Limit Theorems for the Frontier of a One-Dimensional Branching Diffusion

Abstract

Let $R_t$ be the position of the rightmost particle at time $t$ in a time-homogeneous one-dimensional branching diffusion process. Let $\gamma(\alpha,t)$ be the $\alpha$th quantile of $R_t$ under $P^0$, where $P^x$ denotes the probability measure of the branching diffusion process starting with a single particle at position $x$. We show that $\gamma(\alpha,t)$ is a limiting quantile of $R_t$ under $P^x$ in the sense that $\lim_{t \rightarrow\infty}P^x\{R_t \leq \gamma(\alpha,t)\}$ exists for all $\alpha \in (0,1)$ and all $x \in \mathbb{R}$. If the underlying diffusion is recurrent, we show that, after an appropriate rescaling of space, the $P^x$ distribution of $R_t - t$ converges weakly to a nontrivial limiting distribution $w_x$.