𝑥1_[𝑠,𝑡](𝑢)+^𝑡_𝑠𝑥²−𝑢^{2𝐻𝑗−3/2}𝑑𝑥1_[0,𝑠](𝑢).(3.32) We obtain Δ𝐾_𝑗1=^𝑇₀Δ𝐾_𝑗(𝑢)𝑑𝑢=𝐶_ℎ𝑗^𝑇₀𝑢^{3/2−𝐻𝑗}𝑑𝑢^𝑡_𝑠𝑥²−𝑢^{2𝐻𝑗−3/2}𝑑𝑥1_[0,𝑠](𝑢)+^𝑇₀𝑢^{3/2−𝐻𝑗}𝑑𝑢^𝑡_𝑢𝑥²−𝑢^{2𝐻𝑗−3/2}𝑑𝑥1_[𝑠,𝑡](𝑢)∶=𝐶_𝐻𝑗𝐼₁+𝐼₂.(3.33) For 𝐼₁, we obtain 𝐼₁=^𝑇₀𝑢^{3/2−𝐻𝑗}𝑑𝑢^𝑡_𝑠𝑥²−𝑢^{2𝐻𝑗−3/2}𝑑𝑥1_[0,𝑠](𝑢)≤^𝑇₀𝑢^{3/2−𝐻𝑗}𝑑𝑢^𝑡_𝑠(𝑥−𝑢)^{𝐻𝑗−3/2}𝑥^{𝐻𝑗−3/2}𝑑𝑥1_[0,𝑠](𝑢)≤^𝑡_𝑠𝑥^{𝐻𝑗−3/2}𝑑𝑥^𝑥₀(𝑥−𝑢)^{𝐻𝑗−3/2}𝑢^{3/2−𝐻𝑗}1_[0,𝑠](𝑢)≤𝐵𝑑𝑢5/2−𝐻_𝑗,𝐻_𝑗−1/2𝐻_𝑗||+1/2𝑡−𝑠^𝐻𝑗+1/2.(3.34) For 𝐼₂, we obtain 𝐼₂=^𝑇₀𝑢^{3/2−𝐻𝑗}𝑑𝑢^𝑡_𝑢𝑥²−𝑢^{2𝐻𝑗−3/2}𝑑𝑥1_[𝑠,𝑡](𝑢)≤^𝑇₀𝑢^{3/2−𝐻𝑗}𝑑𝑢^𝑡_𝑢(𝑥−𝑢)^{𝐻𝑗−3/2}𝑥^{𝐻𝑗−3/2}𝑑𝑥1_[𝑠,𝑡](𝑢)≤^𝑡_𝑠𝑥^{𝐻𝑗−3/2}𝑑𝑥^𝑥₀(𝑥−𝑢)^{𝐻𝑗−3/2}𝑢^{3/2−𝐻𝑗}≤𝐵𝑑𝑢5/2−𝐻_𝑗,𝐻_𝑗−1/2𝐻_𝑗||+1/2𝑡−𝑠^𝐻𝑗+1/2.(3.35) So Δ𝐾_𝑗1≤𝐶_𝐻𝑗|𝑡−𝑠|^𝐻𝑗+1/2.(3.36) Suppose |𝑡−𝑠| is small enough, we get 𝑆𝛿^(𝑁)𝐒^𝐻(𝑡)−𝐒^𝐻(𝑠)𝐟≤()()2𝜋^−(𝑑/2)_{𝐧,𝑛≥𝑁}12^𝑛𝑑_𝑗=1𝐶_𝐻𝑗𝑓_𝑗^2𝑛𝑗_∞𝛽_𝐻𝑗∧1^{𝑛𝑗+(1/2)}|𝑡−𝑠|^{𝑛𝑗(𝐻𝑗−1/2)+𝐻𝑗}⋅1𝑛_𝑗!≤𝐶_𝐻^𝑑_{𝐧,𝑛≥𝑁𝑗=1}1|𝑡−𝑠|^{𝑛𝑗(𝐻𝑗−1/2)+𝐻𝑗}12^𝑛𝑑_𝑗=1𝑓_𝑗^2𝑛𝑗_∞⋅1𝑛_𝑗!≤𝐶_𝐻1|𝑡−𝑠|^{𝑁(𝐻max−1/2)+𝐻∗}1exp2^𝑑_𝑗=1𝑓_𝑗²_∞,(3.37) with 𝐻^∗=∑^𝑑_𝑗=1𝐻_𝑗. Then we have.

