ABSTRACT

We give a proof in semi-group theory based on the Malliavin Calculus of Bismut type in semi-group theory and Wentzel-Freidlin estimates in semi-group of our result giving an expansion of an hypoelliptic heat-kernel outside the cut-locus where Bismut’s non-degeneray condition plays a preominent role.

1. Introduction

Let us consider some vector fields on with bounded derivatives at each order. We consider the generator

(1)

It generates a Markov semi-group P_t acting on bounded continuous f functions on. The natural question is to know if the semi-group has an heat-kernel:

(2)

Let us suppose that the strong Hoermander hypothesis is checked: in such a case Hoermander ([1]) proved the existence of a smooth heat kernel. Malliavin [2] proved again this theorem by using a probabilistic representation of it. A lot of tools of stochastic analysis were translated recently by Léandre in semi-group theory. We refer to the review papers [3]. In particular [4] proved again the existence of the heat kernel by using the Malliavin Calculus of Bismut type in semi-group theory.

Let us recall what is strong Hoermander hypothesis.

Let

(3)

(4)

Strong Hoermander hypothesis in is the following: there exits an l such that

(5)

Under Hoermander hypothesis in x, exists and is smooth in y.

Let h be a path from [0,1] into with finite energy

(6)

The Hilbert space of such that (6) is satisfied is denoted by.

We consider the horizontal curve starting from:

(7)

We consider the control distance

(8)

By standard result of semi-riemannian geometry ([5], [6]), if the Hoermander hypothesis is checked in all, is finite continuous.

Bismut in his seminal book [7] has introduced the notion of cut-locus associated to the sub-riemannian distance. We will recall in the first part what is the cut-locus in sub-riemannian geometry.

Bismut in his seminal book [3] pointed out the relationship between the Malliavin Calculus, Wentzel-Freidlin estimates and short time asymptotics of heat-kernels. This relationship was fully performed by Léandre in [8,9]. In particular, by using probabilistic technics we proved:

Theorem 1. (Léandre [9]). If and are not in the cut-locus of the sub-riemannian distance, we have when

(9)

where.

In the proof we used a mixture between large deviation estimates, the Malliavin Calculus and the Bismutian procedure. Several authors laters ([10,11]) have presented other probabilistic proofs of (9). See [12] in a special case. We refer to [13] for an analytic proof of this result.

Remark. The complement of the cut-locus is an opensubset of: estimate (9) is uniform on any compact set of the complement of the cut-locus.

For readers interested by short time asymptotics of heat-kernels by using probabilistic methods, we refer to the review papers [14-16] and to the book of Baudoin [17]. We refer to the books of Davies [18] and of Varopoulos-Coulhon-Saloff-Coste [19] for analytical methods and to the review of Jerison-Sanchez [20] and Kupka [6].

The object of this paper is to translate in semi-group theory the proof of Theorem 1 of Takanobu-Watanabe [11], by using the tools of stochastic analysis for estimate of heat kernels we have translated in semi-group theory in [21,22] and [23] for Varadhan type estimates.

2. The Cut Locus Associated to a Sub-Riemannian Distance

The material of this part is taken on [7]. But we refer to [11] for a nice introduction to it.

We consider the map starting from. This map is a Frechet smooth function from into. We consider. It satisfied the linear equation starting from:

(10)

We get

(11)

The Gram matrix associate to the map is

(12)

Bismut introduced the question to know if is a submersion. It is fullfilled if and only if the Gram matrix is invertible.

By standard result on Carnot-Caratheodory distance

for some such that

.

Let be the set of h such that

. The main remark of Bismut [7] is the following: if and is invertible, then is in a neighborhood of h a submanifold of by using the implicit function theorem.

We recall the following definition:

Definition 2. (Bismut [7]) We say that are not in the cut-locus of the cut-locus of the sub-riemannian distance if the following 3 conditions are checked:

1) for only one element of.

2) The Gram matrix is invertible.

3) is a non-degenerated minimum of the energy function on.

Condition 3) has a meaning because is a manifold on a neighborhood of.

As traditional in sub-riemannian geometry, we consider the Hamiltonian. It is the function from into

(13)

When there is an Hamiltonian, people introduced classically the Hamilton-Jacobi equation associated. In sub-riemannian geometry, this was introduced by Gaveau [24]. A bicharacteristic is the solution of the ordinary differential equation on starting from:

(14)

We put

(15)

We recall some classical result on sub-riemannian geometry (See [11], p. 204):

(16)

(17)

(18)

Let us recall one of the main result of [7]. If does not belong to the cut locus of, then

for a convenient bicharectiristic.

