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, Online first, 23 p.

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.

Reçu le :
Révisé le :
Accepté le :
Première publication :
DOI : https://doi.org/10.5802/aif.3433
Classification : 35Q92,  45K05,  35D40,  35F21
Mots clés : Équations cinétiques, Équations de Hamilton–Jacobi, Grandes déviations, Méthode de la fonction test perturbée, Processus de Markov déterministes par morceaux.
@unpublished{AIF_0__0_0_A32_0,
     author = {Caillerie, Nils},
     title = {Large deviations of a forced velocity-jump process with a {Hamilton{\textendash}Jacobi} approach},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2021},
     doi = {10.5802/aif.3433},
     language = {en},
     note = {Online first},
}
TY  - UNPB
AU  - Caillerie, Nils
TI  - Large deviations of a forced velocity-jump process with a Hamilton–Jacobi approach
JO  - Annales de l'Institut Fourier
PY  - 2021
DA  - 2021///
PB  - Association des Annales de l’institut Fourier
N1  - Online first
UR  - https://doi.org/10.5802/aif.3433
DO  - 10.5802/aif.3433
LA  - en
ID  - AIF_0__0_0_A32_0
ER  - 
%0 Unpublished Work
%A Caillerie, Nils
%T Large deviations of a forced velocity-jump process with a Hamilton–Jacobi approach
%J Annales de l'Institut Fourier
%D 2021
%I Association des Annales de l’institut Fourier
%Z Online first
%U https://doi.org/10.5802/aif.3433
%R 10.5802/aif.3433
%G en
%F AIF_0__0_0_A32_0
Caillerie, Nils. Large deviations of a forced velocity-jump process with a Hamilton–Jacobi approach. Annales de l'Institut Fourier, Online first, 23 p.

[1] Alt, Wolgang Biased random walk models for chemotaxis and related diffusion approximations, J. Math. Biol., Volume 9 (1980) no. 2, pp. 147-177 | Article | MR 661424 | Zbl 0434.92001

[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 | Article | MR 957658 | Zbl 0674.49027

[4] Bouin, Emeric A Hamilton–Jacobi approach for front propagation in kinetic equations, Kinet. Relat. Models, Volume 8 (2015) no. 2, pp. 255-280 | Article | MR 3356578 | Zbl 1329.35317

[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 | Article | MR 3914146 | Zbl 1433.35415

[6] Bouin, Emeric; Calvez, Vincent A kinetic eikonal equation, C. R. Math. Acad. Sci. Paris, Volume 350 (2012) no. 5, pp. 243-248 | Article | MR 2911931 | Zbl 1252.35124

[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 | Article | MR 2981349 | Zbl 1253.35186

[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 | Article | MR 3355005 | Zbl 1317.35267

[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 | Article | MR 3351436 | Zbl 1351.92040

[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 | Article | MR 3612704 | Zbl 1364.60034

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

[13] Calvez, Vincent Chemotactic waves of bacteria at the mesoscale, J. Eur. Math. Soc., Volume 22 (2019) no. 2, pp. 593-668 | Article | MR 4049225 | Zbl 1436.35301

[14] Coville, Jérôme Singular measure as principal eigenfunction of some nonlocal operators, Appl. Math. Lett., Volume 26 (2013) no. 8, pp. 831-835 | Article | MR 3066699 | Zbl 1311.45002

[15] Davis, Mark H. A. Markov models and optimization, Monographs on Statistics and Applied Probability, 49, Chapman & Hall, 1993 | Article | Zbl 0780.60002

[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 | Article | MR 1007533 | Zbl 0679.35001

[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 | Article | Numdam | MR 781589 | Zbl 0601.60076

[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 | Article | MR 982575 | Zbl 0692.35014

[19] Fleming, Wendell H. Exit probabilities and optimal stochastic control, Appl. Math. Optim., Volume 4 (1977) no. 1, pp. 329-346 | Article | MR 512217 | Zbl 0398.93068

[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 | Numdam | MR 876121 | Zbl 0622.60032

[21] Freidlin, Mark I. Geometric Optics Approach to Reaction-Diffusion Equations, SIAM J. Appl. Math., Volume 46 (1986) no. 2, pp. 222-232 | Article | MR 833475 | Zbl 0626.35047

[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 | Article | MR 3612701 | Zbl 1366.92104

[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 | Article | MR 3420412 | Zbl 1327.35011

[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 | Article | Numdam | MR 3303941 | Zbl 1312.35011

[25] Perthame, Benoît Global existence to the BGK model of Boltzmann equation, J. Differ. Equations, Volume 82 (1989) no. 1, pp. 191-205 | Article | MR 1023307 | Zbl 0694.35134

[26] Rudnicki, Ryszard; Tyran-Kamińska, Marta Piecewise Deterministic Processes in Biological Models, SpringerBriefs in Applied Sciences and Technology, Springer, 2017 | Article

[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 | Article

[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 | Article | Zbl 0282.60047

Cité par Sources :