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.