By using result of [11] pp. 206-207, we can compute the Hessian of the energy in in. It is equal to

(19)

We can compute. It is given by

(20)

3. Scheme of the Proof of Theorem 1

We translate in semi-group the proof of [9] in the way presented in [12].

See [22] for similar considerations for logarithmic estimates of the heat-kernel.

We consider classically and introduce the operator

(21)

Classically

(22)

We consider the unique curve of minimum enegy sucht and we introduce the operator

(23)

This generates a time inhomogeneous semi-group. According the Girsanov formula in semi-group theory of Léandre [4], we introduce the vector field on:

(24)

and the generator written in Itô form

(25)

According [21], p. 207, we have:

(26)

We consider the generator

(27)

It differs from by. This last vector field commute with. We deduce that

(28)

We consider the vector fields

(29)

and the generator

(30)

We have clearly that

(31)

Let us consider the flow associated to the ordinary differential Equation (7). Let us introduce the vector fields

(32)

and the time-dependent generator

(33)

We have the main formula

(34)

where is the map which to z associate. Since, we have only to estimate the density in of the measure which to associates

(35)

We can suppose without any restriction that.

We perform the dilation.

This means that we have to consider the vector fields

(36)

and the generator

(37)

We consider the density ot the measure which to the test function f associates

(38)

The main result of [21] is the following: for some

(39)

The main difference with [21] is in treatment of the term. We refer to [9,10,12] for the treatment of that expression by using stochastic analysis.

In Part 2, and satisfy a system of stochastic differential equations in cascade with associated vector fields. We denote the generic element of. We consider the vector fields

(40)

and the generator

(41)

From (14), (15), (18), the density is equal to the density in 0 of the measure which to f associates

(42)

where is associated to by the procedure of the Part 2. Theorem 1 will follow from Theorem 6.

We consider the generic element of and

(43)

and the generator

(44)

The following lemma is proved in the appendix and was originally proved by stochastic analysis in [12].

Lemma 3. For any positive, there exists a such that

when

The next lemma is due to Bismut [7] and is proved without using stochastic analysis in the appendix:

Lemma 4. Let be very small. There exists a such that

(46)

The remaining part of the scheme of the proof is to apply the Malliavin Calculus of Bismut type depending of a parameter of [21], Part 3 to the the semi-group

. We will apply an improvement of Theorem 1 of [21]. We consider

where is the set on invertible matrices on and the set of symmetric matrices on (is called the Malliavin matrix). We consider if the vector fields on

(47)

and

(48)

Let be the generator

(49)

It generates a time inhomogeneous semi-group. We have Lemma 5. For all positive, the uniform Malliavin condition is checked:

(50)

Theorem 1 is a consequence of the next theorem, (which is an extension of Theorem 1 of [21]) and of (39):

Theorem 6. When, where is the density of the measure which to f associates

(51)

First of all, we recall the Wentzel-Freidlin estimates translated in semi-group theory by Léandre [22,23,25]:

Theorem 7. (Wentzel-Freidlin) Let some time dependent vector fields with bounded derivatives at each order on,. We consider the control distance as in (8) and the diffusion semi-group

. We suppose that the control distance is continuous. Then for any open subset

(52)

Proof of Theorem 6. Let be a smooth function from into equals to 1 and 0 and equals to 0 if. By Wentzel-Freidlin estimates, we can find an such that if.

(53)

By the integration by part of the Malliavin Calculus and the Technical Lemma 5, we have if α is a multi-index

(54)

Therefore we have only to estimate the density in 0 of the measure which to f associate

(55)

By using Lemma 3, Lemma 4, Lemma 5 the density of this measure tends to by using the Malliavin Calculus of Bismut type which depends of a parameter of [21]. □

4. Proof of the Technical Lemmas

Proof of Lemma 3. Let us first show that

(56)

(We will omitt to write later the obvious initial condition which appear in various semi-group later). We introduce a polynomial F of degre less or equal to 2 in and in. Let us compute the Taylor expansion of. We use Lemma 1 of [21].

