2021/06/24 by L. Koralov, Koralov, L., B. Vainberg +1
Computer Science · Mathematics · #35B40 #35C20 #35K10 #60F10 #60J80 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Stochastic processes and statistical mechanics #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2106.13307
openalex publication_date 2021/06/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We obtain the asymptotics, as t + |x| → ∞, 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 → ∞. A formula for the global asymptotics, as t + |x| → ∞, 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.