Fluctuation limits of the super-Brownian motion with a single point catalyst

TL;DR

This paper establishes a fluctuating limit theorem for super-Brownian motions with a point catalyst, showing convergence to an Ornstein-Uhlenbeck type process driven by Brownian motion.

Contribution

It introduces a new fluctuating limit theorem for super-Brownian motions with a point catalyst, characterizing the limiting process as an Ornstein-Uhlenbeck type process.

Findings

Weak convergence of super-Brownian motions to the Ornstein-Uhlenbeck process

Limit process solves a Langevin type equation driven by Brownian motion

Results extend understanding of fluctuation limits in stochastic processes

Abstract

We prove a fluctuating limit theorem of a sequence of super-Brownian motions over $\mbb{R}$ with a single point catalyst. The weak convergence of the processes on the space of Schwarz distributions is established. The limiting process is an Ornstein-Uhlenbeck type process solving a Langevin type equation driven by a one-dimensional Brownian motion.