Probabilistic Potential Theory

Abstract

Consider a random subset K of ℝ ^d . A basic problem in probabilistic potential theory is the following: For what nonrandom sets E is ℙ(K ∩ E ≠ ø) positive ? The archetypal example of such a set K is the range of a random field. Let X = (X _t ; t ∈ ℝ ₊ ^N ) denote an N-parameter stochastic process that takes its values in ℝ ^d and consider the random set K = {X _s : s ∈ ℝ ₊ ^N }.¹ For this particular random set K, the above question translates to the following: When does the random function X ever enter a given nonrandom set E with positive probability? Even though we will study a large class of random fields in the next chapter, the solution to the above problem is sufficiently involved that it is best to start with the easiest one-parameter case, which is the subject of the present chapter. Even in this simpler one-parameter setting, it is not clear, a priori, why such problems are interesting. Thus, our starting point will be the analysis of recurrence phenomena for one-parameter Markov processes that have nice properties. To illustrate the key ideas without having to deal with too many technical issues, our discussion of recurrence concentrates on Lévy processes. The astute reader may recognize Section 1 below as the continuous-time analogue of the results of the first section of Chapter 3.