Book Review: Topics in occupation times and Gaussian free fields
Srinivasa R. S. Varadhan · Bulletin of the American Mathematical Society · 2014
A continuous time Markov process on a finite or countable state space X that is reversible with respect to the counting measure has transition probabilities p(t, x, y) that satisfy p(t, x, y) = p(t, y, x).With a positive killing rate if the process is recurrent, the Green's function g(x, y) = ∞ 0 p(t, x, y)dt is well defined and is a symmetric positive definite function.There is a natural Gaussian measure P G , the distribution of the collection {ξ z } indexed by z ∈ X with E[ξ z ] = 0 and E[ξ x ξ y ] = g(x, y).Its probability density (with respect to Lebesgue measure on R X ) can be written as