On the boundary of the support of super-Brownian motion: with appendices

TL;DR

This paper investigates the behavior of the density of one-dimensional super-Brownian motion, focusing on the probability of small densities and the boundary's Hausdorff dimension, linking results to eigenvalues of an Ornstein-Uhlenbeck operator.

Contribution

It provides new asymptotic estimates for the density's support boundary and connects these to spectral properties of a specific Ornstein-Uhlenbeck generator.

Findings

Asymptotic behavior of P(0<X(t,x)<a) as a approaches 0

Hausdorff dimension of the support boundary

Relation to the lead eigenvalue of an Ornstein-Uhlenbeck operator

Abstract

We study the density X(t,x) of one-dimensional super-Brownian motion and find the asymptotic behaviour of P(0<X(t,x)<a) as a approaches 0, as well as the Hausdorff dimension of the boundary of the support of X(t). The answers are in terms of the lead eigenvalue of the Ornstein-Uhlenbeck generator with a particular killing term. This work is motivated in part by questions of pathwise uniqueness for associated stochastic partial differential equations.