Positive harmonic functions and diffusion : An integrated analytic and probabilistic approach

where aij , bi, V are nice functions in R (say Lipschitz). Fixing a domain D ⊂ R we denote by CL(D) the class of all positive solutions of the equation Lu = 0 in D. Properties of CL(D) may be studied from an analytic point of view, a probabilistic one using diffusions, or with techniques from both fields combined. The point of the book under review is to give an exposition of this last, integrated approach to the subject. The analytic approach has a longer tradition. For example, the maximum principle dates back to Gauss in 1838. The probabilistic approach is a product of this century. Undoubtedly, credit for the foundational step should go to Wiener [19]. This step is the construction of the measure P on the path space Ω = {ω ∈ C([0,∞),Rd) : ω(0) = 0} representing the trajectories of particles undergoing Brownian motion. The function W : [0,∞) × Ω → R defined by Wt(ω) = ω(t) has the distributional property that under P the random variables, Wt1 , Wt2−Wt1 , . . . ,Wtn−Wtn−1 are independent, Gaussian mean 0, with variances t1, t2− t1, . . . , tn− tn−1. It was known almost from the beginning that P gives mass one to paths which are nowhere differentiable. The next, audacious step is due to Itô [10], who gave a representation of the diffusion naturally associated to the operator L0 = L−V . The association between the diffusion X and operator L0 is captured by the requirement that as h→ 0,