Category Archives: Diffusions on foliated manifolds

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 →

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 →

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 →

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 →

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 →

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 →

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 →

Northwestern Summer School in Probability, 11-21 July 2016     Website of the summer school I will be lecturing in Northwestern University from July 11 to July 21. Lectures will be posted on the blog.  The main topic will be … Continue reading →