On the Self-Intersection Local Time of Subfractional Brownian Motion

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 R, for any dimension d ≥ 1, with arbitrary Hurst coefficients in 0, 1 . 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 Brownianmotion, Yan and Shen 13 studied the so-called collision local time T ∫T 0 δ S H1 0 t −S H2 0 t dt 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 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 Brownianmotion fBm in short is one of the best known andmost 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. Andwe 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,R ) ⊂ S′ d R , 2.1 where L2 R,R is the real Hilbert space of all vector-valued square integrable functions with respect to Lebesgue measure on R and Sd 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 by | · |d or if there is no risk of confusion simply by | · | and the dual pairing between S′ d and Sd R by 〈·, ·〉, which is defined as the bilinear extension of the inner product on L2 R,R , that is


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 H 0 {S H 0 t , t ≥ 0} with the covariance function for all s, t ≥ 0. For H 1/2, S H 0 coincides with the standard Brownian motion. S H 0 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 H 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:  where δ 0 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 For H 1/2, the process S H 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 Brownian motion, Varadhan 6 has proved that 1/2 T,ε does not converge in L 2 but it can be renormalized so that 1/2 T,ε − T/2π log 1/ε converges in L 2 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  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 L 2 . 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.

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.

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.

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 where L 2 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 L 2 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 L 2 R, R d , that is 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 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 H j , j 1, 2, . . . , d, is given by Abstract and Applied Analysis 5 We recall the explicit formula for the kernel K H t, s of a one-dimensional subfractional Brownian motion with parameter H ∈ 0, 1

2.5
Especially, for H > 1/2, we have We refer to Bojdecki

Hida Distributions and Characterization Results
Let us now consider the complex Hilbert space L 2 : L 2 S d R , B, μ . This space is canonically isomorphic to the symmetric Fock space of symmetric square integrable functions leading to the chaos expansion of the elements in Abstract and Applied Analysis The norm of F is given by where | · | 2,n is the norm in L 2 R n , dt .
To proceed further, we have to consider a Gelfand triple around the space L 2 . We will use the space S * of Hida distributions or generalized Brownian functionals and the corresponding Gelfand triple S ⊂ L 2 ⊂ S * . Here S is the space of white noise test functions such that its dual space with respect to L 2 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

2.12
We define the S-transform of a Φ ∈ S * by Here ·, · denotes the dual pairing between S * and S which is defined as the bilinear extension of the sesquilinear inner product on L 2 . We observe that the multilinear expansion of 2.13 SΦ f : n F n , f n ⊗n , 2.14 extends the chaos expansion to Φ ∈ S * with distribution valued kernels F n such that 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: 1 for every f 1 , f 2 ∈ S d R and λ ∈ R, the mapping λ → F λf 1 f 2 has an entire extension to λ ∈ C, Abstract and Applied Analysis 7 2 there are constants K 1 , K 2 > 0 such that 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 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.
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 for some continuous · on S 2d R and for

Self-Intersection Local Time
Let

3.4
The last equality was obtained by the definition of μ. Then we obtain which clearly fulfills the measurability condition. Moreover, for all z ∈ C, we find

3.6
Abstract and Applied Analysis 9 where, for each j 1, . . . , d, the corresponding term in the second product is bounded by ΔK j , f j 2 .

3.8
As a result 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
Proof. For every f ∈ S R , we calculate the S-transform of T,ε as follows:

3.16
Comparing with the general form of the chaos expansion, we find that the kernel functions are equal to For simplicity, we assume that the notation F G means that there are positive constants C 1 and C 2 such that 3.18 in the common domain of definition for F and G.

Abstract and Applied Analysis 11
Let us now compute the expectation of the self-intersection local time of the subfractional Brownian motion, E H T,ε , it is just the first chaos. So in view of Theorem 3.3 Moreover, for all s ≤ t, the second moment of increments δ H j t, s E S H j t − S H j s 2 satisfying the following estimate:

3.21
We use the change of variables s ε d/2H * z : α ε z with H * d j 1 H j ,

3.22
We divide the integral in two parts

12
Abstract and Applied Analysis The first integral in braces is bounded. Set I the second integral in braces, so , H * 1.

3.26
If

3.27
If H * < 1, there is no blow up, that is,

3.28
In particular, we have From the above results, if H * < 1, the self-intersection local time 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 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 It follows from 3.2 that the S-transform of δ N is given by

3.31
We need to estimate the L 1 -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

14
Abstract and Applied Analysis We obtain 3.33 For I 1 , we obtain

3.34
For I 2 , we obtain

3.35
So Suppose |t − s| is small enough, we get Comparing with the general form of chaos expansion, the result is proved.
Next we will estimate the L 2 -norm of the chaos of the self-intersection local time of subfractional Brownian motion. Now we state the result.

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