# Category Archives: Diffusions on foliated manifolds

## Lecture 7. Integration by parts formula and log-Sobolev inequality

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 … Continue reading

## Lecture 6. Transverse Weitzenbock formula and heat equation on one-forms

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: … Continue reading

## Lecture 5. Riemannian foliations and horizontal Brownian motion

In many interesting cases, we do not actually have a globally defined Riemannian sumersion but a Riemannian foliation. Definition:  Let be a smooth and connected dimensional manifold. A -dimensional foliation on is defined by a maximal collection of pairs of … Continue reading

## Lecture 4. Horizontal Brownian motions on bundles and Hopf fibrations

Let us now turn to some examples of some horizontal Brownian motions associated with submersions. We come back first to an example studied earlier that encompasses the Heisenberg group. Let be a smooth one-form on and let be a -dimensional … Continue reading

## Lecture 3. Horizontal Brownian motions and submersions

From now on, we will assume knowledge of some basic Riemannian geometry. We start by reminding the definition of Brownian motions on Riemannian manifolds. Let be a smooth and connected Riemannian manifold. In a local orthonormal frame , one can … Continue reading

## Lecture 2. Horizontal Brownian motion on the Heisenberg group

We now study in more details the geometric structure behind the diffusion underlying the Levy area process where , , is a two dimensional Brownian motion started at 0. Let us recall that if we consider the 3-dimensional process then … Continue reading

## Lecture 1. The Paul Levy’s stochastic area formula

When studying functionals of a Brownian motion, it may be useful to embed this functional into a larger dimensional Markov process. Consider the case of the Levy area where , , is a two dimensional Brownian motion started at 0. … Continue reading