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

Since

we obtain

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

Thus,

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

find

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

I would like to point out that stochastically complete condition is not sufficient to imply an $L^1$-Liouville theorem for (positive) harmonic functions. An example was provided by Li-Schoen. But it is sufficient to imply an $L^1$-Liouville theorem for nonnegative superharmonic functions. This was confirmed by Grigor’yan.