Coalescing systems of non-Brownian particles (Q1955837)

The authors consider the behaviour of the set valued processes given by starting an infinite set of Feller processes at time 0 and merging pairs as soon as they are equal. The key question is whether the set of positions becomes instantaneously almost surely locally finite (comes down from infinity locally). A positive answer for Brownian motions started at all points of a closed subset of the real line is due to Arratia; this paper provides a number of extensions. Sufficient conditions for a positive answer for Feller processes on general compact metric spaces are given. For the infinite Sierpinski Gasket a positive answer is provided for Brownian motions. On the real line and unit circle a sufficient condition is shown to be that the process are stable with index strictly greater than one. Concerning non compact position spaces the authors comment ``elements of our argument seem rather specific'' to either the Sierpinski Gasket or the Real line and not readily generalisable. Almost half of the paper is devoted to the Sierpinski Gasket case, especially to showing that the probability of two Brownian motions meeting converges to 1 as their starting positions converge. The reader interested in Feller processes but only on less exotic spaces can skip the Sierpinski Gasket material for a concise account.