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Derivative Formula and Applications for Degenerate Diffusion Semigroups

By using the Malliavin calculus and solving a control problem, Bismut type derivative formulae are established for a class of degenerate diffusion semigroups with non-linear drifts. As applications, explicit gradient estimates and Harnack inequalities are derived.


Introduction
The Bismut derivative formula introduced in [4], also known as Bismut-Elworthy-Li formula due to [6], is a powerful tool to derive regularity estimates on diffusion semigroups. In the elliptic case this formula can be expressed by using the intrinsic curvature induced by the generator. But in the degenerate case the required curvature lower bound is no longer available. Of course, the Malliavin calculus works also for the hypoelliptic case as shown in e.g. [1] on Riemannian manifolds. In this case the pull-back operator involved in the formula is normally less explicit, so that it is hard for one to derive explicit gradient estimates. Nevertheless, as shown in [1, §6], in some concrete degenerate cases the derivative formula can be explicitly established by solving certain control problems.
Recently, explicit derivative formulae for damping stochastic Hamiltonian systems have been established in [16] and [5] by using Malliavin calculus and coupling respectively, where the degenerate part is linear. In this case successful couplings with control can be constructed in a very explicit way, so that some known arguments developed in the elliptic setting can be applied. However, when the degenerate part is non-linear, the study becomes much more complicated. The main purpose of this paper is to extend results derived in [16,5] to the non-linear degenerate case.
Consider the following degenerate stochastic differential equation on R m × R d : t )dt, dX where X (1) t and X (2) t take values in R m and R d respectively, σ is an invertible d×d-matrix, B t is a d-dimensional Brownian motion, Z (1) ∈ C 2 (R m+d ; R m ) and Z (2) ∈ C 1 (R m+d ; R d ). Let X t = (X (1) t , X (2) t ), Z = (Z (1) , Z (2) ). Then the equation can be formulated as (1.2) dX t = Z(X t )dt + (0, σdB t ).
We assume that the solution is non-explosive, which is ensured by (H1) below. Our purpose is to establish an explicit derivative formula for the associated Markov semigroup P t : where X t (x) is the solution of (1.2) with X 0 = x, and B b (R m+d ) is the set of all bounded measurable functions on R m+d . When m = d, σ = I d×d and Z (1) (x, y) = ∇H(x, ·)(y), Z (2) (x, y) = −∇H(·, y)(x) − F (x, y)∇H(x, ·)(y) for some functions H and F , (1.1) goes back to the stochastic Hamiltonian system with Hamiltonian function H. See e.g. [10] for the physical background and applications in mechanics of the model, and see [11] for exponential convergence of the system to the invariant probability measure. In particular, if H(x, y) = V (x) + 1 2 |y| 2 and F ≡ c for some constant c, (1.3) is associated to the "kinetic Fokker-Planck equation" in PDE, see e.g. [12] where the hypocoercivity and related regularization estimates w.r.t. the invariant probability measure are studied; and is known as "stochastic damping Hamiltonian system" in probability theory, see e.g. [3,15] where some long time behaviors of the system have been investigated.
Following the line of two recent papers [16,5] where Bismut formula and Harnack inequalities are derived for P t associated to (1.1) with Z (1) (x, y) = Ay for some m × dmatrix A, we aim to derive explicit point-wise derivative estimates of P t for more general settings where Z (1) (x, y) might be non-linear and depend on both variables x and y, so that some typical examples for the physical model (1.3) are covered (see Example 4.1 below).
To compare the present equation with those investigated in [16,5] where Z (1) is linear, let us recall some simple notations. Firstly, we write the gradient operator on R m+d as ∇ = (∇ (1) , ∇ (2) ), where ∇ (1) and ∇ (2) stand for the gradient operators for the first and the second components respectively, so that ∇f : R m+d → R m+d for a differentiable function f on R m+d . Next, for a smooth function ξ = (ξ 1 , · · · , ξ k ) : Finally, we will use · to denote the operator norm for linear operators, for instance, M = sup |v|=1 |Mv|.
When Z (1) (x (1) , x (2) ) depends only on x (2) and ∇ (2) Z (1) is a constant matrix with rank m, then equation (1.1) reduces back to the one studied in [5] (and also in [16] for m = d). In this case we are able to construct very explicit successful couplings with control, which imply the desired derivative formula and Harnack inequalities as in the elliptic case. But when Z (1) is non-linear, it seems very hard to construct such couplings. The idea of this paper is to split Z (1) into a linear term and a non-linear term, and to derive an explicit derivative formula by controlling the non-linear part using the linear part in a reasonable way. More precisely, let where B 0 is a constant m × d-matrix. We will be able to establish derivative formulae for P t provided B is dominated by B 0 in the sense that In this case we write F ∈ D(D) and call DF the Malliavin gradient of F . It is well known that (D, D(D)) is a closed operator in L 2 (Ω; µ), whose adjoint operator (δ, D(δ)) is called the divergence operator. That is, For any s ≥ 0, let {K(t, s)} t≥s solve the following random ODE on R m ⊗ R m : We assume (H) The matrix σ ∈ R d ⊗ R d is invertible, and there exists W ∈ C 2 (R m+d ) with W ≥ 1 and lim |x|→∞ W (x) = ∞ such that for some constants C, l 2 ≥ 0 and l 1 ∈ [0, 1], For any v = (v (1) , v (2) ) ∈ R m+d with |v| = 1, we aim to search for h = h(v) ∈ D(δ) such that holds. To construct h, for an H-valued random variable α = (α s ) s∈[0,T ] , let (1.8) We will show that h satisfies (1.7) provided it is in D(δ) and α 0 = v (2) , α T = 0, g T = 0, see Theorem 2.1 below for details. In particular, it is the case for α s given in the following result. Then (2) Let h be determined by (1.8) for (1.11) Then for any p ≥ 2, there exists a constant T p ∈ (0, ∞) if l 1 = 1 and T p = ∞ if l 1 < 1, such that for any T ∈ (0, T p ), (1.7) holds with E|δ(h)| p < ∞.
(3) For any p > 1 there exist constants The remainder of the paper is organized as follows. In Section 2 we present a general result on the derivative formula by using Malliavin calculus, from which we are able to prove Theorem 1.1 in Section 3. In Section 4 we will verify (1.9) for the following two cases respectively: In both cases the L p -gradient estimate (1.12) is derived with specific ξ, while in Case (II) the Harnack inequality introduced in [13] is established provided ∇Z (1) is constant, which extends the corresponding Harnack inequality obtained in [5] for ∇ (1) Z (1) = 0 and ∇ (2) Z (1) is constant with rank m. This type of Harnack inequality has been applied in the study of heat kernel estimates and contractivity properties of Markov semigroups, see e.g. [5] and references therein.

