Long-Time Behaviors of Branching-Diffusion Processes via Spectral Analysis

TL;DR

This paper analyzes the long-term behavior of branching-diffusion processes using spectral analysis, establishing exponential convergence and characterizing quasi-stationary distributions with novel techniques.

Contribution

It introduces a new spectral analysis transformation and heat kernel estimates to study these processes, improving understanding even in one-dimensional cases.

Findings

Exponential convergence rates for total mass established.

Characterization of quasi-stationary distributions achieved.

Results are new even in one-dimensional settings.

Abstract

We study long-time behaviors for branching-diffusion process corresponding to the drifted Schr\"odinger operator $\mathcal{L} = \frac{1}{2} \Delta + \langle \nabla V,\nabla \rangle - K$, where $K$ represents the reduction rate of a population dynamics and $\nabla V$ is a given drift term. In particular, we establish exponential convergence rates for the total mass of this process and characterize its quasi-stationary distribution. The proof is based on a novel transformation in spectral analysis, and heat kernel estimates for Schr\"odinger operators with unbounded potentials. The result is new even in the one-dimensional setting, which especially improves the recent work \cite{CMS}.