TALK by P. Fitzsimmons
Souther California Probability Symposium
December 3, 2005
Intersections and Infinite Divisibility of Regenerative Sets
Pat Fitzsimmons (UCSD)
A regenerative set is the set of times when a strong Markov process visits a fixed
state. By an obvious construction, the intersection of two independent regenerative sets
is itself a regenerative set. Less obvious is the manner in which the "characteristics" of the factor sets combine
to form those of the intersection. A solution of this "intersection problem",
under a mild condition, has been found by J. Bertoin. The general case remains open, and we offer
a conjecture.
A regenerative set is infinitely divisible if for each positive integer n
it has the same distribution as the intersection of n iid regenerative sets.
(Example: the zero set of Brownian motion.) A characterization of the class of
infinitely divisible regenerative sets has proved elusive. We present such a characterization
under a mild additional hypothesis, and offer
evidence that the characterization persists in the general case.