The set of intersections of several independent Brownian motions on Carnot group

Abstract

In this paper the existence of intersections for functions of several Brownian motions on the Carnot group is studied. A condition is presented for the existence of such intersections with Probability 1, which is in the form of the asymptotics of a measure on a specific family of small balls. The measure is arbitrary but can be chosen as a surface measure on the manifold related to intersections, and the balls are constructed using the distances related to the processes. Additionally, if the same condition holds in a weaker form, it is shown that there is a Hausdorff measure, such that the value of this Hausdorff measure on the set of intersection points is finite with Probability 1.

Keywords:

• Brownian motion on Lie group
• intersection of independent random processes
• hitting probabilities of Markov processes

2020 Mathematics Subject Classifications:

• 60J25
• 60J45

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by a grant from the Simons Foundation [grant number 1030291, O.V.R.].