Large deviations of a forced velocity-jump process with a Hamilton–Jacobi approach
[Grandes déviations pour un processus à sauts de vitesse contraint avec une approche Hamilton–Jacobi]
Annales de l'Institut Fourier, Tome 71 (2021) no. 4, pp. 1733-1755.

Nous nous intéressons à la dispersion d’une particule dont le mouvement peut être décrit par un processus à sauts de vitesse contraint par un force extérieure. Pour établir des résultats de grandes déviations, nous étudions l’équation de Kolmogorov après rééchelonnement hyperbolique t,x,vt/ε,x/ε,v, puis nous effectuons une transformée de Hopf–Cole qui nous donne une équation cinétique suivie par un potentiel. Nous montrons la convergence pour ε0 de ce potentiel vers la solution de viscosité d’une équation de Hamilton–Jacobi. Le hamiltonien peut présenter une singularité 𝒞 1 , comme il a déjà été constaté dans ce type d’études. Ceci est un travail préliminaire avant d’étudier des résultats de propagation pour des processus plus réalistes.

We study the dispersion of a particle whose motion dynamics can be described by a forced velocity jump process. To investigate large deviations results, we study the Kolmogorov forward equation of this process in the hyperbolic scaling (t,x,v)(t/ε,x/ε,v) and then, perform a Hopf–Cole transform which gives us a kinetic equation on a potential. We prove the convergence of this potential as ε0 to the solution of a Hamilton–Jacobi equation. The hamiltonian can have a C 1 singularity, as was previously observed in this kind of studies. This is a preliminary work before studying spreading results for more realistic processes.

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DOI : 10.5802/aif.3433
Classification : 35Q92, 45K05, 35D40, 35F21
Keywords: Kinetic equations, Hamilton–Jacobi equations, Large deviations, Perturbed test function method, Piecewise Deterministic Markov Process.
Mot clés : Équations cinétiques, Équations de Hamilton–Jacobi, Grandes déviations, Méthode de la fonction test perturbée, Processus de Markov déterministes par morceaux.
Caillerie, Nils 1

1 Georgetown University Department of mathematics and statistics Georgetown University Saint Mary’s Hall 3700 O Street NW Washington, DC 20057 (USA)
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Caillerie, Nils. Large deviations of a forced velocity-jump process with a Hamilton–Jacobi approach. Annales de l'Institut Fourier, Tome 71 (2021) no. 4, pp. 1733-1755. doi : 10.5802/aif.3433. http://www.numdam.org/articles/10.5802/aif.3433/

[1] Alt, Wolgang Biased random walk models for chemotaxis and related diffusion approximations, J. Math. Biol., Volume 9 (1980) no. 2, pp. 147-177 (Accessed 2017-09-21) | DOI | MR | Zbl

[2] Barles, Guy Solutions de viscosité des équations de Hamilton–Jacobi, Mathématiques & Applications, Springer, 1994 no. 17

[3] Barles, Guy; Perthame, Benoît Exit Time Problems in Optimal Control and Vanishing Viscosity Method, SIAM J. Control Optimization, Volume 26 (1988) no. 5, pp. 1133-1148 (Accessed 2017-03-27) | DOI | MR | Zbl

[4] Bouin, Emeric A Hamilton–Jacobi approach for front propagation in kinetic equations, Kinet. Relat. Models, Volume 8 (2015) no. 2, pp. 255-280 (Accessed 2017-03-06) | DOI | MR | Zbl

[5] Bouin, Emeric; Caillerie, Nils Spreading in kinetic reaction–transport equations in higher velocity dimensions, Eur. J. Appl. Math., Volume 30 (2019) no. 2, pp. 219-247 (Accessed 2020-04-08) | DOI | MR | Zbl

[6] Bouin, Emeric; Calvez, Vincent A kinetic eikonal equation, C. R. Math. Acad. Sci. Paris, Volume 350 (2012) no. 5, pp. 243-248 (Accessed 2016-11-09) | DOI | MR | Zbl

[7] Bouin, Emeric; Calvez, Vincent; Grenier, Emmanuel; Nadin, Grégoire Large deviations for velocity-jump processes and non-local Hamilton–Jacobi equations (2016) (https://arxiv.org/abs/1607.03676)

[8] Bouin, Emeric; Calvez, Vincent; Meunier, Nicolas; Mirrahimi, Sepideh; Perthame, Benoît; Raoul, Gaël; Voituriez, Raphaël Invasion fronts with variable motility: Phenotype selection, spatial sorting and wave acceleration, C. R. Math. Acad. Sci. Paris, Volume 350 (2012) no. 15, pp. 761-766 | DOI | MR | Zbl

[9] Bouin, Emeric; Calvez, Vincent; Nadin, Grégoire Propagation in a Kinetic Reaction-Transport Equation: Travelling Waves And Accelerating Fronts, Arch. Ration. Mech. Anal., Volume 217 (2015) no. 2, pp. 571-617 (Accessed 2016-11-09) | DOI | MR | Zbl

