Level sets of multiparameter Brownian motions

We use Girsanov's theorem to establish a conjecture of Khoshnevisan, Xiao and Zhong that phi(r) = r(N-d/2) (log log(1/r))(d/2) is the exact Hausforff 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 phi-measure of the zero set.