A PROBABILISTIC APPROACH TO DIRAC CONCENTRATION IN NONLOCAL MODELS OF ADAPTATION WITH SEVERAL RESOURCES

• 成果类型: Article
• 署名作者: Champagnat, Nicolas; Henry, Benoit
• 署名单位: Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite de Lorraine; IMT - Institut Mines-Telecom; Universite de Lille; IMT Nord Europe
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/18-AAP1446
• 发表日期: 2019
• 页码: 2175-2216
• 关键词: hamilton-jacobi equations; lotka-volterra; DYNAMICS; CONVERGENCE; limit; time

摘要

This work is devoted to the study of scaling limits in small mutations and large time of the solutions u(epsilon) of two deterministic models of phenotypic adaptation, where the parameter epsilon > 0 scales the size or frequency of mutations. The second model is the so-called Lotka-Volterra parabolic PDE in Rd with an arbitrary number of resources and the first one is a version of the second model with finite phenotype space. The solutions of such systems typically concentrate as Dirac masses in the limit epsilon -> 0. Our main results are, in both cases, the representation of the limits of epsilon log u(epsilon) as solutions of variational problems and regularity results for these limits. The method mainly relies on Feynman-Kac-type representations of u(epsilon) and Varadhan's lemma. Our probabilistic approach applies to multiresources situations not covered by standard analytical methods and makes the link between variational limit problems and Hamilton-Jacobi equations with irregular Hamiltonians that arise naturally from analytical methods. The finite case presents substantial difficulties since the rate function of the associated large deviation principle (LDP) has noncompact level sets. In that case, we are also able to obtain uniqueness of the solution of the variational problem and of the associated differential problem which can be interpreted as a Hamilton-Jacobi equation in finite state space.