Positive Harmonic Functions and Diffusion

Abstract

In this book, Professor Pinsky gives a self-contained account of the theory of positive harmonic functions for second order elliptic operators, using an integrated probabilistic and analytic approach. The book begins with a treatment of the construction and basic properties of diffusion processes. This theory then serves as a vehicle for studying positive harmonic funtions. Starting with a rigorous treatment of the spectral theory of elliptic operators with nice coefficients on smooth, bounded domains, the author then develops the theory of the generalized principal eigenvalue, and the related criticality theory for elliptic operators on arbitrary domains. Martin boundary theory is considered, and the Martin boundary is explicitly calculated for several classes of operators. The book provides an array of criteria for determining whether a diffusion process is transient or recurrent. Also introduced are the theory of bounded harmonic functions, and Brownian motion on manifolds of negative curvature. Many results that form the folklore of the subject are here given a rigorous exposition, making this book a useful reference for the specialist, and an excellent guide for the graduate student.

Spectral analysis of a class of nonlocal elliptic operators related to Brownian motion with random jumps

“…By the Krein-Rutman theorem, one deduces that L γ,µ possesses a principal eigenvalue, λ 0 (γ, µ); that is, λ 0 (γ, µ) is real and simple and satisfies λ 0 (γ, µ) = inf{Re(λ) : λ ∈ σ(L γ,µ )} [14]. It is known that λ ∈ σ(L γ,µ ) if and only if exp(−λt) ∈ σ(T γ,µ t ) [12].…”

Efficient Estimation of One-Dimensional Diffusion First Passage Time Densities via Monte Carlo Simulation

“…In fact, the following result implies that the function λ x is frequently bounded on R + -this is the case, for example, when γ is bounded from below. the rate function λ x may be approximated by the rate functions corresponding to τ n 0 , n ∈ N. It follows from [5] that the density function p n x of τ n 0 has the eigenvalue expansion (see also [12] and [19,Chapter 5] for similar problems)…”

Large time behavior of Dirichlet heat kernels on unbounded domains above the graph of a bounded Lipschitz function

“…Let x 0 = 0 , b + 100 ∈ D and y n = 0 , b + n ∈ D. Then, Theorem 1.5 on page 337 in [5] implies that for any x ∈ D,…”