If the degree of in is 2, the two first terms of the Taylor expansion are 0 and the term of order 2 is

(57)

where we take partial derivatives in the first component. If the polynomial is of degree 1 in, the term of order 1 is

(58)

and the term of order two is

(59)

Lemma 3 will arise from the translation in semi-group theory of Lemma 3.4 of [12].

For all there exists a such that

(60)

The proof follows slightly the line of Lemma 3.4 of [12]. We don’t write the convenient enlarged semigroups when we enlarge the space. We follow the notation of [12], being replaced by and being replaced by.

We introduce the new coordinate

(61)

We use the Itô formula in semi-group theory of [25]. This leads to introduce extra coordinates in the vector fields:

1).

2)

We introduce the new variable which is associated to the extra component vector fields 3).

We use another time the Itô formula in semi-group theory of [25] (11). This leads to introduce the vector field associated to another variable.

4)

We introduce an extra variable associated to another component in the drift which is.

We get for another enlarged semi-group

an extension of formula 3.44 of [12], but with instead of.

Lemma 8. For all, there exists such that

(62)

We postpone later the proof of this lemma which is an analog of the quasi-continuity lemma of [25].

Next we consider another enlarged semi-group to look the couple and together. We use the Itô formula in semi-group theory of [25] (11), [22,23]. We introduce 1)

.

By introducing a cascade of vector fields, we can translate in semi-group theory (3.45) of [12]. We introduce a variable associate to the new component in the drift and we can state an analog of Lemma 8 for a convenient enlarged semi-group.

For every, there exists a small such that

(63)

which is the analog of (3.46) in [12] where we have replaced by.

Let be and associated to the extracomponent vector fields:

1) for the diffusion part.

2)

for the drift part.

We use another time the Itô formula in semi-group theory of [25] (11) for a convenient enlarged semi-group established to study together and. This allow to study and we conclude exactly as in pages 29, 30 of [12] with a small improvement of Lemma 8 to study (3.46), (3.47) of [12]. □

Proof of Lemma 4. We assemble the semi-group

and the semi-group together in a total semi-group. We have some variables and. We have

(64)

Let be small and be very small. We use the exponential inequality in semi-group theory of Lemma 8. For a small and a small, we have (we omitt to write the obvious initial values in the considered semigroups)

(65)

We choose a small and a very small. The exponential inequalities of the proof of Lemma 8 show

(66)

It remains to estimate. We scale the vector fields by and by. We get a generator and a new Markov semi-group

. By a scaling argument, we recognize in

(67)

By a simple improvement of the large deviation estimates of Theorem 7, we get

(68)

We chose a small and we use (20) and the fact don’t belong to the cut-locus in part 2. We deduce that if is very small, that there exists a such that

(69)

Remark. This result is traditionnally hold by using the theory of Fredholm determinant.

Proof of Lemma 5. We assemble together the semigroup and in a global generator We get therefore a total semi-group. We get the Malliavin matrix and

. But is nothing else that

which is invertible because don’t belong to the cut-locus of the subriemannian geometry.

Moreover, by omitting to write the obvious starting conditions, we get for a small:

(70)

for all p. Therefore for a small:

(71)

Since is constant invertible, is bounded independent of if is small enough. By the results of [22,23], there exist such that:

(72)

By Hoelder inequality, we deduce that is bounded independent of. □

Proof of Lemma 8. This follows clearly the line of the quasi-continuity lemma for Wentzel-Freidlin estimates in semi-group theory of [25]. We sketch the proof.

We recall the elementary Kolmogorov lemma of the theory of stochastic processes ([26,27]).

Let be a family of random variables parametrized by with values in equals to 0 or 1 in such that

(73)

for. There exists a continuous version of and the norm of can be estimated only in terms of the data (73).

Let us recall that is a time dependent generator. For there is a time inhomogeneous semigroup. By the Burkholder-DaviesGundy inequality in semi-group theory of [16] (19), we have

(74)

There we can define a continuous stochastic process with probability measure associated to.

We use the Paul Levy martingale exponential in semigroup theory of [25] (33), (46). We get

(75)

By the Kolmogorov lemma, we get

(76)

By standard computations, we deduce that

(77)

But is bounded, and by the same type of argument we deduce that

(78)

But

(79)

such that

(80)

5. Conclusion

We have translated in semi-group theory some classical result of stochastic analysis for subelliptic heat-kernels where Bismutian non degeneracy condition [7] plays a preominent role.

REFERENCES