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

We study long-time behaviors for branching-diffusion process corresponding to the drifted Schrödinger operator $\mathcal{L}=\frac{1}{2}\mathrm{\Delta }+⟨\mathrm{\nabla }V,\mathrm{\nabla }⟩-K$, where $K$ represents the reduction rate of a population dynamics and $\mathrm{\nabla }V$ 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 Schrödinger operators with unbounded potentials. The result is new even in the one-dimensional setting, which especially improves the recent work \cite{CMS}.