Kolmogorov's Theorem for path continuity

Kolmogorov's Theorem gives a simple condition for Hölder continn a complete separable metric space. Since we only use it in Chapter 2 for processes on Rd, we simplify matters and consider only processes on [0, 1]d. This result is interesting from a historical perspective since it contains the germs of the much deeper continuity conditions obtained in Chapter 5 (actually, in Chapter 5, we only consider Gaussian processes, but the methods developed have a far larger scope, as is shown in Ledoux and Talagrand (1991)).

Let Dm be the set of d-dimensional vectors in [0, 1]d with components of the form i/2m for some integer i ∈ [0, 2m]. Let D denote the set of dyadic numbers in [0, 1]d, that is,.

Theorem 14.1.1Let X = {Xt, t ∈ [0, 1]d} be a stochastic process satisfying

for constants c, r > 0 and h ≥ 1. Then, for any α < r/h,

for some random variable C(ω) < ∞ almost surely.

Proof Let Nm be the set of nearest neighbors in Dm. This is the set of pairs s, t ∈ Dm with |s − t| = 2−m. Using (14.1) and the fact that |Nm| ≥ 2d2dm, we have

For any s ∈ D, let sm be the vector in Dm with sm ≤ s that is closest to s. Then, for any m, we have that either sm = sm+1 or else the pair {sm, sm+1} ∈ Nm+1. Now let s, t ∈ D with |s − t| ≤ 2−m.

We found it difficult to choose a title for this book. Clearly we are not covering the theory of Markov processes, Gaussian processes, and local times in one volume. A more descriptive title would have been “A Study of the Local Times of Strongly Symmetric Markov Processes Employing Isomorphisms That Relate Them to Certain Associated Gaussian Processes.” The innovation here is that we can use the well-developed theory of Gaussian processes to obtain new results about local times.

Even with the more restricted title there is a lot of material to cover. Since we want this book to be accessible to advanced graduate students, we try to provided a self-contained development of the Markov process theory that we require. Next, since the crux of our approach is that we can use sophisticated results about the sample path properties of Gaussian processes to obtain similar sample path properties of the associated local times, we need to present this aspect of the theory of Gaussian processes. Furthermore, interesting questions about local times lead us to focus on some properties of Gaussian processes that are not usually featured in standard texts, such as processes with spectral densities or those that have infinitely divisible squares. Occasionally, as in the study of the p-variation of sample paths, we obtain new results about Gaussian processes.

Our third concern is to present the wonderful, mysterious isomorphism theorems that relate the local times of strongly symmetric Markov processes to associated mean zero Gaussian processes.

In Section 7.4 we develop and exploit some special properties of Gaussian processes that are associated with Borel right processes. In this chapter we consider the question of characterizing associated Gaussian processes. In order to present these results in their proper generality, we must leave the familiar framework of Borel right processes and consider local Borel right process, which are introduced in the final sections of Chapter 4. The reader should note that this is the first place in this book, after Chapter 4, that we mention local Borel right processes. We remind the reader that Borel right processes are local Borel right processes, and for compact state spaces, there is no difference between local Borel right processes and Borel right processes.

Let S be a locally compact space with a countable base. A Gaussian process {Gx ; x ∈ S} is said to be associated with a strongly symmetric transient local Borel right process X on S, with reference measure m, if the covariance Γ = Γ(x, y) = E(GxGy) is the 0-potential density of X for all x, y ∈ S. Not all Gaussian processes are associated. It is remarkable that some very elementary observations about the 0-potential density of a strongly symmetric transient Borel right process show what is special about associated Gaussian processes.

One obvious condition is that Γ(x, y) ≥ 0 for all x, y ∈ S, since the 0-potential density of a strongly symmetric transient Borel right process is nonnegative (see Remark (3.3.5)).

So far in this book, we have simply assumed that we are given a strongly symmetric Borel right process with continuous α-potential densities uα(x, y), α > 0 and also u(x, y) when the 0-potential exists. In general, constructing such processes is not trivial. However, given additional conditions on transition semigroups or potentials, we can construct special classes of Borel right processes. In this chapter we show how to construct Feller and Lévy processes. (For references to the general question of establishing the existence of Borel right processes, see Section 3.11.) In Sections 4.7–4.8, we show how to construct certain strongly symmetric right continuous processes with continuous α-potential densities that generalize the notion of Borel right processes and are used in Chapter 13.

In Sections 4.4–4.5 we present certain material, on quasi left continuity and killing at a terminal time, which is of interest in its own right and is needed for Sections 4.7–4.10. In Section 4.6 we tie up a loose end by showing that if a strongly symmetric Borel right process has a jointly continuous local time, then the potential densities {uα(x, y), (x, y) ∈ S × S} are continuous.

In Section 4.10 we present an extension theorem of general interest which is needed for Chapter 13.

Feller processes

A Feller process is a Borel right process with transition semigroup {Pt; t ≥ 0} such that, for each t ≥ 0, Pt : C0(S) ↦ C0(S). Such a semigroup is called a Feller semigroup. We consider C0(S) as a Banach space in the uniform or sup norm, that is, ∥f∥ = supx ∈ S |f(x)|.