Self-intersections of random fields. (English) Zbl 0536.60066

Rosen, Jay

Ann. Probab. 12, 108-119 (1984).

This paper gives Hausdorff dimension results for the r-multiple times of Brownian motions, Brownian sheets and the general multiparametric Gaussian process \(W^{\beta}(t)\) with mean zero and \(E(W^{\beta}(s)- W^{\beta}(t))^ 2=| s-t|^{2\beta}\), \(0<\beta<1\). The main result is that for index \(\beta\) processes from \(R^ N\to R^ d\), the Hausdorff dimension of the r-multiple points is Nr-\(\beta\) d(r-1) when this quantity is positive.

Reviewer: J.Cuzick

MSC:

60G60	Random fields
60G17	Sample path properties

Keywords:

local time; self intersection; Hausdorff dimension; Brownian sheets; multiple points