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

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}.