Bessel processes, the Brownian snake and super-Brownian motion

TL;DR

This paper establishes that the historical path at the minimal spatial position for one-dimensional Brownian snake and super-Brownian motion is a Bessel process of dimension -5, and explores a spine decomposition conditioned on this path.

Contribution

It identifies the minimal spatial path as a Bessel process of dimension -5 and introduces a spine decomposition for the Brownian snake conditioned on this path.

Findings

Minimal path is a Bessel process of dimension -5

Spine decomposition for Brownian snake conditioned on minimal path

Applicable to one-dimensional super-Brownian motion

Abstract

We prove that, both for the Brownian snake and for super-Brownian motion in dimension one, the historical path corresponding to the minimal spatial position is a Bessel process of dimension -5. We also discuss a spine decomposition for the Brownian snake conditioned on the minimizing path.