An integral inequality for the invariant measure of some finite dimensional stochastic differential equation

We prove an integral inequality for the invariant measure $\nu$ of a stochastic differential equation with additive noise in a finite dimensional space $H=\R^d$. As a consequence, we show that there exists the Fomin derivative of $\nu$ in any direction $z\in H$ and that it is given by $v_z=\langle D\log\rho,z\rangle$, where $\rho$ is the density of $\nu$ with respect to the Lebesgue measure. Moreover, we prove that $v_z\in L^p(H,\nu)$ for any $p\in[1,\infty)$. Also we study some properties of the gradient operator in $L^p(H,\nu)$ and of his adjoint.


Introduction and preliminaries
In the recent paper [DaDe14] the following inequality involving the invariant measure ν of the Burgers equation was proved H RDϕ, z dν ≤ C p ϕ L p (H,ν) |z|, (1.1) Identity (1.2) implies that ν is Fomin differentiable in all directions of the range of R(H) of R. We recall that if ν = N Q (the Gaussian measure of mean 0 and covariance Q) identity (1.2) is well known in Malliavin Calculus. In this case the adjoint (Q 1/2 D) * of Q 1/2 D is called the Skorhood operator.
The aim of the present paper is to show that the inequality (1.1), with R replaced by the identity operator, can also be proved for the invariant measures of some stochastic differential equations in H = R d of the form    dX(t) = b(X(t))dt + dW (t), where W is an R d -valued standard Brownian motion and b fulfills the following assumptions.
(1.4) (ii) b : H → H is continuously differentiable and there exists K > 0, N ∈ N such that (1.5) By (ii) it follows that b is Lipschitz continuous on bounded sets of H, whereas (i) allows to estimate |X(t, x)| 2 by Itô' formula; therefore existence and uniqueness of a strong solution X(·, x) of (1.3) is classical, see e.g. the monograph [Kr95]. We shall denote by P t the transition semigroup P t ϕ(x) = E[ϕ(X(t, x))], t ≥ 0, x ∈ H, ϕ ∈ B b (H) (1.6) For proving (1.1) we argue as in [DaDe14] starting from the elementary identity, see (3.2) Then we prove suitable estimates for DP t ϕ and their integrals with respect to ν. These estimates require some work because, due to the polynomial growth of the derivative of b, see (1.5), we cannot exploit the classical Bismut-Elworthy-Li formula, see [El92]. To overcome this problem we shall argue as in [DaDe03], [DaDe07] and [DaDe14], introducing a suitable potential (in the present case V (x) = K(1 + |x| 2N )) and the Feynman-Kac semigroup (1.7) We shall first estimate DS t ϕ(x), h then DP t ϕ(x), h , by taking advantage of the identity which follows from the variation of constants formula, see Section 2 below.
In Section 3 we prove that inequality (1.1) and identity (1.2) hold with R = I. Moreover, for any z ∈ H we show that the Fomin derivative v z in the direction z ∈ H is given by D log ρ, z , where ρ is the density of ν with respect to the Lebesgue measure. Moreover v z ∈ L p (H, ν) for all p ∈ [1, ∞). Finally, we prove a formula for the adjoint D * of D and also for the elliptic operator − 1 2 D * D which can be seen as a generalisation of the Ornstein-Uhlenbeck operator.
We end this section with some notations. We set H = R d , d ≥ 1 (norm | · |, inner product ·, · ) and denote by L(H) the space of all linear bounded operators from H into H. Moreover, C b (H) is the space of all real continuous and bounded mappings ϕ : H → R endowed with the sup norm 2 Estimates of the derivative of the transition semigroup Let us start by giving an estimate of E(|X(t, x)| 2m ), m ∈ N. The following lemma is standard, we shall give some details of the proof for the reader's convenience.
Lemma 2.1. Assume Hypothesis 1.1(i). Then for any m ∈ N there exists a m > 0 such that Proof. Let first consider the case m = 1. Then by Itô's formula, taking into account (1.4) we find We deduce that By a standard comparison result it follows that Now let m > 1 and ϕ m (x) = |x| 2m . Then we have where I represents identity in H. Consequently Then again by Itô's formula we have It follows that The conclusion follows easily by recurrence.
Now we are going to prove an estimate for the derivative D x X(t, x)h, which we denote by η h (t, x), h ∈ H. As well known η h (t, x) is a solution to the random equation Lemma 2.2. Assume Hypothesis 1.1. Then the following estimate holds Proof. By (2.3) we deduce, taking into account (1.5), that So, the conclusion follows from Gronwall's lemma.
Now we are going to estimate of D x P t ϕ.