Theorem 3.6. Let 𝑇>0 and 𝑛∈ℕ, suppose that 𝐻^∗+𝑁𝐻_max−12<1,(3.38) then ℓ_𝑇^𝐻,(𝑁)=^𝑇₀^𝑡₀𝛿^(𝑁)𝐒^𝐻(𝑡)−𝐒^𝐻(𝑠)𝑑𝑠𝑑𝑡,(3.39) is well defined as an element of (𝑆)^∗ and lim_𝜀→0ℓ^𝐻,(𝑁)_𝑇,𝜀=ℓ_𝑇^𝐻,(𝑁),𝑖𝑛(𝑆)^∗.(3.40)

Theorem 3.7. For 𝐻∈(0,1), every positive integer 𝑑≥1, and 𝜀>0, suppose that (3.38) holds, then the truncated self-intersection local time ℓ_𝑇^𝐻,(𝑁) has the following chaos expansion: ℓ_𝑇^𝐻,(𝑁)=1(2𝜋)^∞𝑑/2_𝑚=01−1𝐦!2^𝐦∶𝜔^⊗2𝐦∶,𝐆_𝐦,(3.41) where 𝐆_𝐦=^𝑇₀^𝑡₀𝑑𝑠𝑑𝑡^𝑑_𝑗=11𝛿^𝐻𝑗(𝑡,𝑠)^{𝑚𝑗+(1/2)}Δ𝐾^⊗2𝐦,(3.42) for each 𝐦∈ℕ^𝑑 and 𝑚≥𝑁. All other kernel functions 𝐆_𝐦 are identically equal to zero.

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,…,𝑓_𝑑)∈𝒮_𝑑(ℝ), we have 𝑆ℓ_𝑇^𝐻,(𝑁)(𝐟)=(2𝜋)^{−(𝑑/2)𝑇}₀^𝑡₀𝑑𝑠𝑑𝑡_{𝐧,𝑛≥𝑁}−12^𝐧𝑑_𝑗=1𝑓_𝑗,Δ𝐾_𝑗^2𝑛𝑗𝛿^𝐻𝑗(𝑡,𝑠)^{𝑛𝑗+(1/2)}⋅1𝑛_𝑗!.(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.

Theorem 3.8. Suppose all 𝐻_𝑗=𝐻, let 𝑇>0, 𝐧≠0 and 𝑑≥2. (1)If 𝐻𝑑=1, then lim_𝜀→01𝐺log𝜀^{2𝐧,𝜀2,2𝑛}(3.44) exists.(2)If 1<𝑑𝐻<3/2, then lim_𝜀→0𝐺_{2𝐧,𝜀2,2𝑛}(3.45) exists.(3)If 𝐻>3/2, then lim_𝜀→0𝜀^{𝑑/2−(3/4𝐻)}|𝐺_2𝐧,𝜀|_2,2𝑛 exists.

Proof. For 𝐧∈ℕ^𝑑, the 2𝐧th chaos is given by ℓ_{𝑇,2𝐧,𝜀}=(2𝜋)^−𝑑/211𝐧!2^𝑛∶𝜔^⊗2𝐧∶,𝐺_2𝐧,𝜀,(3.46) where 𝐺_2𝐧,𝜀=^𝑇₀𝑑𝑡^𝑡₀𝑑𝑠^𝑑_𝑗=1Δ𝐾_𝑗^⊗2𝐧𝛿^𝐻𝑗(𝑡,𝑠)+𝜀^𝑛𝑗+1/2.(3.47) So 𝔼ℓ_{𝑇,2𝐧,𝜀}²=(2𝐧)!(2𝜋)^𝑑(𝐧!)²2^2𝑛𝐺_2𝐧,𝜀²_{𝐿2(ℝ𝑛)}.