Lecture Notes: Brownian Chen series and Gauss-Bonnet-Chern theorem

Posted on December 30, 2019 by Fabrice Baudoin

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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.

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Lecture notes: Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian foliations

Posted on December 29, 2019 by Fabrice Baudoin

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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.

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Lecture notes: Heat semigroups methods in Riemannian geometry

Posted on December 29, 2019 by Fabrice Baudoin

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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.

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Lecture notes: An introduction to the geometry of stochastic flows

Posted on December 29, 2019 by Fabrice Baudoin

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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.

 

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Lecture notes: Rough paths theory

Posted on December 29, 2019 by Fabrice Baudoin

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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.

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Lecture Notes: Stochastic differential equations driven by fractional Brownian motions

Posted on December 27, 2019 by Fabrice Baudoin

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Those are lecture notes on stochastic differential equations driven by fractional Brownian motions. It only deals with the case H >1/2, 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.

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Lecture Notes: Stochastic calculus and Diffusion semigroups

Posted on December 27, 2019 by Fabrice Baudoin

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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.

 

 

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Lecture notes: Modelling anticipations in financial markets

Posted on December 26, 2019 by Fabrice Baudoin

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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.

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Caccioppoli sets, Part II

Posted on October 9, 2019 by Fabrice Baudoin

After the general introduction to the theory of Caccioppoli sets that was presented in the previous post. I will now sketch some elements of the theory that was developed in our works:

Dirichlet spaces

Let X be a good measurable space (like a Polish space) equipped with a \sigma-finite measure \mu.  Let (\mathcal{E},\mathcal{F}=\mathbf{dom}(\mathcal{E})) be a densely defined closed symmetric form on L^2(X,\mu).  A function v on X is called a normal contraction of the function u if for almost every x,y \in X
| v(x)-v(y)| \le |u(x) -u(y)| \text{ and } |v(x)| \le |u(x)|.
The form \mathcal{E} is called a Dirichlet form if it is Markovian, that is, has the property that if u \in \mathcal{F} and v is a normal contraction of u then v \in \mathcal{F} and \mathcal{E}(v,v) \le \mathcal{E} (u,u).

Heat semigroup

Let \{P_{t}\}_{t\in[0,\infty)} denote the self-adjoint heat semigroup on L^2(X,\mu) associated with the Dirichlet space (X,\mu,\mathcal{E},\mathcal{F}):

\mathcal{E}(f,f)=\lim_{t\to 0^+}\frac{1}{t}\langle (I-P_t)f,f\rangle.

As is well-known, P_t: L^2(X,\mu) \cap L^p (X,\mu) \to L^p (X,\mu), 1 \le p \le \infty, can be extended into a contraction semigroup P_t : L^p (X,\mu) \to L^p (X,\mu).

We always assume P_t1=1.

BV space

For \alpha >0, consider the L^1 Besov type space
\mathbf{B}^{1,\alpha}(\mathcal{E})=\left\{ f \in L^1(X,\mu), \limsup_{t\to 0} \frac{1}{t^\alpha} \int_X P_t ( | f - f(y)|) d\mu(y) <+\infty \right\} and \alpha^\#_1(\mathcal{E})=\sup \{ \alpha >0\, :\, \mathbf{B}^{1,\alpha}(\mathcal{E}) \text{ contains non a.e. constant functions} \}.

Definition: The space of bounded variation functions associated to the Dirichlet form \mathcal{E} is defined as BV(\mathcal{E})=\mathbf{B}^{1,\alpha^\#}(\mathcal{E}). For f \in BV(\mathcal{E}), one defines its variation as
\mathbf{Var}_\mathcal{E} (f)=\liminf_{t\to 0} \frac{1}{t^\alpha} \int_X P_t ( | f - f(y)|) d\mu(y)
A set E \subset X is called a \mathcal{E}-Caccioppoli set if 1_E \in BV(\mathcal{E}). In that case, its \mathcal{E}-perimeter is defined as P_\mathcal{E}(E)=\mathbf{Var}_\mathcal{E} (1_E).

Examples

Example 1: Euclidean space

The following can be deduced from M. Miranda Jr, D. Pallara, F. Paronetto, M. Preunkert, 2007. Assume that \mathcal{E} is the standard Dirichlet form on \mathbb{R}^n,
\mathcal{E}(f,f)=\int_{\mathbb{R}^n} \| \nabla f \|^2 \, dx, \quad f \in W^{1,2}(\mathbb{R}^n),
then \alpha^\#_1(\mathcal{E})=\frac{1}{2},  BV(\mathcal{E})=\mathbf{BV}(\mathbb{R}^n) and for f \in BV(\mathcal{E}), \mathbf{Var}_\mathcal{E} (f)=\frac{2}{\sqrt{\pi}} \| Df \| (\mathbb{R}^n).

