I am currently looking for a postdoc starting in August 2017 in the Mathematics department at the University of Connecticut.

The position is described here:

Please apply through mathjobs.org and contact me for further details.

I am currently looking for a postdoc starting in August 2017 in the Mathematics department at the University of Connecticut.

The position is described here:

Please apply through mathjobs.org and contact me for further details.

Posted in Uncategorized
Leave a comment

**Exercise 1. ***Let be a locally subelliptic and essentially self-adjoint diffusion operator. Let be the semigroup generated by . By using the maximum principle for parabolic pdes, prove that if is in , then .*

**Exercise 2: ***Let be an essentially self-adjoint diffusion operator. Denote by the semigroup generated by in .*

*Show that for each , the -valued map is continuous.**Show that for each , , the -valued map is continuous.**Finally, by using the reflexivity of , show that for each and every , the -valued map is continuous.*

Posted in Uncategorized
Leave a comment

In the next few lectures, we will show that the diffusion semigroups theory we developed may actually be extended without difficulties to a manifold setting. As a motivation, we start with a very simple example.

We first study the heat semigroup on the simplest (non Euclidean) Riemannian manifold: the circle The Laplace operator on , is the canonical diffusion operator on . A natural question to be asked is: in the same way, is there a canonical diffusion operator on . A first step, of course, is to understand what is a diffusion operator on . We characterized diffusion operators as linear operators on the space of smooth functions that satisfy the maximum principle. Once a notion of smooth functions on is understood, this maximum principle property can be taken as a definition. The circle may be identified with the quotient space . More precisely, it is easily shown that a smooth function, which is periodic, i.e. can be written as for some function . Conversely, any function defines a periodic function on by setting So, with this in mind, we simply say that is a smooth function if is. With this identification between the set of smooth periodic functions on and the set of smooth functions on , it then immediate that the canonical diffusion operator on should write, The corresponding diffusion semigroup is also easily computed from the heat semigroup on . Indeed, a natural computation leads to

,

where . This allows to define the heat semigroup on as the family of operators defined by The natural domain of this operator is where is the measure on which is defined through the property The reader may then check the following properties for this semigroup of operators:

- (Semigroup property) ;
- (Strong continuity) The map is continuous for the operator norm on ;
- (Contraction property) ;
- (Self-adjointness) For ,
- (Markov property) If is such that , then .

**Exercise.**

- Prove the Poisson summation formula: If is a smooth and rapidly decreasing function, then
- Deduce that the heat kernel on may also be written

**Exercise.*** From the previous exercise, the heat kernel on is given by .*

- By using the subordination identity show that for ,
- The Bernoulli numbers are defined via the series expansion By using the previous identity show that for , ,

**Exercise.*** Show that the heat kernel on the torus is given by *

Posted in Diffusions on manifolds
Leave a comment

In the previous lectures, we have seen that if L is an essentially self-adjoint diffusion operator with respect to a measure, then by using the spectral theorem one can define a self-adjoint strongly continuous contraction semigroup on L2 with generator L. This semigroup is moreover positivity preserving and a contraction on the space of bounded square integrable functions. Our goal, in this lecture, is to define, for , the semigroup on . This can be done by using the Riesz–Thorin interpolation theorem that we remind below. In this subsection, in order to simplify the notations we simply denote by . We first start with general comments about semigroups in Banach spaces.

Let be a Banach space (which for us will be , ).

We first have the following basic definition.

**Definition:** *A family of bounded operators on is called a contraction semigroup if:*

- and for , ;
- For each and , .

A contraction semigroup on is moreover said to be strongly continuous if for each , the map is continuous.

In this Lecture, we will prove the following result:

**Theorem:*** let be an essentially self-adjoint diffusion operator. Denote by the self-adjoint strongly continuous semigroup associated to and constructed on thanks to the spectral theorem. Let . On , there exists a unique contraction semigroup such that for , . The semigroup is moreover strongly continuous for . *

This theorem can be proved by using two sets of methods: Hille-Yosida theory on one hand and interpolation theory on the other hand. We shall use interpolation theory in space which takes advantage of the fact that only needs to be constructed on and . The Hille-Yosida method starts from the generator and we sketch it below.

**Definition:**

Let be a strongly continuous contraction semigroup on a Banach space . There exists a closed and densely defined operator where such that for , The operator is called the generator of the semigroup . We also say that generates .

The following important theorem is due to Hille and Yosida and provides, through spectral properties, a characterization of closed operators that are generators of contraction semigroups.

Let be a densely defined closed operator. A constant is said to be in the spectrum of if the the operator is not bijective. In that case, it is a consequence of the closed graph theorem that if is not in the spectrum of , then the operator has a bounded inverse. The spectrum of an operator shall be denoted .