(3.48) In view of (3.48), we need to estimate |𝐺_2𝐧,𝜀|²_{𝐿2(ℝ𝑛)}=|𝐺_2𝐧,𝜀|²_2,2𝑛, where 𝐺_2𝐧,𝜀²_2,2𝑛=_ℝ2𝑛𝑑^2𝑛𝑢^𝑇₀^𝑡₀^𝑇₀^𝑡′₀𝑑𝑡𝑑𝑠𝑑𝑡^′𝑑𝑠^′𝑑_𝑗=1Δ𝐾_𝑗(𝑡,𝑠)^⊗2𝑛𝑗Δ𝐾_𝑗𝑡^′,𝑠^{′⊗2𝑛𝑗}𝛿^𝐻𝑗(𝑡,𝑠)+𝜀𝛿^𝐻𝑗𝑡^′,𝑠^′+𝜀^𝑛𝑗+1/2.(3.49) By using Fubini theorem, we first get _ℝ2𝑛𝑑^2𝑛𝑢^𝑑_𝑗=1Δ𝐾_𝑗(𝑡,𝑠)^⊗2𝑛𝑗Δ𝐾_𝑗𝑡^′,𝑠^{′⊗2𝑛𝑗}=^𝑑_𝑗=1^2𝑛𝑗_𝑖=1ℝ𝑑𝑢^𝑗_𝑖Δ𝐾_𝑗(𝑡,𝑠)(𝑡,𝑠)Δ𝐾_𝑗(𝑡,𝑠)𝑡^′,𝑠^′=^𝑑_𝑗=1^2𝑛𝑗_𝑖=1𝔼𝑆^𝐻𝑗(𝑡)−𝑆^𝐻𝑗(𝑠)𝑆^𝐻𝑗𝑡^′−𝑆^𝐻𝑗𝑠^′=^𝑑_𝑗=1𝐶^𝐻𝑗_{𝑡,𝑠,𝑡′𝑠′}^2𝑛𝑗,(3.50) with 𝐶^𝐻𝑗_{𝑡,𝑠,𝑡′𝑠′}=𝔼(𝑆^𝐻𝑗(𝑡)−𝑆^𝐻𝑗(𝑠))(𝑆^𝐻𝑗(𝑡^′)−𝑆^𝐻𝑗(𝑠^′)). Moreover denote by 𝑅^𝐻𝑗_{𝑡,𝑠,𝑡′𝑠′}=𝔼(𝐵^𝐻𝑗(𝑡)−𝐵^𝐻𝑗(𝑠))(𝐵^𝐻𝑗(𝑡^′)−𝐵^𝐻𝑗(𝑠^′)) with 𝐵^𝐻 a fractional Brownian motion with Hurst index 𝐻∈(0,1). Then, from Bojdecki et al. [1], we know that 0<𝐶^𝐻𝑗_{𝑡,𝑠,𝑡′𝑠′}<𝑅^𝐻𝑗_{𝑡,𝑠,𝑡′𝑠′}1,𝐻>2,𝑅^𝐻𝑗_{𝑡,𝑠,𝑡′𝑠′}<𝐶^𝐻𝑗_{𝑡,𝑠,𝑡′𝑠′}1<0,𝐻<2.(3.51) So 𝐺_2𝐧,𝜀²_2,2𝑛≤^𝑇₀^𝑡₀^𝑇₀^𝑡′₀𝑑𝑡𝑑𝑠𝑑𝑡^′𝑑𝑠^′𝑑_𝑗=1𝑅^𝐻𝑗_{𝑡,𝑠,𝑡′𝑠′}^2𝑛𝑗𝛿^𝐻𝑗(𝑡,𝑠)+𝜀𝛿^𝐻𝑗𝑡^′,𝑠^′+𝜀^𝑛𝑗+1/2.(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^′ and 𝑥+𝑦+𝑧=𝑡−𝑠^′<𝑡, where 𝑡 is considered as a parameter. Set |𝐺^(𝑖)_2𝐧,𝜀|²_2,2𝑛 the integral over 𝒯_𝑖,𝑖=1,2,3.
Step  1. For |𝐺⁽¹⁾_2𝐧,𝜀|²_2,2𝑛, we obtain 𝐺⁽¹⁾_2𝐧,𝜀²_2,2𝑛≤𝐶_𝐻^𝑇₀𝑑𝑡_{0≤𝑥+𝑦+𝑧≤𝑡}⋅𝑑𝑥𝑑𝑦𝑑𝑧_𝑑𝑗=1(𝑥+𝑧)^2𝐻𝑗+𝑦+𝑧^2𝐻𝑗−𝑥+𝑦+𝑧^2𝐻𝑗−𝑧^{2𝐻𝑗2𝑛𝑗}𝑥^2𝐻𝑗+𝜀𝑦^2𝐻𝑗+𝜀^{𝑛𝑗+(1/2)}.(3.54) It is almost possible to compute the integral when all 𝐻_𝑗 are different, so let us suppose that all 𝐻_𝑗 are equal to some 𝐻.