Example 2: Sierpinski triangle

Consider on the Sierpinski triangle  SG the Dirichlet form
\mathcal{E}(f) \simeq \limsup_{r\to 0^+}\frac{1}{r^{d_W}}\int_{SG} \int_{B(x,r)}\frac{|f(y)-f(x)|^2}{ \mu(B(x,r))}\, d\mu(y)\, d\mu(x)
where d_W is the walk dimension of the Sierpinski triangle.

Then \alpha^\#_1(\mathcal{E})=\frac{d_H}{d_W}, where d_H is the Hausdorff dimension of the Sierpinski triangle and
\mathbf{Var}_\mathcal{E} (f) \simeq \liminf_{r\to 0^+}\int_{SG} \int_{B(x,r)}\frac{|f(y)-f(x)|}{r^{d_H} \mu(B(x,r))}\, d\mu(y)\, d\mu(x)
A set E \subset SG is a \mathcal{E}-Caccioppoli set if its boundary is finite.

Example 3: Product of Sierpinski triangles

The space BV behaves nicely with respect to tensorization. Consider the product Dirichlet space (SG^n, \mathcal{E}^{\otimes n}, \mu^{\otimes n}).
Then \alpha^\#_1(\mathcal{E}^{\otimes n})=\frac{d_H(SG)}{d_W(SG)} and
\mathbf{Var}_{\mathcal{E}^{\otimes n}} (f) \simeq \liminf_{r\to 0^+}\int_{SG^n} \int_{B(x,r)}\frac{|f(y)-f(x)|}{r^{d_H(SG)} \mu(B(x,r))}\, d\mu^{\otimes n}(y)\, d\mu^{\otimes n}(x)
Therefore, \mathcal{E}^{\otimes n}-Caccioppoli sets have Hausdorff co-dimension d_H(SG) .

Example 4: Riemannian manifolds

Assume that \mathcal{E} is the standard Dirichlet form on a complete Riemannian manifold \mathbb{M} with Ricci curvature bounded from below
\mathcal{E}(f,f)=\int_{\mathbb{M}} \| \nabla f \|^2 \, dx, \quad f \in W^{1,2}(\mathbb{M}), then \alpha^\#_1(\mathcal{E})=\frac{1}{2},  BV(\mathcal{E})=\mathbf{BV}(\mathbb{M})  and for f \in BV(\mathcal{E}),
\mathbf{Var}_\mathcal{E} (f) \simeq \| Df \| (\mathbb{M}).

In the case of Riemannian manifolds, the space \mathbf{BV}(\mathbb{M}) and the associated notion of variation \| Df \| (\mathbb{M})  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.

Example 5: Carnot groups

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 \mathcal{E} is the Dirichlet form associated to a sub-Laplacian on a Carnot group \mathbb{G}
\mathcal{E}(f,f)=\int_{\mathbb{G}} \| \nabla_{\mathcal H} f \|^2 \, dx, \quad f \in W_{\mathcal H}^{1,2}(\mathbb{G}),
then \alpha^\#_1(\mathcal{E})=\frac{1}{2}, BV(\mathcal E)=\mathbf{BV}_\mathcal{H} (\mathbb{G}) and for f \in BV(\mathcal{E}),
\mathbf{Var}_\mathcal{E} (f) =\int_\mathbb{G} \sigma (x) d D_{\mathcal H}f (x) \simeq \| D_{\mathcal H}f \| (\mathbb{G}).

Locality in time property

Let (X,\mu,\mathcal{E},\mathcal{F}) be a Dirichlet space. We consider the following property:
\mathcal{P}_\infty: \quad \quad \quad \quad \mathbf{Var}_{\mathcal{E}} (f) \simeq \sup_{t >0} \frac{1}{t^{\alpha^\#_1(\mathcal{E})}} \int_X P_t ( | f - f(y)|) d\mu(y)

 

Theorem: (Weak Bakry-Emery estimates I)
Let (X,\mu,\mathcal{E},\mathcal{F}) 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 C>0 such that
\| |\nabla P_t f | \|_{L^\infty (X,\mu)} \le \frac{C}{\sqrt{t}} \| f \|_{L^\infty (X,\mu)}, \quad t >0.
Then, \alpha^\#_1(\mathcal{E})=\frac{1}{2} and \mathcal{P}_\infty is satisfied.
The theorem applies to RCD(0,\infty) 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 (X,\mu,\mathcal{E},\mathcal{F}) be a metric Dirichlet space with a heat kernel admitting sub-Gaussian estimates.  If there exists a constant C>0 such that
| P_t f(x)-P_tf(y)| \le C \frac{d(x,y)^\kappa}{t^{\kappa/d_W}} \| f \|_{L^\infty (X,\mu)}, \quad t >0.
where \kappa=d_W (1-\alpha^\#_1(\mathcal{E})) then \mathcal{P}_\infty 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.

