no. 4
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]
Caillerie, Nils1
1Georgetown University Department of mathematics and statistics Georgetown University Saint Mary’s Hall 3700 O Street NW Washington, DC 20057 (USA)
Annales de l'Institut Fourier, Tome 71 (2021) no. 4, pp. 1733-1755

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 ε→0 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.

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,v→t/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 ε→0 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.

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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.
Mots-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)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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
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