Denote by 𝜃_𝑡(𝜀)=𝜀^−(1/2𝐻)𝑡 and make the following change of variables (𝑥,𝑦,𝑧)=𝜀^−(1/2𝐻)(𝑥^′,𝑦^′,𝑧^′), we get 𝐺⁽¹⁾_2𝐧,𝜀²_2,2𝑛≤𝐶_𝐻𝜀^{(3/2𝐻)−𝑑𝑇}₀𝑑𝑡_{0≤𝑥+𝑦+𝑧≤𝜃𝑡(𝜀)}⋅)𝑑𝑥𝑑𝑦𝑑𝑧(𝑥+𝑧^2𝐻+𝑦+𝑧^2𝐻−𝑥+𝑦+𝑧^2𝐻−𝑧^2𝐻2𝑛𝑥^2𝐻+1𝑦^2𝐻+1^{𝑛+(𝑑/2)}=2𝐶_𝐻𝜀^{(3/2𝐻)−𝑑𝑇}₀𝑑𝑡_{0≤𝑥+𝑦+𝑧≤𝜃𝑡(𝜀),0≤𝑥≤𝑦≤𝜃𝑡(𝜀)}⋅)𝑑𝑥𝑑𝑦𝑑𝑧(𝑥+𝑧^2𝐻+𝑦+𝑧^2𝐻−𝑥+𝑦+𝑧^2𝐻−𝑧^2𝐻2𝑛𝑥^2𝐻+1𝑦^2𝐻+1^{𝑛+(𝑑/2)}.(3.55) Denote by 𝑓_𝐻𝑥,𝑦,𝑧=(𝑥+𝑧)^2𝐻+𝑦+𝑧^2𝐻−𝑥+𝑦+𝑧^2𝐻−𝑧^2𝐻,(3.56) and by 𝑓(𝑡,𝐻,𝑑,𝑛,𝜀)=_{0≤𝑥+𝑦+𝑧≤𝜃𝑡(𝜀),0≤𝑥≤𝑦≤𝜃𝑡(𝜀)}𝑓𝑑𝑥𝑑𝑦𝑑𝑧^𝐻𝑥,𝑦,𝑧^2𝑛𝑥^2𝐻+1𝑦^2𝐻+1^{𝑛+(𝑑/2)}.(3.57) First note that |𝑓_𝐻(𝑥,𝑦,𝑧)|≤𝑥^2𝐻 and for 𝑑𝐻>1, _{0≤𝑥≤𝑦}𝑥𝑑𝑥𝑑𝑦^4𝐻𝑛𝑥^2𝐻+1𝑦^2𝐻+1^{𝑛+(𝑑/2)}<∞.(3.58) This implies that for every 𝑥∈ℝ₊, _{0≤𝑥≤𝑦}𝑓𝑑𝑥𝑑𝑦^𝐻𝑥,𝑦,𝑧^2𝑛𝑥^2𝐻+1𝑦^2𝐻+1^{𝑛+(𝑑/2)},(3.59) is finite. On the other hand, for 𝛼∈(1,2−2𝐻), 𝑧^𝛼𝑓_𝐻(𝑥,𝑦,𝑧)^2𝑛 decreases to zero when 𝑧 tends to zero, then 𝑓(𝐻,𝑑,𝑛)=_ℝ+𝑑𝑧_{0≤𝑥≤𝑦}𝑓𝑑𝑥𝑑𝑦^𝐻𝑥,𝑦,𝑧^2𝑛𝑥^2𝐻+1𝑦^2𝐻+1^{𝑛+(𝑑/2)}<∞,(3.60) and so that lim_𝜀→0𝑓(𝑡,𝐻,𝑑,𝑛,𝜀)=𝑓(𝐻,𝑑,𝑛).(3.61) Therefore, we obtain, if 𝐻𝑑∈(1,3/2), lim_𝜀→0𝐺⁽¹⁾_2𝐧,𝜀²_2,2𝑛=0,(3.62) and if 𝐻𝑑≥3/2, lim_𝜀→0𝜀^{𝑑−(3/2𝐻)}𝐺⁽¹⁾_2𝐧,𝜀²_2,2𝑛=2𝑇𝑓(𝑑,𝐻,𝑛).