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.

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Junfeng Liu 1 and Zhihang Peng 2 and Donglei Tang 1 and Yuquan Cang 1

Recommended by Ahmed El-Sayed

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

Received 16 May 2012; Accepted 24 October 2012

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.

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 0 H} = { ^{S 0 H} (t ) ,t ...5;0 } with the covariance function [figure omitted; refer to PDF] for all s ,t ...5;0 . For H = 1 / 2 , ^{S 0 H} coincides with the standard Brownian motion. ^{S 0 H} 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 0 H} . 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 ...4;t , we have the following estimates: [figure omitted; refer to PDF]

Let ^{S H} = { ^{S H} (t ) ,t ∈ _{... +} } be a d -dimensional subfractional Brownian motion with multiparameters H = ( _{H 1} , _{H 2} , ... , _{H d} ) . Suppose that d ...5;2 , we are interested in, when it exists, the self-intersection local time of subfractional Brownian motion ^{S H} which is formally defined as [figure omitted; refer to PDF] 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 [straight epsilon] tends to zero, of the following sequence of processes [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

For H =1 /2 , the process ^{S 0 H} 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 ^{[cursive l] T , [straight epsilon] 1 /2} does not converge in ^{L 2} but it can be renormalized so that ^{[cursive l] T , [straight epsilon] 1 /2} - ( T / 2 π ) log (1 / [straight epsilon] ) converges in ^{L 2} as [straight epsilon] 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 , ^{[cursive l] T , [straight epsilon] H ,fBm} - _{C H} T ^{[straight epsilon] -1 + ( 1 / 2H )} converges in ^{L 2} as [straight epsilon] 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 ^{L 2} 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 ^{L 2} 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 ...5; 3 / 2d , r ( [straight epsilon] ) ^{[cursive l] T , [straight epsilon] H ,fBm} converges in distribution to a normal law N (0 ,T ^{σ 2} ) , as [straight epsilon] tends to zero, and r ( [straight epsilon] ) = |log [straight epsilon] ^{| -1} if H =3 / (2d ) , and r ( [straight epsilon] ) = ^{[straight epsilon] 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 ^{... d} , for any dimension d ...5;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 ^{L 2} was also derived. For the case of subfractional Brownian motion, Yan and Shen [ 13] studied the so-called collision local time _{[cursive l] T} = ^{∫ 0 T} ... δ ( ^{S 0 _{H 1}} (t ) - ^{S 0 _{H 2}} (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 _{[cursive l] T} is smooth in the sense of Meyer-Watanabe if and only if min ( _{H 1} , _{H 2} ) <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 ^{L 2} . We have organized our paper as follows: Section 2contains 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 ...5;1 , subfractional Brownian motion is the real Gélfand triple [figure omitted; refer to PDF] where ^{L 2} ( ... , ^{... d} ) is the real Hilbert space of all vector-valued square integrable functions with respect to Lebesgue measure on ... and _{...AE; d} ( ... ) , ^{...AE; d [variant prime]} ( ... ) are the Schwartz spaces of the vectors valued test functions and tempered distributions, respectively. Denote the norm in ^{L 2} ( ... , ^{... d} ) by | · _{| d} or if there is no risk of confusion simply by | · | and the dual pairing between ^{...AE; d [variant prime]} and _{...AE; d} ( ... ) by Y9; · , · YA; , which is defined as the bilinear extension of the inner product on ^{L 2} ( ... , ^{... d} ) , that is [figure omitted; refer to PDF] for all g = ( _{g 1} , _{g 2} , ... _{g d} ) ∈ ^{L 2} ( ... , ^{... d} ) and all f = ( _{f 1} , _{f 2} , ... , _{f d} ) ∈ _{...AE; d} ( ... ) . By the Minlos theorem, there is a unique probability measure μ on the σ -algebra [Bernoulli] generated by the cylinder sets on ^{...AE; d [variant prime]} ( ... ) with characteristic function given by [figure omitted; refer to PDF] In this way, we have defined the white noise measure space ( ^{...AE; d [variant prime]} ( ... ) , [Bernoulli] , μ ) . Then a realization of vector of independent subfractional Brownian motion ^{S _{H j}} , j =1,2 , ... ,d , is given by [figure omitted; refer to PDF] We recall the explicit formula for the kernel _{K H} (t ,s ) of a one-dimensional subfractional Brownian motion with parameter H ∈ (0,1 ) [figure omitted; refer to PDF] Especially, for H >1 /2 , we have [figure omitted; refer to PDF]

