Non-degeneracy of some Sobolev Pseudo-norms of fractional Brownian motion

Applying an upper bound estimate for small $L^{2}$ ball probability for fractional Brownian motion (fBm), we prove the non-degeneracy of some Sobolev pseudo-norms of fBm.


Introduction
Let B H = B H t : t ∈ [0, 1] be a fractional Brownian motion (fBm) on (Ω, F , P ). That is, B H t : t ≥ 0 is a centered Gaussian process with covariance where H ∈ (0, 1) is the Hurst parameter. Consider the random variable F given by a functional of B H : where p, q ≥ 0 satisfy (2p − 2)H > q − 1.
In the case of H = 1 2 , B H is a Brownian motion, and the random variable F is the Sobolev norm on the Wiener space considered by Airault and Malliavin in [1]. This norm plays a central role in the construction of surface measures on the Wiener space. Fang [4] showed that F is non-degenerate in the sense of Malliavin calculus (see the definition below). Then it follows from the well-known criteria on regularity of densities that the law of F has a smooth density.
The purpose of this note is to extend this result to the case H = 1 2 and to show that F is non-degenerate.
In order to state our result precisely, we need some notations from Malliavin calculus (for which we refer to Nualart [9, Section 1.2]). Denote by E the set of all step functions on [0, 1]. Let H be the Hilbert space defined as closure of E with respect to the scalar product Then the mapping 1 [0,t] → B H t extends to a linear isometry between H and the Gaussian space spanned by B H . We denote this isometry by B H . Then, for any h, g ∈ H, B H (f ) and B H (g) are two centered Gaussian random variables with E[B H (h)B H (g)] = h, g H . We define the space D 1,2 as the closure of the set of smooth and cylindrical random variable of the form with h i ∈ H, f ∈ C ∞ p (R n ) (f and all its partial derivatives has polynomial growth) under the norm We say that a random vector V = (V 1 , . . . , V d ) whose components are in D 1,2 is nondegenerate if its Malliavin matrix γ V = DV i , DV j H is invertible a.s. and (det γ V ) −1 ∈ L p (Ω), for all p ≥ 1 (see for instance [9, Definition 2.1.1]). Our main result is the following theorem.
Theorem 1 For all H ∈ (0, 1), the functional F of a fBm B H given in (2) is non-degenerate. That is, We shall follow the same scheme introduced in [4] to prove Theorem 1. That is, it suffices to prove that for any integer n, there exists a constant C n such that for all ε small. This kind of inequality is called upper bound estimate in small deviation theory (also called small ball probability theory, for which we refer to [6] and the reference therein). To prove (4), we will need an upper bound estimate of the small deviation for the path variance of the fBm, which is introduced in the following section. We comment that Li and Shao [5,Theorem 4] proved that for p > 0, 0 ≤ q < 1 + 2pH, q = 1 and β = 1/(pH − max {0, q − 1}. But (5) gives the small ball probability of F , not of DF H .

An estimate on the path variance of fBm
In this section we show the following useful lemma.
Lemma 2 (Estimate of the path variance of the fBm) Actually, we will only need lim sup In the case of H = 1 2 , this estimate of the path variance for Brownian motion was introduced by Malliavin [7, Lemma 3.3.2], using the following Payley-Wiener expansion of Brownian motion: standard Gaussian random variables. Then the estimate (7) follows by observing that , a sum of χ 2 (1) random variables. The above expansion of Brownian motion can be obtained by integrating an expansion of white noise on the orthonormal basis 1, Payley-Wiener expansion of fBm has been established recently by Dzhaparidze and van Zanten [3]: . are the real zeros of J −H (the Bessel function of the first kind of order −H), and X, X k , Y k , k ∈ N, are independent centered Gaussian random variables with variance is difficult to evaluate in the case H = 1 2 , the techniques of [7, Lemma 3.3.2] to prove (7) no longer work.
Fortunately, recent developments in small deviation theory allow us to derive a simple proof of (6). Proof of Lemma 2. In [8, Theorem 3.1 and Remark 3.1] Nazarov and Nikitin proved that for any square integrable random variable G and any nonnegative function ψ ∈ L 1 [0, 1], .
We comment that Bronski [2] proved (10) for the case G = 0 and ψ ≡ 1 by estimating the asymptotics of the Karhunen-Loeve eigenvalues of fBm. Actually, the assumption G = 0 is not necessary, because a random variable G here doesn't contribute to the asymptotics of the Karhunen-Loeve eigenvalues.

