In this Lecture, we will prove a first interesting consequence of the Bochner’s identity: We will prove that if, on a complete Riemannian manifold , the Ricci curvature is bounded from below, then the heat semigroup is stochastically complete, that is . This result is due to S.T. Yau, and we will see this property is also equivalent to the uniqueness in for solutions of the heat equation. The proof we give is due to D. Bakry.
Let be a complete Riemannian manifold and denote by its Laplace-Beltrami operator. As usual, we denote by the heat semigroup generated by . Throughout the Lecture, we will assume that the Ricci curvature of is bounded from below by . As seen in the previous Lecture, this is equivalent to the fact that for every ,
We start with a technical lemma:
Lemma: If , then for every , the functions and are in .
Proof: It is straightforward to see from the spectral theorem that . Similarly, . Since, , we are let with the problem of proving that . If , then an integration by parts easily yields As a consequence,
and we obtain
Using a density argument, it is then easily proved that for we have
In particular, we deduce that if , then
We will also need the following fundamental parabolic comparison theorem that shall be extensively used throughout these lectures.
Proposition: Let . Let be smooth functions such that:
- For every , and ;
- for some ;
- For every , and for some .
If the inequality
holds on , then we have
Proof: Let , . We claim that we must have
where for every and a measurable , we have let . To establish this, we consider the function
Differentiating we find
Now, we can bound
and for a.e. the integral in the right-hand side is finite. We have thus obtained
As a consequence, we find
which proves what we claimed.
Let now be a sequence such that , and increases to 1.
Using in place of and letting , gives
We observe that the assumption on and Minkowski’s integral inequality guarantee that the function belongs to . We have in fact
Since this must hold for every non negative , we conclude that
which completes the proof
We are in position to prove the first gradient bound for the semigroup .
Proposition: If is a smooth function in , then for every and ,
Proof: We fix and consider the functional
We first assume that on . From the previous lemma, we have . Moreover . So, we have . Therefore, again from the previous proposition , we deduce that . Next, we easily compute that
We can then use the parabolic comparison theorem to infer that
If vanishes on , we consider the functional
where, for ,
Since , an argument similar to that above (details are let to the reader) shows that
Letting , we conclude that
We now prove the promised stochastic completeness result:
Theorem: For , one has .
Proof: Let , we have
By means of the previous Proposition and Cauchy-Schwarz inequality, we
We now apply the previous inequality with , and then let .
Since by Beppo Levi’s monotone convergence theorem we have for every , we see that the left-hand side converges to . We thus reach the conclusion
It follows that
A consequence of the stochastic completeness is the uniqueness in of solutions of the heat equation. More precisely, the following parabolic comparison theorem holds.
Proposition: Let . Let be smooth functions such that for every , , ; If the inequality
holds on , then we have
Proof: Let be the diffusion Markov process with semigroup and started at . From , we deduce that has an infinite lifetime. We have then for ,
where is a local martingale. From the assumption one obtains
Let now be an increasing sequence of stopping times such that almost surely and is a martingale.
From the previous inequality, we find
By using the dominated convergence theorem, we conclude
which yields the conclusion