The purpose of this chapter is to show how one can transfer the results obtained earlier, particularly those in Chapters 3 and 4, from Euclidean space to a differentiable manifold.

As we have already seen in the Euclidean setting, it is important to distinguish between local (short time) and global (long time) aspects of the theory. When dealing with differentiable manifolds, this distinction becomes even more important. Indeed, by definition, all smooth manifolds are locally Euclidean, and so one should expect that, aside from a few technical details, there is no problem about transferring the local theory. When the manifold is compact, global theory is relatively easy. Namely, because it has nowhere to spread, the heat flow quickly equilibrates and so the fundamental solution to a non-degenerate (i.e., a is elliptic) Kolmogorov equation tends rapidly to its stationary state. On the other hand, when the manifold is non-compact, the global theory reflects the geometric growth properties of the particular manifold under consideration, and so the Euclidean case cannot be used to predict long time behavior.

For the reason just given, we will restrict our attention to the local theory. In fact, in order to avoid annoying questions about possible “explosion,” we will restrict our attention to compact manifolds, a restriction which, for the local theory, is convenient but inessential.

Up until now I have assiduously avoided the use of many of the modern analytic techniques which have become essential tools for experts working in partial differential equations. In particular, nearly all my reasoning has been based on the minimum principle, and I have made no use so far of either Sobolev spaces or the theory of pseudodifferential operators. In this concluding chapter, I will attempt to correct this omission by first giving a brief review of the basic theories of Sobolev spaces and pseudodifferential operators and then applying them to derive significant extensions of the sort of hypoellipticity results proved in §3.4.

Because the approach taken in this chapter is such a dramatic departure from what has come before, it may be helpful to explain the origins of the analysis which follows and of the goals toward which it is directed. For this purpose, consider the Laplace operator Δ for ℝN, and ask yourself what you can say about u on the basis of information about Δu. In particular, what can you say about ∂i∂ju? One of the most bedeviling facts with which one has to contend is that (except in one dimension) u need not be twice continuous differentiable just because Δu ∈ C(ℝN;ℂ) in the sense of Schwartz distributions. On the other hand, if one replaces continuity by integrability, this uncomfortable fact disappears.

Thus far, all our results have been about parabolic equations in the whole of Euclidean space, and, particularly in Chapter 4, we took consistent advantage of that fact. However, in many applications it is important to have localized versions of these results, and the purpose of this chapter is to develop some of them.

Because probability theory provides an elegant and ubiquitous localization procedure, we will begin by summarizing a few of the well-known facts about the Markov process determined by an operator L. We will then use that process to obtain a very useful perturbation formula, known as Duhamel's formal. Armed with Duhamel's formula, it will be relatively easy to get localized statements of the global results which we already have, and we will then apply these to prove Nash's Continuity Theorem and the Harnack principle of Di Georgi and Moser.

Diffusion Processes on ℝN

Throughout, we will be assuming that a and b are smooth functions with bounded derivatives of all orders and that a ≥ ∊I for some ∊ > 0. Given such a and b, L will be one of the associated operators given by (1.1.8), (4.4.1), or, when appropriate (4.3.1). Of course, under the hypotheses made about a and b, the choice between using (1.1.8) or (4.4.1) is a simple matter of notation, whereas the ability to write it as in (4.3.1) imposes special conditions of the relationship between b and a.

There are few benefits to growing old, especially if you are a mathematician. However, one of them is that, over the course of time, you accumulate a certain amount of baggage containing information in which, if you are lucky and they are polite, your younger colleagues may express some interest.

Having spent most of my career at the interface between probability and partial differential equations, it is hardly surprising that this is the item in my baggage about which I am asked most often. When I was a student, probabilists were still smitten by the abstract theory of Markov processes which grew out of the beautiful work of G. Hunt, E.B. Dynkin, R.M. Blumenthal, R.K. Getoor, P.A. Meyer, and a host of others. However, as time passed, it became increasingly apparent that the abstract theory would languish if it were not fed a steady diet of hard, analytic facts. As A.N. Kolmogorov showed a long time ago, ultimately partial differential equations are the engine which drives the machinery of Markov processes. Until you solve those equations, the abstract theory remains a collection of “if, then” statements waiting for someone to verify that they are not vacuous.

Unfortunately for probabilists, the verification usually involves ideas and techniques which they find unpalatable. The strength of probability theory is that it deals with probability measures, but this is also its weakness.