Weak convergence to fractional Brownian motion in some anisotropic Besov space

We give some limit theorems for the occupation times of 1-dimensional Brownian motion in some anisotropic Besov space. Our results generalize those obtained by Csaki et al. [4] in continuous functions space.


Introduction
The classical framework for weak convergence of a sequence (X n , n ≥ 1) of stochastic processes is the Skorokhod space D([0, 1]) endowed with the Skorohod topology, for processes having jumps and a space C([0, 1]) of continuous functions equipped with the uniform convergence topology, for continuous processes, (See for instance Billingsley [2] and Jacod and Shiryaev [5]).Relative compactness in the space of probability measures is a key tool in the study of weak convergence.According to Prohorov's theorem, tightness is always a sufficient condition for relative compactness and is also necessary if the metric space S is separable and complet.In many usual cases, the paths of X n and of the limiting process X offer more regularity than the bare continuity.For instance the Donsker Prokhorov's invariance principle establishes the C([0, 1])-weak convergence to the Brownian motion B of the random polygonal lines X n interpolating the partial sums of an i.i.d.sequence.Here the paths of B are almost surely of Hölder regularity α for any α < 12 and those of X n are of Hölder regularity 1.This remark was exploited by Lamperti [9] to prove the same invariance principle in the space C α (0, 1) of α-Hölder continuous functions for any α < 1  2 .As pointed out by Lamperti, this is an improvement of the C([0, 1])-invariance principle since C α (0, 1) is topologically imbeded in C([0, 1]).
The aim of this paper is to provide a rather general framework to the study of Besov weak convergence of processes indexed by R 2 .To prove our main results, we need a characterization of Besov spaces in terms of the coefficients of the expansion of continuous functions in the Faber-Schauder basis.The first characterization of these spaces, in the L p -norm with p < +∞, by coefficients in the Faber-Schauder basis has been established in Ciesielski et al. [3].The multiparameter case was considered by Kamont [6] et [7].
The structure of this paper is as follows.In section 2 we collecte necessary facts about anisotropic Besov spaces.We presente a regularity of Brownian local times and its fractional derivative in section 3.In section 4 we establich a tightness condition in anisotropic Besov spaces.The last section is devoted to the limit theorem for fractional derivative of Brownian local time.
Most of the estimates of this paper contain unspecified constants; we use the same notation of these constants, even when they vary from line to the next.We shall sometimes emphasize the dependance of these constants upon parameters.

The functional framework
Let T = [0, 1] 2 and denote L p (T ) the space of Lebesgue integrable functions with exponent p, (1 ≤ p < ∞).C(T ) stands for the space of R-valued continuous functions on T .
For any function f : T −→ R, any h ∈ R, and i = 1, 2; the progressive difference in direction e i (where e i = (δ 1,i , δ 2,i ) denotes the i th coordinate vector in R 2 ), is defined by: For any

Now, for any Borel function
We are now going to consider some anisotropic generalized Hölder classes in L p -norm.Define the norm: the anisotropic Besov spaces are defined as follows We will consider a separable Banach subspace of Lip p ( − α, β) defined by: where About more on Besov spaces we refer to Peetre [12] or to Bergh and Löfström [1] for an introduction to one-parameter case.The first characterisation of these spaces in the L p -norm with p < ∞ by coefficients in the Faber-Schauder basis have been done in Ciesielski et al. [3].The multiparameter case have been considered recently by Kamont [6] and [7].To prove our main results, we need the characterization of the Besov spaces in terms of the coefficients of the expansion of a continuous function in the basis consisting of tensor products of Faber-Schauder functions.Let f be a continuous function, let us note by {C n (f ), n ≥ 0} the coefficients of the decomposition of f in the basis consisting of tensor products of Faber-Schauder functions.We refer to Kamont [7] for this characterization.The family of Schauder functions on [0, 1] is defined by: We know that for any continuous function on [0, 1] we have the decomposition: This series is uniformly convergent and the coefficients C n (f ) are given by: and for n = 2 j + k, j ≥ 0 and k = 1, ..., )}}.Now for any continuous function f ∈ C(T ), we have the following decomposition: For 1 ≤ p < ∞, j, j ≥ 0, (k, k ) ∈ {1, ..., 2 j } × {1, ..., 2 j } and 0 < α < 1, we shall use the following notations: .
In order to state our main results we need the following characterization theorem (see Kamont [6] and [7]: (1) The anisotropic Besov space Lip p ( − α, β) is a space of continuous functions linearly isomorphic to a sequence spaces, and we have the following equivalence of norms: with l, l = 0, 1.

