Level Sets of Multiparameter Brownian Motions

Eulalia Nualart,Thomas Mountford

doi:10.1214/ejp.v9-169

Abstract

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

Highlights

Introduction and resultsLet X : IRN → IRd be a multiparameter additive Brownian motion, that is, X has the following decomposition NX(t) = Xi(ti), t = (t1, ..., tN ) ∈ IRN, i=1 where the Xi are independent, two sided d-dimensional Brownian motions
The aim of this paper is to establish a conjecture of Khoshnevisan, Xiao and Zhong, c.f. [5, Problem 6.3], that if 2N > d, for any bounded interval I ⊂ IRN, mφ({t : X(t) = 0} ∩ I) < ∞ a.s., where φ(r)
In one dimension the first result of this kind was due to Taylor and Wendel [13], who showed that if X is a one-dimensional Brownian motion, there exists a positive finite constant c such that mφ({s : X(s) = 0, s ≤ t}) = c L(t) a.s., for all t > 0, where L is the local time at zero of X, that is, L(t)

Summary

Introduction and results

Let X : IRN → IRd be a multiparameter additive Brownian motion, that is, X has the following decomposition. In one dimension the first result of this kind was due to Taylor and Wendel [13], who showed that if X is a one-dimensional Brownian motion, there exists a positive finite constant c such that mφ({s : X(s) = 0, s ≤ t}) = c L(t) a.s., for all t > 0, where L is the local time at zero of X, that is, L(t) lim. X an additive Brownian motion, there exists a positive finite constant K such that on any interval I ⊂ IRN mφ({t : X(t) = 0} ∩ I) ≤ K L(I) a.s. in Section 5 we arrive at the proof of Theorem 1.1. For v a vector in IRd and Σ a positive definite d × d matrix, N (v, Σ) will denote the corresponding Gaussian distribution

Local times for Ornstein-Uhlenbeck processes

Girsanov’s theorem

The Hausdorff measure function