 

L^1-Sobolev inequality and isoperimetric inequality

Let (X,\mu,\mathcal{E},\mathcal{F}) be a Dirichlet space.

Theorem: Assume \mathcal{P}_\infty is satisfied and that P_t admits a measurable heat kernel p_t(x,y) satisfying, for some C>0 and \beta >0,
p_{t}(x,y)\leq C t^{-\beta}, \quad t>0.
Then, if 0<\alpha^\#_1(\mathcal{E}) <\beta , there exists a  constant C>0 such that for every f \in BV(\mathcal{E}),
\| f \|_{L^q(X,\mu)} \le C\mathbf{Var}_{\mathcal{E}} (f),
where q=\frac{\beta}{ \beta- \alpha^\#_1(\mathcal{E})}.

Under the assumptions of this theorem, one therefore obtains the following general isoperimetric inequality for Caccioppoli sets in Dirichlet spaces
\mu(E)^{\frac{ \beta- \alpha^\#_1(\mathcal{E})}{\beta}}\le C P_\mathcal{E} (E).

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.

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Caccioppoli sets, Part I

Posted on October 6, 2019 by Fabrice Baudoin

In the next two posts, longer than usual, I will explain some ideas of  recent works written in collaboration with Patricia Alonso-Ruiz, Li Chen,  Luke Rogers, Nageswari Shanmugalingam  and Alexander Teplyaev about the study of bounded variation functions in the context of Dirichlet spaces.

A basic motivating geometric question for those works was: What is the good mathematical structure on a space that allows to define an intuitively reasonable  notion of  perimeter for “good” sets ? The question is of course a little vague, but deeply thinking about it from different viewpoints is certainly fruitful and yields an interesting mathematical adventure.

In our works we argue that Dirichlet spaces  provide a good framework in which we can define sets of finite perimeter and prove theorems generalizing in an elegant way classical results from the Euclidean space, like the classical isoperimetric inequality. A key guiding insight which comes from potential theory is to think of the perimeter of a set as a L^1 capacity which is concentrated on the essential boundary of this set.

 

The isoperimetric inequality in the plane

 

Dido was, according to ancient Greek and Roman sources, the founder and first Queen of Carthage (in modern-day Tunisia).

 

dido

Aeneas recounting the Trojan War to Dido, a painting by Pierre-Narcisse Guérin.

The legend says that when Dido arrived in 814BC on the coast of Tunisia, she asked for a piece of land. Her request was satisfied provided that the land could be encompassed by an ox-hide. This land became Carthage and Dido became the queen. But:

What is the shape of the land chosen by Queen Dido ?

Mathematically, we can first simplify and rephrase this problem as:

In the Euclidean plane, what is the curve which encloses the maximum area A for a given perimeter P ?

This is the isoperimetric problem in the plane. If we restrict ourselves to smooth closed curves and understand the perimeter of a set as the length of its boundary, the problem can be solved using basic calculus: the method of Lagrange multipliers. As is intuitively clear, the curve which encloses the maximum area A for a given perimeter P is the circle and one has the isoperimetric inequality

A \le \frac{1}{4 \pi} P^2.

So the city limits of Carthage formed a (half) circle.  We can still visit today the ruins of Carthage, and a small museum there  explains the isoperimetric story of Queen Dido.

 

Modern footprints

 

As mathematicians, we like to formulate problems at different levels of generality and in a form that can be generalized to different settings. To formulate the isoperimetric problem in the n-dimensional Euclidean space  \mathbb{R}^n, one needs a notion of volume of a set and a notion of perimeter of a set.  The rigorous and deep understanding of those two notions has motivated many of the spectacular developments in geometric measure theory throughout the 20th and beginning of the 21st century.

Our modern and current understanding of the notion of volume is largely based on the seminal works by H. Lebesgue (1901-1902): Nowadays,  one  understands volumes in the category of measure spaces.