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 ( ^{L 2} ) : = ^{L 2} ( ^{...AE; d [variant prime]} ( ... ) , [Bernoulli] , μ ) . This space is canonically isomorphic to the symmetric Fock space of symmetric square integrable functions [figure omitted; refer to PDF] leading to the chaos expansion of the elements in ^{L 2} ( ^{...AE; d [variant prime]} ( ... ) , [Bernoulli] , μ ) [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] which reduces expansion ( 2.8) to [figure omitted; refer to PDF] The norm of F is given by [figure omitted; refer to PDF] where | · _{| 2 ,n} is the norm in ^{L 2} ( ^{... 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 2.2below we characterize this space through its S -transform. We recall that given a f ∈ _{...AE; d} ( ... ) , let us consider the Wick exponential [figure omitted; refer to PDF] We define the S -transform of a Φ ∈ (S ^{) *} by [figure omitted; refer to PDF] Here Y9; Y9; · , · YA; YA; 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) [figure omitted; refer to PDF] extends the chaos expansion to Φ ∈ (S ^{) *} with distribution valued kernels _{F n} such that [figure omitted; refer to PDF] for every generalized test function [straight phi] ∈ (S ) with kernel function _{[straight phi] n} .

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

Definition 2.1.

A function F : _{...AE; d} ( ... ) [arrow right] ... is called a U -functional whenever

(1) for every _{f 1} , _{f 2} ∈ _{...AE; d} ( ... ) and λ ∈ ... , the mapping λ [arrow right]F ( λ _{f 1} + _{f 2} ) has an entire extension to λ ∈ ... ,

(2) there are constants _{K 1} , _{K 2} >0 such that [figure omitted; refer to PDF] for some continuous norm || · || on _{...AE; d} ( ... ) .

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 ∈ ... ) be a sequence in (S ^{) *} such that

(1) for all f ∈ _{...AE; d} ( ... ) , ( (S _{Φ n} ) (f ) _{) n ∈ ...} is Cauchy sequence in ... ;

(2) there are _{K 1} , _{K 2} >0 such that for some continuous norm || · || on _{...AE; d} ( ... ) one has [figure omitted; refer to PDF]

then ( _{Φ n} ,n ∈ ... ) converges strongly in (S ^{) *} to a unique Hida distribution.

Corollary 2.4.

Let ( Ω , [Bernoulli] ,m ) be a measure space and λ [arrow right] _{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 [arrow right] (S _{[varphi] λ} ) (f ) is measurable for every f ∈ _{...AE; d} ( ... ) ,

(2) the (S _{Φ λ} ) (f ) obeys a U -estimate [figure omitted; refer to PDF]

for some continuous || · || on _{...AE; 2d} ( ... ) and for _{C 1} ∈ ^{L 1} ( Ω ,m ) , _{C 2} ∈ ^{L ∞} ( Ω ,m ) . Then [figure omitted; refer to PDF]

3. Self-Intersection Local Time

Let us now consider the d -dimensional subfractional Brownian motion ^{S H} (t ) with parameter H = ( _{H 1} , _{H 2} , ... , _{H d} ) . In view of ( 2.4), for j =1 , ... ,d , ^{S _{H j}} (t ) - ^{S _{H j}} (s ) = Y9; Δ _{K j} , _{ω j} YA; with Δ _{K j} = _{K H j} (t ,u ) - _{K H j} (s ,u ) .

Proposition 3.1.

For each t and s strictly positive real numbers, the Bochner integral [figure omitted; refer to PDF] is a Hida distribution with S -transform given by [figure omitted; refer to PDF] for all f = ( _{f 1} , ... , _{f d} ) ∈ _{...AE; d} ( ... ) and ^{δ _{H j}} (t ,s ) = | Δ _{K j} ^{| 2 2} .

Proof.

