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 will assume that is bounded from below and that and are bounded from above.

In that case, for every .

As before, we denote by the horizontal Brownian motion. The stochastic parallel transport for the connection along the paths of will be denoted by . Since the connection is horizontal, the map is an isometry that preserves the horizontal bundle, that is, if , then . We see then that the anti-development of ,

is a Brownian motion in the horizontal space . The following integration by parts formula will play an important role in the sequel.

**Lemma: **Let . For any adapted process such that

and any ,

**Proof:**

We consider the martingale process

We have then for ,

where we integrated by parts in the last equality.

As an immediate consequence of the integration by parts formula, we obtain the following Clark-Ocone type representation.

**Proposition: **Let . For every , and every ,

where is the natural filtration of .

**Proof:**

Let . From Ito’s integral representation theorem, we can write

for some adapted and square integrable . Using the integration by parts formula, we obtain therefore,

Since is arbitrary, we obtain that

We deduce first the following Poincare inequality for the heat kernel measure.

**Proposition: **For every , , , ,

**Proof: **From the previous proposition we have

We also get the log-Sobolev inequality for the heat kernel measure.

**Proposition: **For every , , , ,

**Proof: **The method for proving the log-Sobolev inequality from the representation theorem is due to Capitaine-Hsu-Ledoux and the argument is easy to reproduce in our setting. Denote and consider the martingale . Applying now Ito’s formula to and taking expectation yields

where is the quadratic variation of . From the Clark-Ocone representation theorem applied with , we have

Thus we have from Cauchy-Schwarz inequality