Measure of the multiple self-intersection set of a markov process
Authors:
Simeon M. Berman
Abstract:
Let X(t), t ≧ 0, be a Markov process in Rm with homogeneous transition density p(t; x, y). For a closed bounded set B ? Rm, X is said to have a self-intersection of order r ≧ 2 in B if there are distinct points t1 < … < tr such that X(t1) ∈ B and X(tj) = X(t1), for j = 2,…, r. The focus of this work is the Hausdorff measure, suitably defined, of the set of such r-tuples. The main result is that under general conditions on p(t; x, y) as well as the specific condition there is a measure function M(t), defined in terms of the integral above, such that the corresponding Hausdorff measure of self-intersection set is positive, with positive probability. The results are applied to Lévy and diffusion processes, and are shown to extend recent results in this area.