A General Result
In this section we will make use of the following assumption.
be an H-valued random variable such that α 0 = v (2) and α T = 0, and let g t and h t be given in (1.8). If g T = 0 and h ∈ D(δ), then (1.7) holds.
Proof. For simplicity, we will drop the initial data of the solution by writing X t (x) = X t . By (H ′ ) and (1.2) we have X t ∈ D(D), and due to the chain rule and the definition of h t , holds for t ∈ [0, T ]. Next, it is easy to see that Combining this with (2.1) we obtain On the other hand, the directional derivative process Thus, by the uniqueness of the ODE we conclude that In particular, since (g T , α T ) = 0, we have and due to (H ′ ) and (2.2), Combining this with (1.5) and letting f ∈ C 1 b (R m+d ), we are able to adopt the dominated convergence theorem to obtain Remark 2.1. Using the same argument as above, we also have the following derivative formula: where (e j ) is the canonical basis of R m+d , and h(e j ) is defined by (1.8) which implies (2.5) by the integration by parts formula.
Remark 2.2. For the higher order derivative formula, under further regularity assumptions, for any v 1 , · · · , v j ∈ R m+d and f ∈ C 1 b (R m+d ), we have where h(v) is defined by (1.8). In fact, as in the proof of Theorem 2.1, we have The higher derivatives can be obtained by induction.

