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.