Brownian Local times and fractional derivatives
Throughout the paper, {B t , t ≥ 0} will denote a real-valued Brownian motion, with a jointly continuous family of local times {L x t , x ∈ R, t ≥ 0}.The local times satisfy the so called occupation density formula for any bounded or nonnegative Borel function f .It is well known (see McKean [11] and Trotter [14]) that x −→ L x t may be chosen to be Hölder continuous of order η < 1  2 , uniformly in t varying in a compact interval.This allows us to define the fractional derivative of order γ for x −→ L x t for all 0 < γ < 1  2 .Definition 3.1: Let 0 < δ < 1 and f : R −→ R be a function that belongs to C δ (R) ∩ L 1 (R) where C δ (R) is the space of locally δ-Hölder continuous functions on R. For 0 < γ < δ we define D γ ± f by (see e.g.Samko et al. [8]) dy .
The operators D γ + and D γ − are called respectively, right-handed and lefthanded Marchaud fractional derivatives of order γ.

Tightness in the anisotropic Besov space
In the sequel it is more convenient to work with lip * p (

As the canonical injection of lip
) is continuous, weak convergence in the former implies weak convergence in the latter.The following theorem due to Prohorov, plays a crucial role in the sequel.

Theorem 4.1: Suppose a general metric space S is separable and complete. A family Π of probability measures in S is relatively compact if and only if it is tight.
A sufficient condition for the tightness in lip * p ( − α, β) is given by Theorem 4.2: Let (X n s,t , (s, t) ∈ T ) n≥1 be a sequence of stochastic processes satisfying: for some ) for all β > 2 p and all p ≥ 2. The proof of this theorem is based on the following lemmas: .
Proof of Lemma 4.3 Note that (1) implies in particular that F is bounded in L p .Recall that for α i > 1 p , i = 1, 2 the Besov space is a space of continuous functions; by the Fréchet-Kolmogorov's theorem (See Brézis p.72), it is easy to check that F is relatively compact in L p .Hence, for any sequence (f n , n ≥ 1) of F there exists a subsequence (Also denoted by (f n )) converging in L pnorm to some function f ∈ L p .To finish the proof it suffices to prove the two following conditions: Let us choose a subsequence of (f n ) that converges almost surely to f .By Fatou's lemma we get Therefore, for all Then In the same way we can prove that sup 0<t≤1 ωp, < ∞ and sup 0<t≤1 ωp, this shows that f ∈ Lip p ( − α, β).On the other hand, from (2), we have ∀ε > 0, ∃δ 0 such that ∀n ≥ 1 : As a consequence, condition (4.1) gives ω p (f, In the same way we prove that: This completes the proof of (a).
To prove (b); let n, n ≥ 0, we have The estimate of the term sup .
The two other terms can be estimated in the same way.This ends the proof of Lemma 4.3.

It is clear that if
p , which gives (1).To show (2), we have The convergence holds in the anisotropic Besov space lip * p ((( 2β .The proof of these results uses tension criterion established in Theorem 4.2.
Proof.We introduce the following notation: The convergence of the finite-dimensional distributions follows from the results of Csaki et al. [4].It suffices to prove that the sequence A ε (t, x) is tight in the anisotropic Besov space lip * p (((3 2 − α)

Go to show (ii).
It follows now from the Markov property and the additive property of fractional derivative of local time that ∀m ≥ 1, (x, y) ∈ [0, 1]2 and (t, s) ∈ [0, 1] 2 Where θ denote the translation operator.
To finish the proof it suffices to prove that

1
and β ∈ R, we will consider the real valued function ω − α β (.) defined on T by: