A probabilistic formula for gradients of solutions of hypoelliptic Dirichlet problems

We prove a new probabilistic formula for the gradient of the Dirichlet semigroup associated with a class of hypoelliptic operators in a bounded subset of Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^d$$\end{document}.


Introduction and setting of the problem
We are here concerned with the following Cauchy-Dirichlet problem in H = R d : where A and C are d × d matrices, C being symmetric and semi-definite positive, O is a regular subset of R d with boundary ∂O. Formally, a solution of problem (1) is given by the stopped semigroup u(t, x) . . = R O T ϕ(x), T 0, defined by where Here W A is the stochastic convolution and W (t), t 0, is an H -valued standard Wiener process defined on a probability space ( , F, P).
Our basic assumption on A and C is the following: The matrix Q t . . = t 0 e s A Ce s A * ds is non-singular for all t > 0. Hypothesis 1.1 arises in the null controllability for the deterministic system D t ξ = Aξ + √ C u, where u is a control, see e.g. [12]; as is well known it implies that the operator The goal of this paper is to prove an explicit probabilistic formula for the gradient of R t ϕ, t > 0, when ϕ is only bounded and Borel. The main tool is the construction of a suitable translation for which the Cameron-Martin formula applies, see Sect. 2.
We notice that for all t > 0, C ∞ regularity of u(t, x) for more general hypoelliptic equations on O r was proved by Cattiaux [4,5], using Malliavin calculus; however under condition det C > 0.
To our knowledge, the existence of the gradient of the solution u(t, x) for t > 0 under Hypotheses 1.1 and 1.2 is not known.
We believe that our method could be generalised to Kolmogorov operators of the form where b : H → H is a suitable nonlinear mapping. This will be the object of future work.

A Cameron-Martin formula
First we start from an obvious consequence of (2), where Q T , see e.g. [8,Theorem 2.23]. Now we take advantage of the special form of a(x, · ) to simplify identity (3). We write and Note that both F and G are regular. Now (4) becomes so that (3) can be written as where In the integral (6) the variable x no longer appears under the argument of ϕ. Since the As expected, this will produce a surface integral which, unfortunately, does not fulfill the classical assumptions of the theory of Airault-Malliavin [1], see also [7]. To overcome this difficulty, we introduce in Sect. 3 an approximating family of operators R O r T ,n , T > 0, for all decompositions t j = j T 2 n , j = 0, 1, . . . , 2 n of [0, T ], namely by approximating any function h from E by step functions. Then we arrive to an identity for R O r T ,n ϕ(x) (see (17)) that can be easily differentiated with respect to x, see identity (18). It remains to let n → ∞; this is not easy, however, due to the factor which appears in identity (18) because d x (x, · ) y does not belong to the Cameron-Martin space of N Q T . Therefore, some additional work is required, based on the Ehrhard inequality for the Gaussian measure N Q T and Helly's selection principle, see Sect. 5. After some manipulations, we arrive at the representation formula (31) which is the main result of the paper. Our procedure was partially inspired by the paper of Linde [11], which was dealing, however, with a completely different situation.
We end this section with some notation. For any T > 0 we consider the law of X ( · , x) both in the Banach space E = C([0, T ]; H ) and in the Hilbert space X = L 2 (0, T ; H ) (in the second case it is concentrated on E which is a Borel subset of X ). We shall denote by | · | X (resp. | · | E ) the norm of X (resp. of E). The scalar product from two elements x, y ∈ H (resp. X ) will be denoted either by x, y H (resp. x, y X ) or by x · y. If ϕ ∈ C 1 b (E) and η ∈ E we denote by Dϕ(h) · η the derivative of ϕ at h in the direction η.
In what follows several integrals with respect to d N Q will be considered, according to the convenience, both in X and in E.

Strong Feller property
We first recall some properties of the Gaussian measure N Q T . The following lemma is well known, see e.g. [8,Theorem 5.2].
We note that there exists an orthonormal basis (e j ) on X and a sequence (λ j ) of nonnegative numbers such that and an integer k 0 0 such that λ 1 = λ 2 = · · · = λ k 0 = 0, λ j > 0 for all j > k 0 .
We shall denote by L T the linear operator from X into itself defined by Its adjoint L * T is given by It is easily checked that Q T = L T L * T . Moreover, the Cameron-Martin space of N Q T is given by both in E and in X , see [8,Corollary B5].

Remark 3.2
If det C = 0 one easily checks that the Gaussian measure Q T is degenerate and We shall denote by Q −1 T (resp. Q −1/2 T ) the pseudo-inverse of Q T (resp. the pseudoinverse of Q 1/2 T ). 1 Clearly, the domain of Q −1 T is equal to Q T (X ) and h ∈ Q −1 T (X ) if and only if the following series is convergent in X : Similar assertion holds for Q −1/2 T . Now, to introduce the required translation, we first need a lemma.
It follows that det U T 2 e T Tr A det Q T /2 > 0, as claimed.
and define a(x, · ) . . = Q T u(x, · ). Then a(x, T ) = e T A x. Moreover, there are c T , c 1,T > 0 such that and Proof Write so that (9) and (10) follow easily.
Now we prove the first new result of the paper.

Proposition 3.5 Under Hypotheses
Taking into account (9) we have

It follows that
Since the integral above is finite, we have lim x→x 0 A 1 = 0. Concerning A 2 , we have lim x→x 0 A 2 = 0 by the continuity of d(x, · ) and the Dominated Convergence Theorem.

Approximating family of operators
We define an approximating family of operators where F(x), x ∈ O r , is defined by (5), d(x, t) by (7) and n by Proof (i) follows from Hypothesis 1.2 (ii) and (ii) is a well-known consequence of the local Lipschitzianity of n .

Proposition 4.2 Under Hypotheses
Taking into account (9), we have The conclusion follows from the Dominated Convergence Theorem.
It is useful to write an expression of R O r T ,n ϕ as a finite dimensional integral. To this purpose we consider the linear mapping t j = j T 2 n , j = 0, 1, . . . , 2 n , whose law is obviously Gaussian, say N Q T ,n . Then for any n ∈ N and any ϕ : H 2 n → R bounded and Borel we have Now we can write the approximating operator as an integral over H 2 n , namely

Proposition 4.3 Q T ,n has a bounded inverse for all n
Proof Let n ∈ N, then by (15) we have We claim that if Q T ,n ξ = 0 then ξ = 0. In fact, if Q T ,n ξ = 0, we have E e iλ 2 n h=1 ξ h ,W A (t h ) H = 1 and so, Multiplying both sides by ρ 1 + ρ 2 + · · · + ρ 2 n , L(t 1 ) and taking expectation, yields , v and Q t 1 is non-singular by Hypothesis 1.1.

Differentiating the approximating family of operators
First note that by (12) we have and with and (ψ j ) is the standard orthogonal basis of H 2 n .
Proof We first write identity (11) as Then we drop the dependence on x under the domain of integration by making the translation ξ → ξ − d(x, · ) and recalling that d(x, T ) = 0; we write So, using again the Cameron-Martin Theorem (this is possible thanks to Proposition 4.3), we have where We now can differentiate R O r T ,n ϕ(x) in any given direction y ∈ H . Taking into account that for any x, y ∈ H we have Here F x and G x denote the derivatives with respect to x of F and G respectively, whereas G ξ is the derivative with respect to ξ . Now making the opposite translation ξ j → ξ j + d(x, t j ), j = 0, 1, . . . , 2 n , we obtain Finally, we arrive at the conclusion making the change of variables (14).
In Sect. 4 we shall easily prove the existence of the limit of M 1 (n, x, y) as n → ∞. However, a problem arises, as stated in the introduction, for the term M 2 (n, x, y) due to the factor because d x (x, · ) y does not belong to Q 1/2 T (X ). So, in the next Lemma 5.6 we look for a different expression of M 2 (n, x, y) that does not contain this term. To this purpose, we recall the definition and some properties of the Sobolev space W 1, p (E, N Q T ). We shall need a result which is a straightforward generalisation of [6, Proposition 6.1.5].

whereφ(s) is the extension by oddness of ϕ(s), for s ∈ (−T , 0) and s ∈ (T , 2T ).
Then it is easy to check that (ϕ n ) fulfills (i) and (ii).
The following result is similar to [3, Proposition 4.2].

Proposition 5.3
For all ϕ ∈ C 1 b (E) and any η ∈ Q 1/2 (X ) ⊂ E the following integration by parts formula holds: Proof Let ϕ n ∈ C 1 b (X ) be a sequence as in Lemma 5.2, then we have The conclusion follows letting n → ∞.

(21)
Proof By (19) we have Let us first assume in addition that ϕ ∈ C 1 (O r ). Then we argue as in [3,Proposition 4.5] defining a mapping θ : R → R, Then we approximate M 2 (n, x, y) by setting so that lim →0 M 2 (n, x, y) exists and is given by Now, by a classical integration by parts formula, see e.g. [2], we have Taking into account that the first integral vanishes, because d x (x, T )y = 0, and that we deduce by (22), letting → 0, that Then the conclusion of the lemma follows (by the change of variables (14)) when ϕ ∈ C 1 (O r ). The case ϕ ∈ C(O r ) can be handled by a uniform approximation of ϕ by C 1 (O r ) functions. Finally, if ϕ ∈ B b (O r ) we conclude using the strong Feller property of the semigroup, see Proposition 3.5.
We still need to compute the limit in identity (21). This will require the Ehrhard inequality.

