Let be a smooth, connected manifold with dimension . We assume that is equipped with a Riemannian foliation with bundle like metric and totally geodesic -dimensional leaves.

We define the canonical variation of as the one-parameter family of Riemannian metrics:

We now introduce some tensors and definitions that will play an important role in the sequel.

For , there is a unique skew-symmetric endomorphism such that for all horizontal vector fields and ,

where is the torsion tensor of . We then extend to be 0 on . If is a local vertical frame, the operator does not depend on the choice of the frame and shall concisely be denoted by . For instance, if is a K-contact manifold equipped with the Reeb foliation, then is an almost complex structure, .

The horizontal divergence of the torsion is the tensor which is defined in a local horizontal frame by

The -adjoint of will be denoted .

In the sequel, we shall need to perform computations on one-forms. For that purpose we introduce some definitions and notations on the cotangent bundle.

We say that a one-form to be horizontal (resp. vertical) if it vanishes on the vertical bundle (resp. on the horizontal bundle ). We thus have a splitting of the cotangent space

The metric induces then a metric on the cotangent bundle which we still denote . By using similar notations and conventions as before we have for every in ,

By using the duality given by the metric , tensors can also be seen as linear maps on the cotangent bundle . More precisely, if is a tensor, we will still denote by the fiberwise linear map on the cotangent bundle which is defined as the -adjoint of the dual map of . The same convention will be made for any tensor.

We define then the horizontal Ricci curvature as the fiberwise symmetric linear map on one-forms such that for every smooth functions ,

where is the Ricci curvature of the connection .

If is a horizontal vector field and , we consider the fiberwise linear map from the space of one-forms into itself which is given for and by

We observe that is skew-symmetric for the metric so that is a -metric connection.

If is a one-form, we define the horizontal gradient of in a local frame as the tensor

Similarly, we will use the notation

Finally, we will still denote by the covariant extension on one-forms of the horizontal Laplacian. In a local horizontal frame, we have thus

For , we consider the following operator which is defined on one-forms by

where the adjoint is understood with respect to the metric . It is easily seen that, in a local horizontal frame,

We can also consider the operator which is defined on one-forms by

It is clear that for every smooth one-form on and every the following holds

The following theorem that was proved in this paper is the main result of the lecture:

**Theorem: **Let . For every , we have

**Proof:**

We only sketch the proof and refer to the original paper for the details. If is a local vertical frame of the leaves, we denote

where is the the projection of onto the vertical cotangent bundle. It does not depend on the choice of the frame and therefore defines a globally defined tensor.

Also, let us consider the map which is given in a local coframe ,

A direct computation shows then that

Thus, we just need to prove that if is the operator defined on one-forms by

then for any ,

A computation in local frame shows that

which completes the proof

We also have the following Bochner’s type identity.

**Theorem: **For any ,

We now turn to probabilistic applications.

We denote by the horizontal Brownian motion on . The lifetime of the process is denoted by . We assume that the metric is complete and is bracket generating. As a consequence, one can define the heat semigroup associated to as being the semigroup generated by the self-adjoint extension of .

We define a process by the formula

where the process is the stochastic parallel transport with respect to the connection along the paths of . The multiplicative functional is defined as the solution of the following ordinary differential equation

Observe that the process is a solution of the following covariant Stratonovitch stochastic differential equation:

where is any smooth one-form.

From Gronwall’s lemma and the fact that is an isometry, we easily deduce that

**Lemma: **Let . Assume that there exists a constant such that for every ,

Then, there exists a constant , such that for every ,

For , as usual we will denote

**Theorem: **Assume that there exists a constant such that for every ,

Let be a one-form on which is smooth and compactly supported. The unique solution in of the Cauchy problem:

is given by

**Proof:**

This is Feynman-Kac formula.

Sketch of the proof.

It is proved in this course, that the operator

is essentially self-adjoint on the space of smooth and compactly supported one-forms. Thus, from the assumption, is the generator of a bounded semigroup in that uniquely solves the above Cauchy problem.

From the Bochner’s identity, one has

.

From Shigekawa (L^p contraction for vector valued semigroups), this implies the a priori pointwise bound

.

We now claim that the process

,

is a local martingale. Indeed, from Ito’s formula and the definition of , we have

We now conclude from the fact that the bounded variation part of

is given by .

Form the previous estimates, we conclude that is a martingale

**Corollary: **Let . Assume that there exists a constant such that for every ,

Then, for , and

As a consequence, .

**Proof: **Let .

We have

Thus, from the previous theorem

This representation implies the bound

It is well-known that this type of gradient bound implies the stochastic completeness of . More precisely, we can adapt an argument of Bakry. Let , we have

By means of Cauchy-Schwarz inequality we

find

We now apply the previous inequality with , where is an increasing sequence in , , such that on , and , as .

By monotone convergence theorem we have for every . We conclude that the

left-hand side converges to . Since the right-hand side converges to zero, we reach the conclusion

Since it is true for every , it follows that .