**Theorem:*** A necessary and sufficient condition that a densely defined closed operator $A$ generates a strongly continuous contraction semigroup is that:*

- ;
- for all .

These two conditions are unfortunately difficult to directly check for diffusion operators.

We can bypass the study of the closure in of a diffusion operator by using interpolation theory.

**Theorem: (Riesz-Thorin interpolation theorem)** *Let , and . Define by If is a linear map such that and then, for every , Hence extends uniquely as a bounded map from to with
*

The statement that is a linear map such that and means that there exists a map with and In such a case, can be uniquely extended to bounded linear maps , . With a slight abuse of notation, these two maps are both denoted by in the theorem.

The proof of the theorem can be found in this post by Tao.

One of the (numerous) beautiful applications of the Riesz-Thorin theorem is to construct diffusion semigroups on by interpolation. More precisely, let be an essentially self-adjoint diffusion operator. We denote by the self-adjoint strongly continuous semigroup associated to constructed on thanks to the spectral theorem. We recall that satisfies the submarkov property: That is, if is a function in , then .

**Theorem:** *The space is invariant under and may be extended from to a contraction semigroup on for all : For , These semigroups are consistent in the sense that for , *

**Proof:** If which is a subset of , then This implies The conclusion follows then from the Riesz-Thorin interpolation theorem

**Exercise:** *Show that if and with then, *

**Exercise:**

- Show that for each , the -valued map is continuous.
- Show that for each , , the -valued map is continuous.
- Finally, by using the reflexivity of , show that for each and every , the -valued map is continuous.

We mention, that in general, the valued map is not continuous.

Due to the consistency property, we always remove the subscript from and only use the notation .

To finish this Lecture, we finally connect the heat semigroup in to solutions of the heat equation.

**Proposition:*** Let , , and let Then, if is elliptic with smooth coefficients, is smooth on and is a strong solution of the Cauchy problem
*

**Proof:** The proof is identical to the case. For , we have

Therefore is a weak solution of the equation . Since is smooth it is also a strong solution .

We now address the uniqueness of solutions. As in the case, we assume that is elliptic with smooth coefficients and that there is a sequence , , such that on , and , as .

**Proposition:*** Let be a non negative function such that and such that for every , , where . Then .*

**Proof:** Let and . Since is a subsolution with the zero initial data, for any ,

On the other hand, integrating by parts yields

Observing that

we obtain the following estimate.

Combining with the previous conclusion we obtain ,

By using the previous inequality with an increasing sequence , , such that on , and , as , and letting , we obtain

thus .

As a consequence of this result, any solution in , of the heat equation is uniquely determined by its initial condition, and is therefore of the form . We stress that without further conditions, this result fails when or .

Posted in Diffusions on manifolds
Leave a comment

**Exercise:*** Let be an elliptic diffusion operator with smooth coefficients that is essentially self-adjoint with respect to a measure . Let be a multi-index. If is a compact set of , show that there exists a positive constant such that for , where is the smallest integer larger than *

Posted in Uncategorized
Leave a comment

In the previous lectures, we have proved that if L is a diffusion operator that is essentially self-adjoint then, by using the spectral theorem, we can define a self-adjoint strongly continuous contraction semigroup with generator L and this semigroup is unique. A remarkable property of the semigroup is that it preserves the positivity of functions.

More precisely, we are going to prove that if is am essentially self-adjoint operator with respect to a measure then, by denoting the semigroup generated by : If satisfies , then , . This property is called the submarkov property of the semigroup . The terminology stems from the connection with probability theory where is interpreted as the transition semigroup of a sub-Markov process.

As a first step, we prove the positivity preserving property, which is a consequence of the following functional inequality satisfied by diffusion operators:

**Lemma:*** (Kato’s inequality for diffusion operators). Let be a diffusion operator on which is symmetric with respect to a measure . Let . Define and In the sense of distributions, we have the following inequality *

**Proof:** If is a smooth and convex function and if is assumed to be smooth, it is readily checked that By choosing for the function , we deduce that for every smooth function , As a consequence this inequality holds in the sense of distributions, that is for every , , . Letting gives the expected result

We are now in position to state and prove the positivity preserving theorem.

**Proposition:*** Let be an essentially self-adjoint diffusion operator on . If is almost surely nonnegative , then we have for every , almost surely.*

**Proof:** The main idea is to prove that for , the resolvent operator which is well defined due to essential self adjointness preserves the positivity of function. Then, we may conclude by the fact that, as it is seen from spectral theorem, .

We first extend Kato’s inequality to a larger class of functions.

