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.
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,
- 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
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
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 .