The Annals of Applied Probability

A probabilistic approach to Dirac concentration in nonlocal models of adaptation with several resources

Nicolas Champagnat and Benoit Henry

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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.

Article information

Ann. Appl. Probab., Volume 29, Number 4 (2019), 2175-2216.

Received: June 2018
Revised: November 2018
First available in Project Euclid: 23 July 2019

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Mathematical Reviews number (MathSciNet)

Primary: 60F10: Large deviations 35K57: Reaction-diffusion equations
Secondary: 49L20: Dynamic programming method 92D15: Problems related to evolution 35B25: Singular perturbations 47G20: Integro-differential operators [See also 34K30, 35R09, 35R10, 45Jxx, 45Kxx]

Adaptive dynamics Dirac concentration large deviations principles Hamilton–Jacobi equations Varadhan’s lemma Lotka–Volterra parabolic equation Feynman–Kac representation


Champagnat, Nicolas; Henry, Benoit. A probabilistic approach to Dirac concentration in nonlocal models of adaptation with several resources. Ann. Appl. Probab. 29 (2019), no. 4, 2175--2216. doi:10.1214/18-AAP1446.

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