Global limit theorem for parabolic equations with a potential

TL;DR

This paper derives the global asymptotic behavior of the fundamental solution to a heat equation with a compactly supported potential, revealing different regimes inside and outside a conical region, with implications for branching diffusion processes.

Contribution

It provides a comprehensive global asymptotic formula for the heat equation with potential, including probabilistic interpretations for branching diffusions.

Findings

Asymptotics inside the conical region are governed by the principal eigenvalue.

Outside the cone, the solution behaves like the unperturbed fundamental solution.

The results describe the density decay of particles in branching diffusion with potentials.

Abstract

We obtain the asymptotics, as $t + |x| \rightarrow \infty$, of the fundamental solution to the heat equation with a compactly supported potential. It is assumed that the corresponding stationary operator has at least one positive eigenvalue. Two regions with different types of behavior are distinguished: inside a certain conical surface in the $(t,x)$ space, the asymptotics is determined by the principal eigenvalue and the corresponding eigenfunction; outside of the conical surface, the main term of the asymptotics is a product of a bounded function and the fundamental solution of the unperturbed operator, with the contribution from the potential becoming negligible if $|x|/t \rightarrow \infty$. A formula for the global asymptotics, as $t + |x| \rightarrow \infty$, of the solution in the entire half-space $t > 0$ is provided. In probabilistic terms, the result describes the asymptotics of the density of particles in a branching diffusion with compactly supported branching and killing potentials.