Level sets of multiparameter brownian motions

Abstract

We use Girsanov’s theorem to establish a conjecture of Khoshnevisan, Xiao and Zhong that ϕ(r)=rN−d/2(log log(1/r))d/2is the exact Hausdorff measure function for the zero level set of an N-parameter d-dimensional additive Brownian motion. We extend this result to a natural multiparameter version of Taylor and Wendel’s theorem on the relationship between Brownian local time and the Hausdorff ϕ-measure of the zero set. © 2004 Applied Probability Trust.