(3.63) Suppose now 𝐻𝑑=1, 𝐺⁽¹⁾_2𝐧,𝜀²_2,2𝑛=2𝜀^1/2𝐻𝑇₀𝑓(𝑡,𝐻,𝑑,𝑛,𝜀)𝑑𝑡,(3.64) with 𝑓(𝑡,𝐻,𝑑,𝑛,𝜀)=𝜃_𝑡(𝜀)_{0≤𝑥≤𝑦≤𝜃𝑡(𝜀)}𝑥𝑑𝑥𝑑𝑦^4𝐻𝑛𝑥^2𝐻+1𝑦^2𝐻+1^{𝑛+(𝑑/2)},(3.65) and then 𝑓(𝑡,𝐻,𝑑,𝑛,𝜀)log𝜀≤𝜃_𝑡(𝜀𝑎())𝐻,𝑑,𝑛(,log𝜀+𝑏𝐻,𝑑,𝑛)(3.66) where 𝑎(𝐻,𝑑,𝑛)=_{0≤𝑥≤𝑦,𝑥≤1}𝑥𝑑𝑥𝑑𝑦^4𝐻𝑛𝑥^2𝐻+1𝑦^2𝐻+1^{𝑛+(𝑑/2)},𝑏(𝐻,𝑑,𝑛)=sup_{0≤𝑡≤𝑇}sup_𝜀>01log𝜀_{1≤𝑥≤𝑦≤𝜃𝑡(𝜀)}𝑥𝑑𝑥𝑑𝑦^4𝐻𝑛𝑥^2𝐻+1𝑦^2𝐻+1^{𝑛+(𝑑/2)}.(3.67) So we obtain 1𝐺log𝜀^{(1)𝑇,2𝐧,𝜀22,2𝑛}≤𝑇²𝑎(𝐻,𝑑,𝑛)(.log𝜀+𝑏𝐻,𝑑,𝑛)(3.68)
Step  2. Let us now treat |𝐺⁽²⁾_2𝐧,𝜀|²_2,2𝑛. 𝐺⁽²⁾_2𝐧,𝜀²_2,2𝑛=2𝜀^{(3/2𝐻)−𝑑𝑇}₀𝑑𝑡𝑔(𝑡,𝐻,𝑑,𝑛,𝜀),(3.69) where 𝑔(𝑡,𝐻,𝑑,𝑛,𝜀)≤𝐶_{𝐻0≤𝑥+𝑦−𝑧≤𝜃𝑡(𝜀),0≤𝑧≤𝑥≤𝑦≤𝜃𝑡(𝜀)}𝑔𝑑𝑥𝑑𝑦𝑑𝑧^𝐻𝑥,𝑦,𝑧^2𝑛𝑥^2𝐻+1𝑦^2𝐻+1^{𝑛+(𝑑/2)},(3.70) where 𝑔_𝐻𝑥,𝑦,𝑧=(𝑥−𝑧)^2𝐻+𝑦+𝑧^2𝐻−𝑥+𝑦−𝑧^2𝐻−𝑧^2𝐻.(3.71) We have that |𝑔_𝐻(𝑥,𝑦,𝑧)|≤2𝑥^2𝐻 and so 𝑧→_{0≤𝑧≤𝑥≤𝑦}𝑔𝑑𝑥𝑑𝑦^𝐻𝑥,𝑦,𝑧^2𝑛𝑥^2𝐻+1𝑦^2𝐻+1^{𝑛+(𝑑/2)}=𝑧^4𝐻𝑛+2_{1≤𝑥≤𝑦}𝑔𝑑𝑥𝑑𝑦^𝐻𝑥,𝑦,𝑧^2𝑛(𝑧^2𝐻𝑥^2𝐻+1)(𝑧^2𝐻𝑦^2𝐻+1)^{𝑛+(𝑑/2)},(3.72) is well defined on ℝ₊ and for 𝐻𝑑≥3/2, one can choose 𝛼∈(1,2−2𝐻) such that when 𝑧→+∞𝑧^𝛼_{0≤𝑧≤𝑥≤𝑦}𝑔𝑑𝑥𝑑𝑦^𝐻𝑥,𝑦,𝑧^2𝑛𝑥^2𝐻+1𝑦^2𝐻+1^{𝑛+(𝑑/2)}→0.(3.73) And then _{ℝ+0≤𝑧≤𝑥≤𝑦}𝑔𝑑𝑥𝑑𝑦^𝐻𝑥,𝑦,𝑧^2𝑛𝑥^2𝐻+1𝑦^2𝐻+1^{𝑛+(𝑑/2)}=𝑔(𝐻,𝑑,𝑛)<∞.(3.74) So that lim_𝜀→0𝑔(𝑡,𝐻,𝑑,𝑛,𝜀)=𝑔(𝐻,𝑑,𝑛).(3.75) Suppose 𝐻𝑑=1, the same computations as in the case of |𝐺⁽¹⁾_2𝐧,𝜀|²_2,2𝑛 leads to 1𝐺log𝜀^{(2)2𝐧,𝜀22,2𝑛}≤2^2𝑛𝑇²𝑎(𝐻,𝑑,𝑛)(.log𝜀+𝑏𝐻,𝑑,𝑛)(3.76) Suppose now 𝐻𝑑∈(1,3/2), we have 𝑔(𝑡,𝐻,𝑑,𝑛,𝜀)≤2^{2𝑛𝜃𝑡}₀^(𝜀)𝑑𝑧𝑧^{2−2𝐻𝑑}_{1≤𝑥≤𝑦}𝑥𝑑𝑥𝑑𝑦^4𝐻𝑛𝑥𝑦^{𝐻(2𝑛+𝑑)}∶=2^2𝑛𝐶(𝐻,𝑑,𝑛).