Super-Brownian motion with reflecting historical paths

Abstract

We consider super-Brownian motion whose historical paths reflect from each other, unlike those of the usual historical super-Brownian motion. We prove tightness for the family of distributions corresponding to a sequence of discrete approximations but we leave the problem of uniqueness of the limit open. We prove a few results about path behavior for processes under any limit distribution. In particular, we show that for any [gamma] > 0, a "typical" increment of a reflecting historical path over a small time interval [delta] t is not greater than [delta] (t) [to the power of] (3/4− [gamma]).