The notion of perimeter is more elusive and can be understood from different viewpoints, yielding different generalizations. The Italian school in geometric measure theory played a major role in advancing the theory.

A first stepstone: R. Caccioppoli’s basic definitions, 1920’s, 1950’s

 

The first idea to define the perimeter of a Borel set E \subset \mathbb{R}^n is a very geometric and intuitive one based on the generalization of the notion of rectifiability for a curve. One defines the perimeter of E as

P(E)=\inf \left\{ \liminf_{n \to +\infty} \text{Area} (\partial E_n), \, E_n \text{ polyhedra}, \, E_n \to E \text{ in } L^1_{loc} \right\}

and say that E has a finite perimeter if the right hand side above is finite. This is a perfectly reasonable and visionary take on what quantity should the perimeter of a set actually represent. However, this definition is somehow rigid: It makes a strong use of the Euclidean structure of the space, using the notion of polyhedron and its area. Also, this definition is difficult to reconcile with the variational interpretation of a perimeter that is given by the calculus of variations (Gauss-Green formula).

A (somehow dual) definition for the perimeter of a set which coincides with the previous one is given by

P(E)=\sup \left\{ \int_{E} \mathbf{div} (\phi) dx, \, \phi \in C_0^\infty( \mathbb{R}^n,\mathbb{R}^n), \, \| \phi \|_\infty \le 1 \right\}

This definition only requires a differential structure and a Riemannian metric to define the divergence of vector fields; it can therefore be extended to any Riemannian manifold.

 

A second stepstone: E. De Giorgi, 1950’s

 

If E is a Borel set in \mathbb{R}^n, then one has

P(E)=\lim_{t \to 0} \int_{\mathbb{R}^n} \| \nabla P_t 1_E \| dx

where P_t is the so-called heat semigroup in \mathbb{R}^n which is defined for f \in L^1 by

P_t f(x)=\frac{1}{(4\pi t)^{n/2}} \int_{\mathbb{R}^n} e^{-\frac{\|y-x\|^2}{4t}} f(y) dy

This characterization of sets of finite perimeter is far reaching, convenient to work with, and may actually serve as a definition in a large class of spaces.  It only requires a heat semigroup, an object which is well defined in the category of Dirichlet spaces, and a length of gradient \| \nabla f \|, an object which is well defined in a large class of metric spaces admitting an upper gradient structure.

 

A third stepstone: H. Federer, 1960’s

 

 

Given a subset E \subset \mathbb{R}^n, define

\partial^*E =  \left\{ x \in \mathbb{R}^n, \, \limsup_{r \to 0} \min \left\{ \frac{\mathcal{L}^{n}(B_r(x) \cap E) }{ \mathcal{L}^{n}(B_r(x))},\frac{\mathcal{L}^{n}(B_r(x) \setminus E) }{ \mathcal{L}^{n}(B_r(x))} \right\} >0 \right\}

Then, P(E) can be computed as the n-1 dimensional Hausdorff measure of  \partial^*E, i.e.
P(E) =\mathcal{H}^{n-1} (\partial^* E).

This characterization of sets of finite perimeter is striking, it shows that the perimeter of a set can be understood using only a metric and a measure.

 

A fourth stepstone: M. Ledoux, 1990’s

 

If E is a Borel set in \mathbb{R}^n, then one has

P(E)=\liminf_{t \to 0} \frac{\sqrt{\pi}}{2\sqrt{t}} \int_{\mathbb{R}^n}\int_{\mathbb{R}^n} |1_E (x) -1_E(y)| p_t(x,y) dx \, dy

where p_t(x,y)= \frac{1}{(4\pi t)^{n/2}}e^{-\frac{\|y-x\|^2}{4t}} is the Euclidean heat kernel. Note that in that case, \liminf is actually also a limit, however it is part of the result that the set is of finite perimeter if and only if the \liminf is finite.

This characterization of sets of finite perimeter, which is due to M. Ledoux for balls and  in the general case to M Miranda Jr, D Pallara, F Paronetto & M Preunkert, 2007, only requires a heat semigroup. As noted above, heat semigroups are well-defined in Dirichlet spaces.

Thus, sets of finite perimeter may perfectly be well defined in any Dirichlet space and we do not need a gradient or a distance. This is the point of view we will take on the next post and will explain the associated theory. A key point is the understanding of the normalizing factor \frac{1}{\sqrt{t}} which, in the Euclidean space reflects its “smooth” structure, but which has to be modified to understand sets of finite perimeter in non-smooth spaces like the Sierpinski gasket.

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