Weighted Local Times of a Sub-fractional Brownian Motion as Hida Distributions

The sub-fractional Brownian motion is a Gaussian extension of the Brownian motion. It has the properties of self-similarity, continuity of the sample paths, and short-range dependence, among others. The increments of sub-fractional Brownian motion is neither independent nor stationary. In this paper we study the sub-fractional Brownian motion using a white noise analysis approach. We recall the represention of sub-fractional Brownian motion on the white noise probability space and show that Donsker’s delta functional of a sub-fractional Brownian motion is a Hida distribution. As a main result, we prove the existence of the weighted local times of a d-dimensional sub-fractional Brownian motion as Hida distributions.


Introduction
It is well-known that fractional Brownian (fBm) motion is an extension of the Brownian motion which satisfies Gaussian properties, self-similarity, long/short-range dependence, and stationarity of increments. Due to these properties fBm has been used as an important tool for stochastic modeling in hydrology, telecommunication, turbulence, image processing, finance, etc. A comprehensive study of fBm can be found in [1,11] and references therein. Another extension of the Brownian motion is the so-called sub-fractional Brownian motion (sub-fBm). It was introduced by Bojdecki et al. [2] while studying occupation time fluctuations of branching particle systems with Poisson initial condition. The sub-fBm with parameter H ∈ (0, 1) is defined as a centered Gaussian process S H = S H (t) t≥0 with S H (0) = 0 a.s. and covariance function When H = 1 2 one recovers the standard Brownian motion. S H is neither a semimartingale nor a Markov process unless H = 1 2 . This implies that the powerful techniques from classical stochastic calculus are not available when dealing with S H . The sub-fBm shares many properties to those of fBm such as self-similarity and long-range dependence. On the contrary the increments of sub-fBm are not stationary. However, S H satisfies the inequalities, for any s, t ≥ 0 with t > s, Hence, by the Kolmogorov's continuity criterion, sample paths of sub-fBm is Hölder continuous of order γ for any γ < H. As a conclusion sub-fBm is suitable for modeling of random phenomenon which posseses self-similarity, long/short dependence, continuous sample paths but non-stationary increments. Some works on sub-fBm can be found, for example, in [3,4,5,10,13,14,15,16,17,18,19].
In this paper we deal with the problem of existence of the weighted local times of a sub-fBm in higher dimensions. Our approach will be based on the theory of white noise analysis. The organization of the paper is as follow. In section 2 we summarize the basics of white noise theory including a realization of sub-fBm on the white noise probability space. Section 3 contains the main results of the paper on the existence of the weighted local times of a sub-fBm as Hida distributions.

WHITE NOISE THEORY
In this section we give some pertinent results of white noise analysis used throughout this paper. For a more comprehensive discussions including various applications of white noise theory we refer to [7,9,12] and references therein. We start with the Gelfand triple is the real Hilbert space of all R d -valued Lebesgue squareintegrable functions. Next, we construct a probability space (S d (R), C, µ) where C is the Borel σ-algebra generated by weak topology on S d (R) and the probability measure µ is uniquely determined through the Bochner-Minlos theorem by fixing the characteristic function Here |·| 0 denotes the usual norm in the , and ·, · denotes the dual pairing between S d (R) and S d (R). The dual pairing is considered as the bilinear extension of the inner product on L 2 d (R), i.e.
. This probability space is known as the R d -valued white noise space since it contains the sample paths of the d-dimensional Gaussian white noise. In the white noise analysis setting a d-dimensional Brownian motion can be represented by a continuous modification of the stochastic process B = (B t ) t≥0 with such that for independent d-tuples of Gaussian white noise ω = (ω 1 , . . . , ω d ) ∈ S d (R) where 1 A denotes the indicator function of a set A ⊂ R.
In order to represent sub-fBm on the white noise space, we use of the following operator sin πHΓ(2H) dan Γ denotes the gamma function. Here I β − f , 0 < β < 1 is the Weyl's type fractional integral operator defined by and D β − f , 0 < β < 1 is the Marchaud's type fractional derivative operator defined by For any Borel function f on [0, ∞) we define its odd extension f • by where B = (B(t)) t∈R is a one-dimensional two-sided Brownian motion.
As a consequence, the d-dimensional sub-fBm can be represented on the white noise space by a continuous modification of the stochastic process S H = (S(t)) t≥0 with such that for independent d-tuples of Gaussian white noise ω = (ω 1 , . . . , ω d ) ∈ S d (R) where 1 • A denotes the odd extension of the indicator function of a set A ⊂ R. In the sequel we will use the Gel'fand triple where (S) is the space of white noise test functions obtained by taking the intersection of a family of Hilbert subspaces of L 2 (µ). The space of white noise distributions (S) * is defined as the topological dual space of (S). Elements of (S) and (S) * are also known as Hida test functions and Hida distributions, respectively. The S-transform of an element Φ ∈ (S) * is defined as where : exp ·, f ::= is the so-called Wick exponential and ·, · denotes the dual pairing between (S) * and (S). We define this dual pairing as the bilinear extension of the sesquilinear inner product on L 2 (µ). The S-transform provides a convenient way to identify a Hida distribution Φ ∈ (S) * , in particular, when it is hard to find the explicit form for the Wiener-Itô chaos decomposition of Φ. Below is a sufficient condition on the Bochner integrability of a family of Hida distributions which depend on an additional parameter.
Theorem 2.2. [8] Let (Ω, A, ν) be a measure space and λ → Φ λ be a mapping from Ω to (S) * . If the S-transform of Φ λ fulfils the following two conditions: (Ω, A, ν) and a continuous seminorm · on S d (R) such that for all z ∈ C, f ∈ S d (R) then Φ λ is Bochner integrable with respect to some Hilbertian norm which topologizing (S) * .

Main Results
By using a limit argument and an Itô formula, Yan et al in [19] obtained a Tanaka formula for one-dimensional sub-fBm involving the so-called weighted local time of the form Our aim is to define multidimensional version of L H (x, T ) in the framework of white noise analysis. First we show that the Donsker delta function of a sub-fBm exists in the sense of Hida distribution.
for any ϕ ∈ S d (R).
Proof. For any ϕ ∈ S d (R) we have , a measurable function of λ ∈ R d . Furthermore, for ϕ ∈ S d (R) and z ∈ C we obtain In the last expression the first factor is an integrable function of λ and the second factor is constant. Thus, according to Theorem 2.2 we have δ S H (t) − x ∈ (S) * . Finally, by applying Gaussian integral formula we obtain Further, we assume that for any ϕ = (ϕ 1 , . . . , ϕ d ) ∈ S d (R) there exists j ∈ {1, . . . , d} fulfils Then, the Bochner integral is a Hida distribution with the S-transform given by for any ϕ ∈ S d (R).