The proof of this result follows from an applications of Corollary 2.4to the S -transform of the integrand function [figure omitted; refer to PDF] With respect to the Lebesgue measure on ^{... d} . By symmetry, we assume that s ...4;t . For this purpose, we begin by observing that, for f = ( _{f 1} , ... , _{f d} ) ∈ _{...AE; d} ( ... ) , and all λ ∈ ... , one has [figure omitted; refer to PDF] The last equality was obtained by the definition of μ . Then we obtain [figure omitted; refer to PDF] which clearly fulfills the measurability condition. Moreover, for all z ∈ ... , we find [figure omitted; refer to PDF] where, for each j =1 , ... ,d , the corresponding term in the second product is bounded by [figure omitted; refer to PDF] because [figure omitted; refer to PDF] As a result [figure omitted; refer to PDF] where, as a function of λ , the first exponential is integrable on ^{... 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 ^{[cursive l] T , [straight epsilon] H} given by ( 1.4) and ( 3.5), combining [figure omitted; refer to PDF] and the elementary equality [figure omitted; refer to PDF] then we have the following.

Corollary 3.2.

For each t and s strictly positive real numbers, [figure omitted; refer to PDF] is a Hida distribution with S -transform given by [figure omitted; refer to PDF] for all f = ( _{f 1} , ... , _{f d} ) ∈ _{...AE; d} ( ... ) and ^{δ _{H j}} (t ,s ) = | Δ _{K j} ^{| 2 2} .

Theorem 3.3.

For H ∈ (0,1 ) , every positive integer d ...5;1 , and [straight epsilon] >0 , the self-intersection local time ^{[cursive l] T , [straight epsilon] H} has the following chaos expansion: [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Proof.

For every f ∈ ...AE; ( ... ) , we calculate the S -transform of _{[cursive l] T , [straight epsilon]} as follows: [figure omitted; refer to PDF] Comparing with the general form of the chaos expansion, we find that the kernel functions are equal to [figure omitted; refer to PDF]

For simplicity, we assume that the notation F [asymptotically =]G means that there are positive constants _{C 1} and _{C 2} such that [figure omitted; refer to PDF] 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 ( ^{[cursive l] T , [straight epsilon] H} ) , it is just the first chaos. So in view of Theorem 3.3 [figure omitted; refer to PDF] Moreover, for all s ...4;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: [figure omitted; refer to PDF] with _{β H j} =2 - ^{2 2 _{H j} -1} . As the integrand in E ( ^{[cursive l] T , [straight epsilon] H} ) is always positive, we have [figure omitted; refer to PDF] We use the change of variables s = ^{[straight epsilon] d / 2 H *} z : = α ( [straight epsilon] )z with ^{H *} = ^{∑ j =1 d} ... _{H j} , [figure omitted; refer to PDF] We divide the integral in two parts [figure omitted; refer to PDF] The first integral in braces is bounded. Set I the second integral in braces, so [figure omitted; refer to PDF]

Proposition 3.4.

Denote by ^{H *} = ^{∑ j =1 d} ... _{H j} . Let T >0 and d ...5;1 , then if ^{H *} ...5;1 , [figure omitted; refer to PDF] Moreover if ^{H *} =1 , [figure omitted; refer to PDF] If ^{H *} >1 [figure omitted; refer to PDF] If ^{H *} <1 , there is no blow up, that is, [figure omitted; refer to PDF]

In particular, we have

Proposition 3.5.

Suppose that all _{H j} =H , let T >0 and d ...5;1 .

(1) If H =1 /d , then E ^{[cursive l] T , [straight epsilon] H} =Cln (1 / [straight epsilon] ) +o ( [straight epsilon] ) .

(2) If 1 /d <H <3 / (2d ) , then E ^{[cursive l] T , [straight epsilon] H} =C ^{[straight epsilon] ( 1 / 2H ) - ( d / 2 )} +o ( [straight epsilon] ) .

From the above results, if ^{H *} <1 , the self-intersection local time ^{[cursive l] T H} 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 *} ...5;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 2.4will 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 [figure omitted; refer to PDF] It follows from ( 3.2) that the S -transform of ^{δ (N )} is given by [figure omitted; refer to PDF] so [figure omitted; refer to PDF] We need to estimate the ^{L 1} -norm of Δ _{K j} , | Δ _{K 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) [figure omitted; refer to PDF] We obtain [figure omitted; refer to PDF] For _{I 1} , we obtain [figure omitted; refer to PDF] For _{I 2} , we obtain [figure omitted; refer to PDF] So [figure omitted; refer to PDF] Suppose |t -s | is small enough, we get [figure omitted; refer to PDF] with ^{H *} = ^{∑ j =1 d} ... _{H j} . Then we have.

Theorem 3.6.

Let T >0 and n ∈ ... , suppose that [figure omitted; refer to PDF] then [figure omitted; refer to PDF] is well defined as an element of (S ^{) *} and [figure omitted; refer to PDF]

Theorem 3.7.