Let . We consider on the norm and denote by the completion of . Our goal will be to show that the Kato’s inequality is also satisfied for . As in the proof of Kato’s inequality, we first consider smooth approximations of the absolute value. For we introduce the function It is easily seen that for , . Let now be a Cauchy sequence for the norm . We claim that the sequence is also a Cauchy sequence. Indeed, since we have, on one hand Now, keeping in mind that is a nonnegative bilinear form and thus satisfies Cauchy-Schwarz-inequality, we have on the other hand

As a consequence, is a Cauchy sequence and thus converges toward an element of . If denotes the limit of in , the limit in of is . As a conclusion, if then .

From the proof of Kato’s inequality, if then for every , , . This may be rewritten as

Let and . We consider a sequence such that for the norm . In particular for the norm , so that by passing to a subsequence we can suppose that pointwise almost surely. Applying the inequality to and letting leads to the conclusion that the inequality also holds for and .

Finally, by using the same type of arguments as above, it is shown for , when , tends to in the norm . Thus, if , .

As a consequence of all this, if , , and moreover for ,

And this last inequality is easily extended to by density of in for the norm . In particular when we get that for , Again by density, this inequality can be extended to every . Since is essentially self-adjoint we can consider the bounded operator that goes from to . For and with , we have

Moreover, from the previous inequality, for ,

By taking in the two above sets of inequalities, we draw the conclusion

The above inequalities are therefore equalities which implies As a conclusion if is , then for every , . Thanks to spectral theorem, in , By passing to a subsequence that converges pointwise almost surely, we deduce that almost surely

**Exercise:** *Let be an elliptic diffusion operator with smooth coefficients that is essentially self-adjoint. Denote by the heat kernel of . Show that . (Remark: It actually possible to prove that ). *

Besides the positivity preserving property, the semigroup is a contraction on . More precisely,

**Proposition:** *Let be an essentially self adjoint diffusion operator on . If , then and *

**Proof:** The proof is close and relies on the same ideas as the proof of the positivity preserving property. So, we only list below the main steps and let the reader fills the details.

As before, for , we consider on the norm and denote by the completion of .

The first step is to show that if , then (minimum between and ) also belongs to and moreover

Let satisfy and put and . According to the first step, and . Now, we observe that:

As a consequence , that is .

The previous step shows that if satisfies then for every , . As in the previous proposition, we deduce that almost surely

Posted in Uncategorized
Leave a comment

In this Lecture, we use the local regularity theory of subelliptic operators, to prove the existence of heat kernels.

**Proposition:*** Let be a locally subelliptic diffusion operator with smooth coefficients that is essentially self-adjoint with respect to a measure . Denote by the corresponding semigroup on .*

- If is a compact set of , there exists a positive constant such that for , where is the smallest integer larger than and .
- For , the function is smooth on .

**Proof:** Let us first observe that from the spectral theorem that if then for every , and Now, let be a compact set of . From the proposition in, there exists therefore a positive constant such that Since it is immediately checked that the bound easily follows. We now turn to the second part. Let . First, we fix . As above, from the spectral theorem, for every , , for any bounded open set . By hypoellipticity of , we deduce therefore that is a smooth function.

Next, we prove joint continuity in the variables . It is enough to prove that if and if is a compact set in , From the previous proposition, there exists a positive constant such that Now, again from the spectral theorem, it is checked that This gives the expected joint continuity in . The joint smoothness in is a consequence of the second part of the previous proposition and the details are let to the reader

**Remark:*** If the bound uniformly holds on , that is if then the semigroup is said to be ultracontractive.*

**Exercise:*** Let be an elliptic diffusion operator with smooth coefficients that is essentially self-adjoint with respect to a measure . Let be a multi-index. If is a compact set of , show that there exists a positive constant such that for , where is the smallest integer larger than .*

We are now in position to prove the following fundamental theorem:

**Theorem:*** Let be a locally subelliptic and essentially self-adjoint diffusion operator. Denote by the corresponding semigroup on . There is a smooth function , , such that for every and , The function is called the heat kernel associated to . It satisfies furthermore:*

- (Symmetry) ;
- (Chapman-Kolmogorov relation) .

**Proof:** Let and . From the previous proposition, the linear form is continuous on , therefore from the Riesz representation theorem, there is a function , such that for , From the fact that is self-adjoint on , we easily deduce the symmetry property And the Chapman-Kolmogorov relation stems from the semigroup property . Finally, from the previous proposition the map is smooth on for the weak topology of . This implies that it is also smooth on for the norm topology. Since, from the Chapman-Kolmogorov relation we conclude that is smooth on

Posted in Diffusions on manifolds
Leave a comment