Over the past 50 years, the body of results concerning questions of existence and characterization of positive harmonic functions for second-order elliptic operators has been nourished by two distinct sources – one rich in analysis, the other less well endowed analytically but amply compensated by generous heapings of probability theory (more specifically, martingales and stopping times). For example, the results appearing in the first seven sections of Chapter 4, which have been developed for the most part over the past decade, have been proved with nary a word about probability, while the results appearing in Chapter 6 have traditionally been formulated and proved using a probabilistic approach. On the other hand, the Martin boundary theory of Chapters 7 and 8 have long been studied by distinct probabilistic and analytic methods. My original intention was to write a monograph which would provide an integrated probabilistic and analytic approach to a host of results and ideas related, at least indirectly, to the existence and/or characterization of positive harmonic functions. When the undertaking was still in its inchoate stages, it became apparent that such a monograph, if executed appropriately, might serve as a graduate text for students working in diffusion processes. This direction also seemed appealing. Indeed, too numerous have been the occasions on which I explained a result or a ‘meta-result’ to a student or colleague and then found myself at a loss when it came to suggesting a reference text. I hope this book might ground some of the folklore. In the end, then, the book has been written with two intentions in mind.

I have endeavoured to keep the book as self-contained as the dictates of good taste permit.