Those lecture notes are associated to a course I taught at the University of Connecticut in Spring 2019. The focus is on the theory of Dirichlet spaces and heat kernels in metric measure spaces.

]]>

The purpose of these notes is to provide a new probabilistic approach to the Gauss-Bonnet-Chern theorem (and more generally to index theory). They correspond to a five hours course given at a Spring school in France (Mons) in June 2009.

]]>

These notes are the basis of a course given at the Institut Henri Poincare in September 2014. We survey some recent results related to the geometric analysis of hypoelliptic diffusion operators on totally geodesic Riemannian foliations. We also give new applications to the study of hypocoercive estimates for Kolmogorov type operators.

]]>

In those lecture notes, we review some applications of heat semigroups methods in Riemannian and sub-Riemannian geometry. The notes contain parts of courses taught at Purdue University, Institut Henri Poincaré, Levico Summer School and Tata Institute.

]]>

Those are the notes corresponding to my book on stochastic flows. Most of them were written in 2003 during my stay as a postdoc at the Technical University of Vienna.

]]>

Those are the notes of a course on rough paths theory taught at Purdue University in Spring 2013. We develop the theory according to its founder Terry Lyons’ point of view and rely on the book by P. Friz and N. Victoir.

]]>

Those are lecture notes on stochastic differential equations driven by fractional Brownian motions. It only deals with the case , so that the equations are understood in the sense of Young’s integration.

Those notes correspond to a mini course given during the Finnish Summer School in Probability 2012.

]]>Those are the lecture notes of the stochastic calculus course I have been teaching at the University of Toulouse (2003-2008) and then at Purdue University. Some parts of this book grew out of the lectures posted on this blog.

]]>

During my Phd thesis (completed in 2002 under the supervision of Marc Yor) I worked on applying stochastic calculus to mathematical finance. I quit doing research on mathematical finance soon after the thesis but was invited to deliver lectures at Princeton University in 2003 on the topics of modeling of anticipations on financial markets.

A published version might be found here.

]]>- Besov class via heat semigroup on Dirichlet spaces I: Sobolev type inequalities
- Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates
- Besov class via heat semigroup on Dirichlet spaces III: BV functions and sub-Gaussian heat kernel estimates

Let be a good measurable space (like a Polish space) equipped with a -finite measure . Let be a densely defined closed symmetric form on . A function on is called a normal contraction of the function if for almost every

The form is called a Dirichlet form if it is Markovian, that is, has the property that if and is a normal contraction of then and .

Let denote the self-adjoint heat semigroup on associated with the Dirichlet space :

As is well-known, , , can be extended into a contraction semigroup .

**We always assume** .

For , consider the Besov type space

and

**Definition: ***The space of bounded variation functions associated to the Dirichlet form is defined as . For , one defines its variation as
A set is called a -Caccioppoli set if . In that case, its -perimeter is defined as .
*

The following can be deduced from M. Miranda Jr, D. Pallara, F. Paronetto, M. Preunkert, 2007. Assume that is the standard Dirichlet form on ,

then , and for , .

Consider on the Sierpinski triangle the Dirichlet form

where is the walk dimension of the Sierpinski triangle.

Then , where is the Hausdorff dimension of the Sierpinski triangle and

A set is a -Caccioppoli set if its boundary is finite.

The space behaves nicely with respect to tensorization. Consider the product Dirichlet space .

Then and

Therefore, -Caccioppoli sets have Hausdorff co-dimension .

Assume that is the standard Dirichlet form on a complete Riemannian manifold with Ricci curvature bounded from below

, then , and for ,

In the case of Riemannian manifolds, the space and the associated notion of variation we are using are for instance presented in the paper: Heat semigroup and functions of bounded variation on Riemannian __manifolds __by M. Miranda Jr, D. Pallara, F. Paronetto & M. Preunkert.

The following can be deduced from the paper Two Characterization of BV Functions on Carnot Groups via the Heat Semigroup by M. Bramanti, M. Miranda Jr. & D. Pallara. Assume that is the Dirichlet form associated to a sub-Laplacian on a Carnot group

then , and for ,

Let be a Dirichlet space. We consider the following property:

**Theorem: (**Weak Bakry-Emery estimates I)

Let be a strictly local metric Dirichlet space that is locally doubling and that locally supports a 2-Poincar\’e inequality on balls.

If there exists a constant such that

Then, and is satisfied.

The theorem applies to spaces, Carnot groups and large classes of sub-Riemannian manifolds with non-negative Ricci curvature in the sense of Baudoin-Garofalo.

**Theorem: (**Weak Bakry-Emery estimates II)

Let be a metric Dirichlet space with a heat kernel admitting sub-Gaussian estimates. If there exists a constant such that

where then is satisfied.

This applies to the unbounded Sierpinski triangle and their products and large classes of fractals or products of fractals. This is however a conjecture on the Sierpinski carpet.

Let be a Dirichlet space.

**Theorem**: Assume is satisfied and that admits a measurable heat kernel satisfying, for some and ,

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

where .

Under the assumptions of this theorem, one therefore obtains the following general isoperimetric inequality for Caccioppoli sets in Dirichlet spaces

It generalizes the isoperimetric inequality which was known in Riemannian manifolds or Carnot groups (due to N. Varopoulos) but also applies to new situations like fractals.

]]>