Proof of Theorem 1.1
The idea of the proof is to apply Theorem 2.1 for the given process α s . Obviously, (H1) implies that for any l ≥ 1, there exists a constant C l such that LW l ≤ C l W l , so that EW (X t (x)) l ≤ e C l t W (x) l and thus, the process is non-explosive; while (H2) imply that ∇Z + ∇ 2 Z ≤ CW l 1 ∨l 2 holds for some C > 0, so that Consequently, (H2) imply that U := E exp[2 T 0 ∇Z (X t )dt] is locally bounded on R m+d if either l 1 < 1 or l 1 = 1 but T 2 C 2 σ 2 e 4+2CT ≤ 1.
Proof. It suffices to prove the first assertion. By the Itô formula and (H1), we have Thus, letting τ n = inf{t ≥ 0 : W (X t ) ≥ n}, for any n ≥ 1 and λ > 0 we have where the second inequality is due to the exponential martingale and (H1). By taking This completes the proof by letting n → ∞.
To ensure that E|δ(h)| p < ∞, we need the following two lemmas.
Consequently, if l 1 < 1, then for any p ≥ 1, and if l 1 = 1, then for any p ≥ 1 there exists a constant T p > 0 such that Proof. Due to Lemma 3.1, it suffices to prove (3.2). From (1.2) we see that for any h ∈ H, D h X t solves the following random ODE: (∇Z)(X s )D h X s ds + (0, σh(t)).
which yields the first estimate in (3.3) by Gronwall's inequality. Moreover, noticing that The second estimate in (3.
This implies that Q t is invertible and (1.10) holds.

Two Specific Cases
As indicated in the end of Section 1, we intend to apply Theorem 1.1 to Case (I) and Case (II) respectively with concrete functions ξ satisfying (1.9). Consequently, for any p > 1 there exist two constants c 1 (p), c 2 (p) ≥ 0, where c 2 (p) = 0 if l 1 = l 2 = 0, such that Proof. It is easy to see that the desired gradient estimate follows from (1.12) for the claimed ξ with φ(t) = t(T −t) T 2 , we only prove the first assertion. Since ∇ (1) Z (1) is bounded, there exists a constant C > 0 such that If Rank[B 0 ] = m, then |B * 0 a| ≥ c ′ |a| holds for some constant c ′ > 0 and all a ∈ R m . Therefore, This completes the proof.
(1) Let According to [9, §3], the limit exists and is an invertible matrix, where (Γ t ) t>0 is a family of projection matrices. Thus, U t ≥ c(t ∧ 1) 2k+1 I m×m holds for some constant c > 0 and all t > 0. Then there exist constants c 1 , c 2 > 0 such that for any t ∈ (0, T 2 ], holds. This proves the first assertion. (3) By the semigroup property and the Jensen inequality, we assume that T ∈ (0, 1]. Let ∇ (2) Z (1) = B 0 be constant. Then h given in Theorem 1.1 is adapted such that Moreover, it is easy to see that for ξ(t) given in (1) and T ∈ (0, 1], holds for some constant c 1 > 0 independent of T . Thus, for any λ > 0, On the other hand, since l 1 ∈ [0, 1], by Lemma 3.1 and the Jensen inequality, there exist two constants c 3 , c 4 > 0 such that Moreover, since 2l 1 < 1, there exists a constant c 5 > 0 such that Combining this with (4.3) and (4.4), we conclude that holds for some constant c > 0. This completes the proof of (3) by (1.7) and the Young inequality (see [2,Lemma 2.4 (4) Again, we only consider T ∈ (0, 1]. Let c 2 and C be in (4.3) and Lemma 3.1 respectively. Then there exists a constant c > 0 such that for any T ∈ (0, 1], λ ≥ c Thus, by (4.3) and Lemma 3.
holds for some constant c ′ > 0 independent of T . Combining this with (4.5) we finish the proof.
To derive the Harnack inequality of P T from Theorem 4.2 (3) and (4), let us recall a result of [5]. If there exist a constant λ 0 > 0 and a positive measurable function γ : holds for some constant λ 0 ∈ (0, ∞] and all f ∈ B + b (R m+d ), then by [5, Proposition 4.1], holds for all f ∈ B + b (R m+d ) and p ≥ 1 + λ 0 . Then we have the following consequence of  (1) If l 1 ∈ [0, 1/2), then there exists a constant c > 0 such that holds for all x, v ∈ R m+d , T > 0, p > 1 and f ∈ B + b (R m+d ).
Then the proof is completed by (4.7).
Finally, according to e.g. [14, §4.2], the Harnack inequalities presented above imply explicit heat kernel estimates and entropy-cost inequalities for the invariant probability measure (if exists).
Since there exist many non-trivial examples of A and B 0 such that (A) holds (see [7]), it is easy to construct corresponding examples to illustrate Theorem 4.2. For instance, for Theorem 4.2 (3) and (4) only simply consider (1.3) with H(x, y) = Ax, y + W (y) such that ∇W = B 0 , and for assertion (2) a small perturbation of W is allowed.