Applying the Ehrhard inequality
Since by Hypothesis 1.2 (i), g is convex, the mapping (· + d(x, · )) (resp. n (· + d(x, · ))) is convex as well. By applying the Ehrhard inequality (see e.g. [2, Theorem 4.4.1]) we see that for any x ∈ O r the real function is concave. Note that −1 maps (0, 1) into (−∞, +∞). As a consequence, x ( · ) (resp. n,x ( · )) is differentiable at any s > 0 up to a discrete set N s where there exist the left and the right derivatives; we shall denote by D + r x (s) (resp. D + r n,x (s)) the right derivative at any discontinuity point, and also (with the same symbol) the derivative at the other points.
It follows that N Q T • ( (h +d(x, · ))) −1 (resp. N Q T • ( n (h +d(x, · ))) −1 ) is absolutely continuous with respect to the Lebesgue measure in R, so that Note that, for any x ∈ H , n,x (s) is increasing in s and decreasing in n. Moreover, n,x (0) = 0 and n,x (s) ↑ 1 as s → ∞. Also, it results Now we are going to estimate D + r n,x (s) independently of n, x and s ∈ [r /2, 3r /2]. Then we shall show that D + r n,x → D + r x as n → ∞.

Lemma 5.7
There exists K r > 0 independent of x, n, s such that Moreover, it results Proof We proceed in three steps.
Step 1. There is l 1 > 0 such that It is enough to show (27) for 2,x (s), because 2,x (s) n,x (s) for n 2. Since the convex set {ξ ∈ H 2 n : 2 (ξ + d(x, · )) < s} is open and non-empty and the measure N Q T ,2 is non-degenerate by Proposition 4.3, it follows that there is l 1 (x) such that

Now
Step 1 follows because O r is compact.
In fact, thanks to Hypothesis 1.2 (ii), there exists M > 0 such that Step 3. Conclusion.
Note first that The sequence (S n,x ( · )) is obviously increasing and also concave by the Ehrhard inequality. Therefore, all elements of (S n,x ( · )) are positive and decreasing; so, they are BV in the interval [r /2, 3r /2]. We claim that the sequence (S n,x (r )) is equi-bounded in [r /2, 3r /2] in BV norm. To show the claim, it is enough to realize that (S n,x (r )) is equi-bounded at r 1 (because it is decreasing). In fact, since S n,x is concave we have if 0 < r /2, Therefore we can apply Helly's selection principle, see e.g. [10,Theorem 5,p. 372] to the sequence (S n,x ( · )) and conclude that there exists a subsequence of (S n,x ( · )) still denoted by (S n,x ( · )) that converges in all points of [r /2, 3r /2] to a function f (x, · ).
We claim that f (x, s) is the derivative of S x (s) in [r /2, 3r /2]. This follows by an elementary argument. Write and recall that the real-valued function S n,x ( · ) is absolutely continuous thanks to [2,Corollary 4.4.2]. By the Dominated Convergence Theorem it follows that for k → ∞ we have Therefore there is a subsequence of (S n,x ) which converges to S x and consequently all the sequence (S n,x ) will converge to S x . Thus x (r ) has the right derivative for r ∈ [r /2, 3r /2], x ∈ O r , and (26) follows.
The next lemma is devoted to the computation of lim →0 M 2,2 (n, x, y), defined by (21).
Proof Let us recall that by (21) we have Taking into account (23) it follows that Note that the existence of a regular distribution of is granted because E is separable, see e.g. [9, 10.2.2]. It follows that by virtue of the Dominated Convergence Theorem.
We are ready now to show the following result.

Proposition 5.9
Assume Hypotheses 1.1 and 1.2 and let n ∈ N. Then we have Moreover, there is c 2,T (r ) > 0 such that the following estimate holds: Proof From Lemmas 5.1, 5.6 and 5.8 we obtain D x R O r T ,n ϕ(x) · y = M 1 (n, x, y) + M 2,1 (n, x, y) + M 2,2 (n, x, y) and so Since letting n → ∞ we obtain, after some simplifications, identity (29). Finally, we prove (30). First by (13) we have Moreover by (12) it follows that and therefore we have Now, by Hypothesis 1.2 (ii) there exists c 1 T > 0 such that Finally, taking into account (25) and Lemma 4.1, we have The result is proved.

Main results
Now we take ϕ ∈ B b (H ), T > 0 and prove a representation formula for D x R O r T ϕ(x). Theorem 6.
Proof We recall that by Proposition 5.9 we have and J (n, x, y) Step 1. Convergence of I (n, x, y) as n → ∞.
For all x ∈ O r and all y ∈ H we have lim n→∞ I (n, x, y) This follows by the Dominated Convergence Theorem arguing as in the proof of Proposition 4.2.
Note that D r n,x (r ) → D r x (r ) as n → ∞ by Lemma 5.
To this aim write Since n → in L 1 (E, N Q ) we have Concerning J 2 (n), note that because n is decreasing to , see e.g. [9, 10.1.7]. Now Step 2 follows from (33) and (34).
Step 3. Existence of D x R O r T ϕ for all ϕ ∈ C b (O r ). Let us recall that by Proposition 4.2 and Steps 1, 2 we know that Note that by (25) we know that D r (x, r ) is uniformly bounded in x. Therefore we can apply Theorem 6.1.