(3.77) Then 𝐺⁽²⁾_2𝐧,𝜀²_2,2𝑛≤𝐶_𝐻2^{2𝑛𝐶(𝐻,𝑑,𝑛)}()𝑇3−2𝐻𝑑)(2−𝐻𝑑^{4−2𝐻𝑑}.(3.78) We know that lim_𝜀→0|𝐺⁽¹⁾_2𝐧,𝜀|²_2,2𝑛=0, then lim_𝜀→0|𝐺⁽²⁾_2𝐧,𝜀|²_2,2𝑛 exists. Let us suppose that 𝐻𝑑=3/2, 𝐺⁽²⁾_2𝐧,𝜀²_2,2𝑛=2^𝑇₀𝑑𝑡ℎ(𝑡,𝐻,𝑑,𝑛,𝜀)(3.79) with ℎ(𝑡,𝐻,𝑑,𝑛,𝜀)≤¹₀𝑑𝑧_{𝑧≤𝑥≤𝑦≤𝜃𝑡(𝜀)}𝑔𝑑𝑥𝑑𝑦^𝐻𝑥,𝑦,𝑧^2𝑛𝑥^2𝐻+1𝑦^2𝐻+1^{𝑛+(𝑑/2)}+^𝜃𝑡₁^(𝜀)𝑑𝑧_{𝑧≤𝑥≤𝑦≤𝜃𝑡(𝜀)}𝑔𝑑𝑥𝑑𝑦^𝐻(𝑥,𝑦,𝑧)^2𝑛𝑥^2𝐻+1𝑦^2𝐻+1^{𝑛+(𝑑/2)}.(3.80) The first term is bounded by 2^2𝑛_{0≤𝑥≤𝑦}𝑥𝑑𝑥𝑑𝑦^4𝐻𝑛𝑥^2𝐻+1𝑦^2𝐻+1^{𝑛+(𝑑/2)}=2^2𝑛𝑒(𝐻,𝑑,𝑛).(3.81) The second term is bounded by ^𝜃𝑡₁^(𝜀)1𝑑𝑧𝑧2^2𝑛_{1≤𝑥≤𝑦}𝑥𝑑𝑥𝑑𝑦^4𝐻𝑛𝑥𝑦^{𝐻(2𝑛+(𝑑/2))}=2^2𝑛𝐶(𝐻,𝑑,𝑛)log𝜃_𝑡(𝜀).(3.82) So 1𝐺log𝜀^{(2)2𝐧,𝜀22,2𝑛}≤2^2𝑛+1𝑒()𝑇+2log𝜀𝐻,𝑑,𝑛_2𝑛𝐻𝐶()2𝐻,𝑑,𝑛𝑇+^2𝑛+1log𝜀^𝑇₀log𝑡𝑑𝑡.(3.83) Therefore 1𝐺log𝜀^{(2)2𝐧,𝜀22,2𝑛}(3.84) has a finite nontrivial limit and 1𝐺log𝜀^{(2)2𝐧,𝜀22,2𝑛}≤2^2𝑛𝐻𝐶(𝐻,𝑑,𝑛)𝑇.(3.85)
Step  3. Finally, let us treat |𝐺⁽³⁾_2𝐧,𝜀|²_2,2𝑛. Let 𝑥=𝑡−𝑡^′,𝑦=𝑡^′−𝑠^′,𝑧=𝑠^′−𝑠. We know 𝐺⁽³⁾_2𝐧,𝜀²_2,2𝑛≤2𝜀^{(3/2𝐻)−𝑑𝑇}₀𝑑𝑡𝑗(𝑡,𝐻,𝑑,𝑛,𝜀),(3.86) where 𝑗(𝑡,𝐻,𝑑,𝑛,𝜀)=_{0≤𝑥+𝑦+𝑧≤𝜃𝑡(𝜀)}𝑘𝑑𝑥𝑑𝑦𝑑𝑧^𝐻𝑥,𝑦,𝑧^2𝑛𝑥^2𝐻+1𝑦^2𝐻+1^{𝑛+(𝑑/2)},(3.87) with 𝑘_𝐻𝑥,𝑦,𝑧=𝑥+𝑦^2𝐻−𝑥^2𝐻+𝑦+𝑧^2𝐻−𝑧^2𝐻.