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August 2019 A probabilistic approach to Dirac concentration in nonlocal models of adaptation with several resources
Nicolas Champagnat, Benoit Henry
Ann. Appl. Probab. 29(4): 2175-2216 (August 2019). DOI: 10.1214/18-AAP1446

Abstract

This work is devoted to the study of scaling limits in small mutations and large time of the solutions $u^{\varepsilon}$ of two deterministic models of phenotypic adaptation, where the parameter $\varepsilon>0$ scales the size or frequency of mutations. The second model is the so-called Lotka–Volterra parabolic PDE in $\mathbb{R}^{d}$ 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 $\varepsilon\to0$. Our main results are, in both cases, the representation of the limits of $\varepsilon\log u^{\varepsilon}$ as solutions of variational problems and regularity results for these limits. The method mainly relies on Feynman–Kac-type representations of $u^{\varepsilon}$ 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.

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Nicolas Champagnat. Benoit Henry. "A probabilistic approach to Dirac concentration in nonlocal models of adaptation with several resources." Ann. Appl. Probab. 29 (4) 2175 - 2216, August 2019. https://doi.org/10.1214/18-AAP1446

Information

Received: 1 June 2018; Revised: 1 November 2018; Published: August 2019
First available in Project Euclid: 23 July 2019

zbMATH: 07120706
MathSciNet: MR3983337
Digital Object Identifier: 10.1214/18-AAP1446

Subjects:
Primary: 35K57 , 60F10
Secondary: 35B25 , 47G20 , 49L20 , 92D15

Keywords: adaptive dynamics , Dirac concentration , Feynman–Kac representation , Hamilton–Jacobi equations , Large deviations principles , Lotka–Volterra parabolic equation , Varadhan’s lemma

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.29 • No. 4 • August 2019
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