[10] Bouin, Emeric; Mirrahimi, Sepideh A Hamilton–Jacobi approach for a model of population structured by space and trait, Commun. Math. Sci., Volume 13 (2015) no. 6, pp. 1431-1452 (Accessed 2017-03-06) | DOI | MR | Zbl

[11] Caillerie, Nils Large deviations of a velocity jump process with a Hamilton–Jacobi approach, C. R. Math. Acad. Sci. Paris, Volume 355 (2017) no. 2, pp. 170-175 (Accessed 2017-02-21) | DOI | MR | Zbl

[12] Caillerie, Nils Stochastic and deterministic kinetic equations in the context of mathematics applied to biology, phdthesis, Université de Lyon (2017) (Accessed 2018-01-17 https://tel.archives-ouvertes.fr/tel-01579877/document)

[13] Calvez, Vincent Chemotactic waves of bacteria at the mesoscale, J. Eur. Math. Soc., Volume 22 (2019) no. 2, pp. 593-668 (Accessed 2020-04-08) | DOI | MR | Zbl

[14] Coville, Jérôme Singular measure as principal eigenfunction of some nonlocal operators, Appl. Math. Lett., Volume 26 (2013) no. 8, pp. 831-835 (Accessed 2016-12-16) | DOI | MR | Zbl

[15] Davis, Mark H. A. Markov models and optimization, Monographs on Statistics and Applied Probability, 49, Chapman & Hall, 1993 | DOI | Zbl

[16] Evans, Lawrence C. The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. R. Soc. Edinb., Sect. A, Math., Volume 111 (1989) no. 3-4, pp. 359-375 (Accessed 2017-02-21) | DOI | MR | Zbl

[17] Evans, Lawrence C.; Ishii, Hitoshi A PDE approach to some asymptotic problems concerning random differential equations with small noise intensities, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 2 (1985) no. 1, pp. 1-20 (Accessed 2020-04-08) | DOI | Numdam | MR | Zbl

[18] Evans, Lawrence C.; Souganidis, Panagiotis E. A PDE Approach to Geometric Optics for Certain Semilinear Parabolic Equations, Indiana Univ. Math. J., Volume 38 (1989) no. 1, pp. 141-172 | DOI | MR | Zbl

[19] Fleming, Wendell H. Exit probabilities and optimal stochastic control, Appl. Math. Optim., Volume 4 (1977) no. 1, pp. 329-346 (Accessed 2020-04-08) | DOI | MR | Zbl

[20] Fleming, Wendell H.; Souganidis, Panagiotis E. PDE-viscosity solution approach to some problems of large deviations, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser., Volume 13 (1986) no. 2, pp. 171-192 (Accessed 2017-01-30) | Numdam | MR | Zbl

[21] Freidlin, Mark I. Geometric Optics Approach to Reaction-Diffusion Equations, SIAM J. Appl. Math., Volume 46 (1986) no. 2, pp. 222-232 (Accessed 2017-04-13) | DOI | MR | Zbl

[22] Gandon, Sylvain; Mirrahimi, Sepideh A Hamilton–Jacobi method to describe the evolutionary equilibria in heterogeneous environments and with non-vanishing effects of mutations, C. R. Math. Acad. Sci. Paris, Volume 355 (2017) no. 2, pp. 155-160 (Accessed 2017-09-22) | DOI | MR | Zbl

[23] Mirrahimi, Sepideh; Perthame, Benoît Asymptotic analysis of a selection model with space, J. Math. Pures Appl., Volume 104 (2015) no. 6, pp. 1108-1118 (Accessed 2020-04-08) | DOI | MR | Zbl

[24] Mirrahimi, Sepideh; Perthame, Benoît; Souganidis, Panagiotis E. Time fluctuations in a population model of adaptive dynamics, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 32 (2015) no. 1, pp. 41-58 (Accessed 2017-09-22) | DOI | Numdam | MR | Zbl

[25] Perthame, Benoît Global existence to the BGK model of Boltzmann equation, J. Differ. Equations, Volume 82 (1989) no. 1, pp. 191-205 (Accessed 2020-04-08) | DOI | MR | Zbl

[26] Rudnicki, Ryszard; Tyran-Kamińska, Marta Piecewise Deterministic Processes in Biological Models, SpringerBriefs in Applied Sciences and Technology, Springer, 2017 (Accessed 2020-04-08) | DOI

[27] Saragosti, Jonathan; Calvez, Vincent; Bournaveas, Nikolaos; Perthame, Benoît; Buguin, Axel; Silberzan, Pascal Directional persistence of chemotactic bacteria in a traveling concentration wave, Proc. Natl. Acad. Sci. USA, Volume 108 (2011) no. 39, pp. 16235-16240 | DOI

[28] Stroock, Daniel W. Some stochastic processes which arise from a model of the motion of a bacterium, Z. Wahrscheinlichkeitstheor. Verw. Geb., Volume 28 (1974) no. 4, pp. 305-315 (Accessed 2017-09-21) | DOI | Zbl

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