For H ∈ (0,1 ) , every positive integer d ...5;1 , and [straight epsilon] >0 , suppose that ( 3.38) holds, then the truncated self-intersection local time ^{[cursive l] T H , (N )} has the following chaos expansion: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] for each m ∈ ^{... d} and m ...5;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 = ( _{f 1} , ... , _{f d} ) ∈ _{...AE; d} ( ... ) , we have [figure omitted; refer to PDF] 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.

Theorem 3.8.

Suppose all _{H j} =H , let T >0 , n ...0;0 and d ...5;2 .

(1) If Hd =1 , then [figure omitted; refer to PDF] exists.

(2) If 1 <dH <3 /2 , then [figure omitted; refer to PDF] exists.

(3) If H >3 /2 , then _{lim [straight epsilon] [arrow right]0} ^{[straight epsilon] d /2 - ( 3 / 4H )} _{| G 2n , [straight epsilon] | 2,2n} exists.

Proof.

For n ∈ ^{... d} , the 2n th chaos is given by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] So [figure omitted; refer to PDF] In view of ( 3.48), we need to estimate | _{G 2n , [straight epsilon]} ^{| L 2 ( ... n ) 2} = | _{G 2n , [straight epsilon]} ^{| 2,2n 2} , where [figure omitted; refer to PDF] By using Fubini theorem, we first get [figure omitted; refer to PDF] with ^{C t ,s ,t 's ' _{H j}} = E ( ^{S _{H j}} (t ) - ^{S _{H j}} (s ) ) ( ^{S _{H j}} (t ' ) - ^{S _{H j}} (s ' ) ) . Moreover denote by ^{R t ,s ,t 's ' _{H j}} = E ( ^{B _{H j}} (t ) - ^{B _{H j}} (s ) ) ( ^{B _{H j}} (t ' ) - ^{B _{H j}} (s ' ) ) with ^{B H} a fractional Brownian motion with Hurst index H ∈ (0,1 ) . Then, from Bojdecki et al. [ 1], we know that [figure omitted; refer to PDF] So [figure omitted; refer to PDF] In view of the symmetry of the domain and integrand function, it suffices to integrate only on [figure omitted; refer to PDF] where _{...AF; 1} = {0 <s ' <t ' <s <t } , _{...AF; 2} = {0 <s ' <s <t ' <t } , _{...AF; 3} = {0 <s <s ' <t ' <t } . Let us first integrate over _{...AF; 1} . 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 2n , [straight epsilon] (i ) | 2,2n 2} the integral over _{...AF; i} , i =1,2 ,3 .

Step 1 . For | ^{G 2n , [straight epsilon] (1 ) | 2,2n 2} , we obtain [figure omitted; refer to PDF] 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} ( [straight epsilon] ) = ^{[straight epsilon] - ( 1 / 2H )} t and make the following change of variables (x ,y ,z ) = ^{[straight epsilon] - ( 1 / 2H )} (x ' ,y ' ,z ' ) , we get [figure omitted; refer to PDF] Denote by [figure omitted; refer to PDF] and by [figure omitted; refer to PDF] First note that | _{f H} (x ,y ,z ) | ...4; ^{x 2H} and for dH >1 , [figure omitted; refer to PDF] This implies that for every x ∈ _{... +} , [figure omitted; refer to PDF] is finite. On the other hand, for α ∈ (1,2 -2H ) , ^{z α} _{f H} (x ,y ,z ^{) 2n} decreases to zero when z tends to zero, then [figure omitted; refer to PDF] and so that [figure omitted; refer to PDF] Therefore, we obtain, if Hd ∈ (1,3 /2 ) , [figure omitted; refer to PDF] and if Hd ...5;3 /2 , [figure omitted; refer to PDF] Suppose now Hd =1 , [figure omitted; refer to PDF] with [figure omitted; refer to PDF] and then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] So we obtain [figure omitted; refer to PDF]