Proof of the main theorem
In this section we prove (3) by estimating P ( DF H ≤ ε) for ε small.
For simplicity, we denote Then the operator Q on L 2 (I) defined by is symmetric positive and compact.
Proof. Compactness follows from Q( t, s) ∈ L 2 (I × I). The function Q( t, s) is symmetric, so is the operator Q. Finally, Q is positive because for any f ∈ L 2 (I), Then, it follows that Q has a sequence of decreasing eigenvalues {λ n } n∈N , i.e. λ 1 ≥ · · · ≥ λ n > 0, and λ n → 0. The corresponding normalized eigen-functions {ϕ n } n∈N form an orthonormal basis of L 2 (I). Each of them is continuous because φ n ( t) = λ −1 n I Q( t, s)φ n ( s)d s and Q( t, s) is continuous. We can write Q( t, s) = n≥1 λ n ϕ n ( t)ϕ n ( s) . (
Since Ψ β is continuous on I, there exists t β = (t ′ β , t β ) ∈ I, δ β and ρ β such that for all Let C = 2 max i∈{1,...,n} sup t∈I ϕ( t) < ∞. Then for any β ′ ∈ S n−1 , Then for any β ′ ∈ S n−1 satisfying β ′ − β ≤ ρ β /2C, one has Then it follows from (17) that for any β ∈ S n−1 , there exists a β i ∈ S n−1 such that Ψ 2 β ( t) ≥ ρ i /2, for all t ∈ A i . Then noticing that |t − t ′ | ≤ 1 and applying Jensen's inequality we obtain Then one obtains (16) by choosing C p,H = c H min 1≤i≤m δ i (ρ i δ i ) 1 2H(p−1) . Remark: At the first glance, it seems that (16) can be obtained by applying (5) to the first inequality in (18). But (5) can only be applied to square interval on the diagonal like [a, b] × [a, b] (after applying the scaling and self-similarity property of fBm), and here the interval Lemma 5 For any integer n, the random vector V = (V 1 , . . . , V n ) defined in (14) is nondegenerate.
Proof. Denote by M = DV i , DV j H the Malliavin matrix of V. We want to show that (det M) −1 ∈ L k , for any k ≥ 1. Note that det M ≥ γ n 1 , where γ 1 > 0 is the smallest eigenvalue of the positive definite matrix M. Then it suffices to show that γ −1 1 ∈ L nk , for any k ≥ 1, for which it is enough to estimate P (γ 1 ≤ ε) for ε small. We have (20) {gamma1} Applying (11) in the computation of the norm (20) yields where in the third inequality we used the fact that n i=1 a 2 Combining (20) and (21) we have For any ε > 0 and 0 < α < where C n is a constant independent of β, β ′ and ε. Note that we can find a finite cover ∪ m i=1 B(β i , exp(− 2 ε α )) of S n−1 with β i ∈ S n−1 and Then on W n , for any β ∈ S n−1 , there exists a β i such that On W β i ∩ W n , applying (22) with A n = (2p − 1) 2 λn n and taking ε small enough, On the other hand, applying Lemma 4, we have 2n ε α exp(− C p,α ε 1/2H(p−1) ) ≤ C exp(− C ε 1/2H(p−1) ).
Also, by Chebyshev's inequality, we can write Then it follows from (23)-(25) that for ε small, This completes the proof of the lemma.