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.

Introduction

In Chapter 3, we developed the spectral theory of elliptic operators L on smooth bounded domains D, concentrating in particular on the principal eigenvalue λ0. This theory hinged on the fact that L possessed a compact resolvent, since only then could the Fredholm theory be applied. In the case of an arbitrary domain D, L does not possess a compact resolvent in general, and the above spectral theory breaks down. Indeed, it is no longer even clear on what space to define L or on what space to define the corresponding semigroup Tt. In this chapter, we develop a generalized spectral theory for elliptic operators L on arbitrary domains. Specifically, we will extend the definition of the principal eigenvalue λ0 via the existence or non-existence of positive harmonic functions for L – λ in D, that is, functions u satisfying (L – λ)u = 0 and u > 0 in D.

In this chapter, we will assume that L satisfies Assumption locally:

Assumptionis defined on a domain D ⊆ Rdand satisfies Assumption (defined in Chapter 3, Section 7) on every subdomain D' ⊂⊂ D.

(In Exercise 4.16, the reader is asked to check that all the theorems in this chapter which involve only L and not hold if L satisfies Assumption H (defined in Chapter 3, Section 2) locally.)

An indispensable tool in this chapter is Harnack's inequality, which we state here for operators in non-divergence form since the formulation is simpler in this case. See the notes at the end of the chapter for references.

Introduction

In this chapter, we will assume that V ≡ 0. In Chapter 4, it was shown that in this case, criticality and subcriticality are equivalent to recurrence and transience, respectively, and that, in the critical (recurrent) case, the product L1 property is equivalent to positive recurrence. It was also shown that the function Φ ≡ 1 is not invariant for the transition measure p(t, x, dy) if and only if the diffusion explodes. In Chapter 5, integral tests were given which allow one to distinguish between transience and recurrence, between null recurrence and positive recurrence, and between explosion and non-explosion in the one-dimensional case. The multidimensional case is much more involved and does not yield a simple solution. In Section 1–3, we develop the Liapunov technique for considerations of transience and recurrence. For diffusions with appropriate coefficients, this technique is used to give sufficiency conditions for transience, for recurrence, for null recurrence and for positive recurrence in terms of integral tests. In Section 4, we present a variational approach to transience and recurrence for diffusions corresponding to symmetric operators. By exploiting the classical Dirichlet principle, a necessary and sufficient condition for transience or recurrence is given in terms of a variational formula. From this, one can extract sufficiency conditions for transience and for recurrence in terms of integral tests. The variational method also reveals an important comparison principle concerning transience and recurrence for such diffusions. In Section 5, we develop a generalized Dirichlet principle for general operators which is then used in Section 6 to give a necessary and sufficient condition for transience or recurrence in terms of a mini-max variational formula.