Local times for grey Brownian motion

In this paper we study the grey Brownian motion, namely its representation and local time. First it is shown that grey Brownian motion may be represented in terms of a standard Brownian motion and then using a criterium of S. Berman, Trans. Amer. Math. Soc., 137, 277–299 (1969), we show that grey Brownian motion admits a λ-square integrable local time almost surely (λ denotes the Lebesgue measure). As a consequence we obtain the occupation formula and state possible generalizations of these results.


Introduction
Grey Brownian motion (gBm) was introduced by W. Schneider 1, 2 as a model for slow anomalous diffusions, i.e., the marginal density function of the gBm is the fundamental solution of the time-fractional diffusion equation, see also Ref. 3. This is a class {B β (t), t ≥ 0, 0 < β ≤ 1} of processes which are self-similar with stationary increments. More recently, this class was extended to the, so called "generalized" grey Brownian motion (ggBm) to include slow and fast anomalous diffusions which contain either Gaussian or non-Gaussian processes e.g., fBm, gBm and others. The time evolution of the marginal density function of this class is described by partial integro-differential equations of fractional type, see Refs. 4,5. In this paper we investigate the class of gBm, namely their representation in terms of standard Brownian motion (Bm) and show the existence of local time.
In Section 2 we recall the construction and certain properties of gBm, in particular the representation of gBm as a product of Bm and an independent positive random variable, see (6) below. Here we would like to emphasize the fact that the representation (6) for gBm allow us to study certain properties of gBm in terms This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 3.0 (CC-BY) License. Further distribution of this work is permitted, provided the original work is properly cited. of those from Bm, e.g., path properties and simulations. Finally, in Section 3 we use the criterium due to S. Berman 6 in order to show that gBm admits a λ-square integrable local time, almost surely, cf. Theorem 3.1 below. As a corollary we obtain the occupation formula.

Grey Brownian motion
In this section we recall the construction of gBm due to W. Schneider. 2 The grey noise space is the probability space (S (R), B(S (R)), µ β ), where S (R) is the space of tempered distributions defined on R, B(S (R)) is the σ-algebra generated by the cylinder sets and µ β is the grey noise measure given by its characteristic functional Here ·, · is the canonical bilinear pairing between S(R) and S (R), · the norm in L 2 (R) and E β is the Mittag-Leffler function of order β defined by In other words, there exists a probability measure ν β on R + which is absolutely continuous with respect to the Lebesgue measure with density M β such that The density M β , the so-called M -Wright probability density function, is related to the fundamental solution of the time-fractional diffusion equation, emerges as a natural generalization of the Gaussian distribution. It is also a special case of the Wright function, namely, Then (2) gives the decomposition which says that the grey noise measure µ β is the mixture of the the Gaussian measures µ τ , τ ≥ 0 with µ 0 = δ 0 the Dirac measure at zero.

Local times for grey Brownian motion
It is easy to show that the random variable X β (1 1 [0,t) )(·) = ·, 1 1 [0,1) , t ≥ 0 is a well defined element in L 2 (S (R), B(S (R)), µ β ) =: L 2 (µ β ) and The gBm B β is then defined as the stochastic process The following properties can be easily derived from (1) and the fact that (1) B β (0) = 0 almost surely. In addition, for each t ≥ 0, the moments of any order are given by Here E denotes the expectation with respect to µ β . (2) For each t, s ≥ 0, the characteristic function of the increments is (3) The covariance function has the form All these properties may be summarized as follows. For any 0 < β ≤ 1, the gBm B β (t), t ≥ 0, is 1 2 -self-similar with stationary increments. It is clear that for β = 1 the gBm coincides with Bm.
It was shown in Ref. 4 that the gBm B β admits the following representation where d = denotes the equality of the finite dimensional distribution and B is Bm. Y β is an independent non-negative random variable with probability density function M β (τ ), τ ≥ 0.

Local times for grey Brownian motion
In this section we prove the existence of local times for gBm using the criterium due to Berman, 6 Lemma 3.1. For the readers convenience we recall the notion of occupation measure as well as occupation density.
is defined in the same way but now the measure µ X·(w) is a random measure, it depends on the sample point w of the probability space. We say that f has an occupation density over [0, T ] if µ f is absolutely continuous with respect to the Lebesgue measure λ and denote this density by L f (·, [0, T ]). In explicit form, for any x ∈ R, Thus, we have A continuous stochastic process X has an occupation density on [0, T ] if, for almost all w ∈ Ω, X · (w) has an occupation density L X (·, [0, T ]), also called local time of X, see Berman. 6 The criteria for the existence of local times for stochastic processes are due to Berman, 6 Section 3. More precisely, a stochastic process X admits a local time if and only if In the following we show that (7) is fulfilled if the stochastic process X is the gBm B β . In fact, from (4) the characteristic function of the increments of gBm B β is given by Using Fubini, and the change of variables r = (2) −1/2 θ|t−s| 1/2 , we have to compute at first The integral in the rhs may be computed using the representation of the Mittag-Leffler function (2), Fubini theorem again and the Gaussian integral, namely .
In the last equality we used the absolute moments of the M -Wright function M β given in (3). Finally, the t, s-integration is performed as follows. Therefore, putting all together, we obtain Thus, we have shown the main result of this subsection which we state in the following theorem.  (1) Theorem (3.1) may be generalized for the so-called "generalized" grey Brownian motion B β,α introduced by A. Mura and F. Mainardi 5 such that for α = 1 we recover the gBm, i.e., B β,1 = B β . Moreover, the local times L B β,α (·, [0, T ]) of B β,α may be weak approximated by the number of crossings of a regularization by convolution of B β,α . For the details see Ref. 9. (2) On the other hand, we may develop the Appell system which is a biorthogonal system of polynomials associated to the grey noise measure µ β in order to construct, describe and characterize test and generalized functions spaces. Then the local time of gBm may be understood as a generalized function in this framework. For the details see Ref. 10.