학술논문
Spectral gap for the growth-fragmentation equation via Harris's theorem.
Document Type
Journal
Author
Cañizo, José A. (E-GRAN-AM) AMS Author Profile; Gabriel, Pierre (F-VER-LM) AMS Author Profile; Yoldaş, Havva (F-LYON-ICJ) AMS Author Profile
Source
Subject
45 Integral equations -- 45K Integro-partial differential equations
45K05Integro-partial differential equations
47Operator theory -- 47D Groups and semigroups of linear operators, their generalizations and applications
47D06One-parameter semigroups and linear evolution equations
92Biology and other natural sciences -- 92D Genetics and population dynamics
92D25Population dynamics
45K05
47
47D06
92
92D25
Language
English
Abstract
Summary: ``We study the long-time behavior of the growth-fragmentation equation, a nonlocal linear evolution equation describing a wide range of phenomena in structured population dynamics. We show the existence of a spectral gap under conditions that generalize those in the literature by using a method based on Harris's theorem, a result coming from the study of equilibration of Markov processes. The difficulty posed by the nonconservativeness of the equation is overcome by performing an $h$-transform, after solving the dual Perron eigenvalue problem. The existence of the direct Perron eigenvector is then a consequence of our methods, which prove exponential contraction of the evolution equation. Moreover the rate of convergence is explicitly quantifiable in terms of the dual eigenfunction and the coefficients of the equation.''