(3.88) It is obvious that |𝑘_𝐻(𝑥,𝑦,𝑧)|≤2𝑦^2𝐻. So 𝑗(𝑡,𝐻,𝑑,𝑛,𝜀)≤2_{0≤𝑥+𝑦+𝑧≤𝜃𝑡(𝜀),0≤𝑥≤𝑦≤𝜃𝑡(𝜀)}𝑑𝑥𝑑𝑦𝑑𝑧2𝑦^2𝐻2𝑛𝑥^2𝐻+1𝑦^2𝐻+1^{𝑛+(𝑑/2)}.(3.89) For 𝐻𝑑>1, _{0≤𝑥≤𝑦}2𝑦^2𝐻2𝑛𝑥^2𝐻+1𝑦^2𝐻+1^{𝑛+(𝑑/2)}𝑑𝑥𝑑𝑦<∞,(3.90) this implies that for every 𝑧∈ℝ₊, _{0≤𝑥≤𝑦}𝑘_𝐻𝑥,𝑦,𝑧^2𝑛𝑥^2𝐻+1𝑦^2𝐻+1^{𝑛+(𝑑/2)}𝑑𝑥𝑑𝑦<∞.(3.91) On the other hand, for 𝛼∈(1,2−2𝐻), 𝑧^𝛼𝑘_𝐻(𝑥,𝑦,𝑧)^2𝑛 decreases to zero when 𝑧 tends to infinity, then 𝑔(𝐻,𝑑,𝑛)=_ℝ+𝑑𝑧_{0≤𝑥≤𝑦}𝑘𝑑𝑥𝑑𝑦^𝐻𝑥,𝑦,𝑧^2𝑛𝑥^2𝐻+1𝑦^2𝐻+1^{𝑛+(𝑑/2)}<∞,(3.92) and so that lim_𝜀→0𝑔(𝑡,𝐻,𝑑,𝑛,𝜀)=𝑔(𝐻,𝑑,𝑛).(3.93) Therefore, if 𝐻𝑑∈(1.3/2), we obtain lim_𝜀→0𝐺⁽³⁾_2𝐧,𝜀²_2,2𝑛=0.(3.94) If 𝐻𝑑≥3/2, lim_𝜀→0𝜀^{𝑑−(3/2𝐻)}𝐺⁽³⁾_2𝐧,𝜀²_2,2𝑛=2𝑇𝑔(𝐻,𝑑,𝑛).(3.95) Suppose now 𝐻𝑑=1, 𝐺⁽³⁾_2𝐧,𝜀²_2,2𝑛=2𝜀^1/2𝐻𝑇₀𝑑𝑡𝑔(𝑡,𝐻,𝑑,𝑛,𝜀),(3.96) where 𝑔(𝑡,𝐻,𝑑,𝑛,𝜀)≤𝜃_𝑡(𝜀)_{0≤𝑥≤𝑦≤𝜃𝑡(𝜀)}𝑑𝑥𝑑𝑦2𝑦^2𝐻2𝑛𝑥^2𝐻+1𝑦^2𝐻+1^{𝑛+(𝑑/2)},(3.97) and then 𝑔(𝑡,𝐻,𝑑,𝑛,𝜀)log𝜀≤𝜃_𝑡(𝜀𝑎())𝐻,𝑑,𝑛(,log𝜀+𝑏𝐻,𝑑,𝑛)(3.98) where 𝑎(𝐻,𝑑,𝑛)=_{0≤𝑥≤𝑦,𝑥≤1}𝑑𝑥𝑑𝑦2𝑦^2𝐻2𝑛𝑥^2𝐻+1𝑦^2𝐻+1^{𝑛+(𝑑/2)},𝑏(𝐻,𝑑,𝑛)=sup_{0≤𝑡≤𝑇}sup_𝜀>01log𝜀_{1≤𝑥≤𝑦≤𝜃𝑡(𝜀)}𝑑𝑥𝑑𝑦2𝑦^2𝐻2𝑛𝑥^2𝐻+1𝑦^2𝐻+1^{𝑛+(𝑑/2)}.(3.99) So we obtain 1𝐺log𝜀^{(3)2𝐧,𝜀22,2𝑛}≤𝑇²𝑎(𝐻,𝑑,𝑛)(.log𝜀+𝑏𝐻,𝑑,𝑛)(3.100)

Theorem 3.9. Assume that 𝐻_𝑖=𝐻,𝑖=1,…,𝑑, then for 𝐻∈(0,1) and 𝑑≥1 satisfying 𝐻𝑑<1, ℓ^𝐻_𝑇,𝜀→ℓ^𝐻_𝑇,𝑖𝑛𝐿²,(3.101) as 𝜀 tends to zero.