Pointwise estimate
As we said in the introduction, we cannot estimate D x P t ϕ for ϕ ∈ C b (H) using the Bismut-Elworthy-Li formula see [El92], because we do not know whether the expectation on the right hand side of (2.4) does exist. For this reason, we introduce the potential We recall that the Bismut-Elworthy-Li formula generalises to S t , see [DaZa97].
In fact for all ϕ ∈ C b (H), setting the following identity holds (2.5) We shall first estimate DS t ϕ(x), h , then DP t ϕ(x), h . In the latter case, we take advantage of the identity which follows from the variation of constants formula; in fact, denoting by L and K the infinitesimal generators of P t and S t respectively, it holds Proof. We start by estimating I 1 . By Hölder's inequality with exponents p, q = p p−1 we have where We now apply Itô's formula to g(z(t)) where g(r) = |r| q , r ∈ R. Since and we find Integrating from 0 to t, yields (2.9) Neglecting the negative first term in the previous identity and taking expectation, we find (2.10) By the Burkholder inequality we have, taking into account Lemma 2.1 |z(r)| q−1 |h|.
(2.11) By Hölder's inequality with exponents q, q q−1 , it follows that (2.12) Now by the Young inequality |z(r)| q + c 1 t q/2 |h| q . (2.14) Concerning A 2 , using again Lemma 2.1, we find By Hölder's inequality with exponents q 2 , q q−2 we have By the Young inequality (2.13) with u = q 2 and v = q q−2 , it follows that there exists c 2 > 0 such that Now let us consider I 2 , and write where 1 p + 1 q = 1 and (2.19) So (2.20) Recalling finally (2.1) we see that there exists c 3 > 0 such that Finally, by (2.5), (2.17) and (2.21), the conclusion follows easily.

The invariant measure ν
We shall denote by π t,x the law of X(t, x) so that for each ϕ ∈ B b (H) we have where a m is the constant in (2.1).
Proof. Let r > 0 and fix x ∈ H. Set B c r = {y ∈ H : |y| ≥ r}. Then, taking into account (2.2) it follows that (2.24) Therefore by the Krylov-Bogoliubov theorem, see e.g [DaZa96], there exists a sequence T n ↑ +∞ such that where ν is an invariant measure of P t . Now we can prove (2.23). By (2.1) we deduce It follows that for any ǫ > 0 Consequently integrating both sides with respect to t over [0, T n ] and dividing by T n , yields for all x ∈ H, t ≥ 0. Finally, letting n → +∞ and taking into account (2.25), we find H |y| 2m 1 + ǫ|y| 2m ν(dy) ≤ a m and the conclusion follows letting ǫ tend to 0.

Integral estimates
Let us start with an estimate of H D x S t ϕ(x), h(x) ν(dx).
Now we are ready to estimate H D x P t ϕ(x), h(x) ν(dx). We start from the identity (2.33) Proposition 2.6. Let p > 1, q > 1, 1 Proof. The first term of (2.33) is bounded by (2.29). Let us estimate the second one. Again by (2.29) we have Now let us chose ǫ > 0 such that Then by Hölder's inequality with exponents p+ǫ ǫ and p+ǫ p it follows that by the invariance of ν. Now by (2.2) there exists a constant C ′ such that (2.37) Non the conclusion follows by the arbitrariness of ǫ, p, q.

The main inequality and its consequences
Theorem 3.1. For all p > 1 there exists a constant C p > 0 such that for all ϕ ∈ L p (H, ν) and all h ∈ H we have Proof.
Step 1. For any ϕ ∈ C 1 b (H) and any h ∈ H the following identity holds.
To prove (3.2) we consider a sequence (b n ) of mappings H → H of class C ∞ such that (i) lim n→∞ b n (x) = b(x), uniformly on bounded sets of H.
To construct (b n ) we first set and f n is sub-linear, then we regularise f n using mollifiers. Now we prove the identity where P n t is the transition semigroup corresponding to b n . It is enough to show (3.3) for each ϕ ∈ C 3 b (H). In such a case set u n (t, x) = P n t ϕ(x) and write (3.4) Now, taking h ∈ H and setting v n (t, x) = Du n (t, x), h we see, by a simple computation, that (3.5) By the variation of constants formula it follows that v n (t, x) = P n t ( Dϕ(x), h ) + t 0 P n t−s Du n (s, x), Ah + b ′ n (x)h ds, (3.6) which coincides with (3.3). Letting n → ∞, yields (3.2).
Integrating (3.2) with respect to ν over H and taking into account the invariance of ν, yields Setting and t = 1 we deduce (3.8) Concerning J 2 we have by (2.34) and taking into account (1.5) (3.9) Finally, recalling (2.23) and setting t = 1 the conclusion follows.

Consequences of the integral inequality (3.1)
The following result can be proved exactly as in [DaDe14], replacing R by I so, we omit the proof. Remark 3.6. The fact that ν has a density ρ with respect to the Lebesgue measure, together with several properties of ρ have already been proved in [MePaRh05], [BoKrRo01] and [BoKrRo05].
Let us finally study some properties of operators D * and D * D. Since, in view of (3.13) (3.17) follows. Now (3.18) follows as well setting F = Dϕ in (3.17).