Conditional Distribution of Independent Brownian Motions to Event of Coalescing Paths

Vitalii Konarovskyi, Victor Marx

Published 2020-08-06Version 1

We consider a family of independent real-valued Brownian motions starting at distinct points and are interested in the description of the conditional distribution of this family to the event that trajectories coalesce, which is of probability zero. We develop a general approach to construct a conditional probability to events of measure zero and apply it first when the above-mentioned family is finite and then when the family is infinite. In both cases, it leads us to the distribution of a modified massive Arratia flow [arXiv:1408.0628] as the conditional distribution.