Step 2 . Let us now treat | ^{G 2n , [straight epsilon] (2 ) | 2,2n 2} . [figure omitted; refer to PDF] where [figure omitted; refer to PDF] where [figure omitted; refer to PDF] We have that | _{g H} (x ,y ,z ) | ...4;2 ^{x 2H} and so [figure omitted; refer to PDF] is well defined on _{... +} and for Hd ...5; 3 / 2 , one can choose α ∈ (1,2 -2H ) such that when z [arrow right] + ∞ [figure omitted; refer to PDF] And then [figure omitted; refer to PDF] So that [figure omitted; refer to PDF] Suppose Hd =1 , the same computations as in the case of | ^{G 2n , [straight epsilon] (1 ) | 2,2n 2} leads to [figure omitted; refer to PDF] Suppose now Hd ∈ (1,3 /2 ) , we have [figure omitted; refer to PDF] Then [figure omitted; refer to PDF] We know that _{lim [straight epsilon] [arrow right]0} | ^{G 2n , [straight epsilon] (1 ) | 2,2n 2} =0 , then _{lim [straight epsilon] [arrow right]0} | ^{G 2n , [straight epsilon] (2 ) | 2,2n 2} exists. Let us suppose that Hd =3 /2 , [figure omitted; refer to PDF] with [figure omitted; refer to PDF] The first term is bounded by [figure omitted; refer to PDF] The second term is bounded by [figure omitted; refer to PDF] So [figure omitted; refer to PDF] Therefore [figure omitted; refer to PDF] has a finite nontrivial limit and [figure omitted; refer to PDF]

Step 3 . Finally, let us treat | ^{G 2n , [straight epsilon] (3 ) | 2,2n 2} . Let x =t -t ' ,y =t ' -s ' ,z =s ' -s . We know [figure omitted; refer to PDF] where [figure omitted; refer to PDF] with [figure omitted; refer to PDF] It is obvious that | _{k H} (x ,y ,z ) | ...4;2 ^{y 2H} . So [figure omitted; refer to PDF] For Hd >1 , [figure omitted; refer to PDF] this implies that for every z ∈ _{... +} , [figure omitted; refer to PDF] On the other hand, for α ∈ (1,2 -2H ) , ^{z α} _{k H} (x ,y ,z ^{) 2n} decreases to zero when z tends to infinity, then [figure omitted; refer to PDF] and so that [figure omitted; refer to PDF] Therefore, if Hd ∈ (1.3 /2 ) , we obtain [figure omitted; refer to PDF] If Hd ...5;3 /2 , [figure omitted; refer to PDF] Suppose now Hd =1 , [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] So we obtain [figure omitted; refer to PDF]

Theorem 3.9.

Assume that _{H i} =H ,i =1 , ... ,d , then for H ∈ (0,1 ) and d ...5;1 satisfying Hd <1 , [figure omitted; refer to PDF] as [straight epsilon] tends to zero.

Proof.

By Theorems 3.3and 3.7, we only need to consider chaos expansion of ^{[cursive l] T , [straight epsilon] H} and ^{[cursive l] T H} . Similar techniques in Oliveira et al. [ 12] allow us to write [figure omitted; refer to PDF] In view of ( 3.102), we need to estimate | _{G 2n} ^{| L 2 ( ... n ) 2} = | _{G 2n} ^{| 2,2n 2} , where [figure omitted; refer to PDF] where [figure omitted; refer to PDF] In order to show |I | < ∞ , we need some preliminaries. Recall [figure omitted; refer to PDF] Without loss of generality, one can assume t <t ' . For any (s ,t ,s ' ,t ' ) ∈ ... , we denote _{... 1} = { 0 ...4;s < ^{s [variant prime]} <t < ^{t [variant prime]} ...4;T } , _{... 2} = { 0 ...4; ^{s [variant prime]} <s <t < ^{t [variant prime]} ...4;T } , _{... 3} = { 0 ...4;s <t < ^{s [variant prime]} < ^{t [variant prime]} ...4;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 ' ) ∈ _{... 1} , [figure omitted; refer to PDF]

(2) for (s ,t ,s ' ,t ' ) ∈ _{... 2} , [figure omitted; refer to PDF]

(3) for (s ,t ,s ' ,t ' ) ∈ _{... 3} , [figure omitted; refer to PDF]

Then we can easily check that, if Hd <1 , |I | < ∞ . Moreover, _{[cursive l] T , [straight epsilon]} converges to ^{[cursive l] T H} in ( ^{L 2} ) as [straight epsilon] tends to zero.

Remark 3.10.

In this work, we only study the existence of self-intersection of subfractional Brownian motion in ^{L 2} under mild conditions. The asymptotic behavior and the central limit theorem of the difference ^{[cursive l] T , [straight epsilon] H} - E ^{[cursive l] T , [straight epsilon] H} in ^{L 2} as [straight epsilon] 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).

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