Proof. By Theorems 3.3 and 3.7, we only need to consider chaos expansion of ℓ^𝐻_𝑇,𝜀 and ℓ^𝐻_𝑇. Similar techniques in Oliveira et al. [12] allow us to write 𝐼=𝔼ℓ^𝐻_𝑇,2𝐧²=(2𝐧)!(2𝜋)^𝑑(𝐧!)²2^2𝑛𝐺_2𝐧,𝜀²_{𝐿2(ℝ𝑛)}.(3.102) In view of (3.102), we need to estimate |𝐺_2𝐧|²_{𝐿2(ℝ𝑛)}=|𝐺_2𝐧|²_2,2𝑛, where 𝐺_2𝐧²_2,2𝑛=_𝐦1𝐦!2𝜋^𝑑12^2𝐦1(𝐦!)²⋅_ℝ2𝑛𝑑^2𝑛𝑢_𝒯𝑑𝑡𝑑𝑠𝑑𝑡^′𝑑𝑠^′𝑑_𝑗=1Δ𝐾_𝑗(𝑡,𝑠)^⊗2𝑛𝑗Δ𝐾_𝑗𝑡^′,𝑠^{′⊗2𝑛𝑗}𝛿^𝐻𝑗(𝑡,𝑠)𝛿^𝐻𝑗𝑡^′,𝑠^{′𝑛𝑗+1/2}=12𝜋^𝑑_𝒯𝜆_𝑡,𝑠𝜆_{𝑡′,𝑠′}−𝜇^2−(𝑑/2)𝑑𝑡𝑑𝑠𝑑𝑡^′𝑑𝑠^′,(3.103) where 𝜆_𝑡,𝑠=|𝑡+𝑠|^2𝐻+|𝑡−𝑠|^2𝐻−2^2𝐻−1𝑡^2𝐻+𝑠^2𝐻,𝜆_{𝑡′,𝑠′}=𝑡^′+𝑠^′2𝐻+𝑡^′−𝑠^′2𝐻−2^2𝐻−1𝑡^′2𝐻+𝑠^′2𝐻,1𝜇=2𝑠−𝑡^′2𝐻+𝑠^′−𝑡^2𝐻−𝑡−𝑡^′2𝐻−𝑠−𝑠^′2𝐻+𝑠−𝑡^′2𝐻+𝑠^′+𝑡^2𝐻−𝑡+𝑡^′2𝐻−𝑠+𝑠^′2𝐻.(3.104) In order to show |𝐼|<∞, we need some preliminaries. Recall 𝕋=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, 𝜆_𝑡,𝑠𝜆_{𝑡′,𝑠′}−𝜇²≥𝜅(𝑡−𝑠)^2𝐻𝑡^′−𝑡^2𝐻+𝑡^′−𝑠^′2𝐻𝑠^′−𝑠^2𝐻;(3.106)(2)for (𝑠,𝑡,𝑠^′,𝑡^′)∈𝕋₂, 𝜆_𝑡,𝑠𝜆_{𝑡′,𝑠′}−𝜇²≥𝜅(𝑡−𝑠)^2𝐻𝑡^′−𝑠^′2𝐻;(3.107)(3)for (𝑠,𝑡,𝑠^′,𝑡^′)∈𝕋₃, 𝜆_𝑡,𝑠𝜆_{𝑡′,𝑠′}−𝜇²≥𝜅(𝑡−𝑠)^2𝐻𝑡^′−𝑠^′2𝐻.(3.108) Then we can easily check that, if 𝐻𝑑<1, |𝐼|<∞. Moreover, ℓ_𝑇,𝜀 converges to ℓ^𝐻_𝑇 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 theorem of the difference ℓ^𝐻_𝑇,𝜀−𝔼ℓ^𝐻_𝑇,𝜀 in 𝐿² as 𝜀 tends to zero will be discussed in future works.

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), 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).