Large deviations of a velocity jump process with a Hamilton–Jacobi approach

Nils Caillerie

doi:10.1016/j.crma.2016.12.011

Partial differential equations

Large deviations of a velocity jump process with a Hamilton–Jacobi approach

the Editorial Board

Nils Caillerie ¹

¹Université de Lyon, Université Claude-Bernard Lyon-1, CNRS UMR 5208, Institut Camille-Jordan, 43 bd du 11-Novembre-1918, 69622 Villeurbanne cedex, France

Comptes Rendus. Mathématique, Volume 355 (2017) no. 2, pp. 170-175

Abstract (Original language)
Résumé

We study a random process on $R^{n}$ moving in straight lines and changing randomly its velocity at random exponential times. We focus more precisely on the Kolmogorov equation in the hyperbolic scale $(t, x, v) \to (\frac{t}{ε}, \frac{x}{ε}, v)$ , with $ε > 0$ , before proceeding to a Hopf–Cole transform, which gives a kinetic equation on a potential. We show convergence as $ε \to 0$ of the potential towards the viscosity solution to a Hamilton–Jacobi equation $\partial_{t} φ + H (\nabla_{x} φ) = 0$ where the Hamiltonian may lack $C^{1}$ regularity, which is quite unseen in this type of studies.

Nous nous intéressons à un processus aléatoire sur $R^{n}$ qui alterne des phases de mouvements rectilignes uniformes et change de vitesse à des temps exponentiels. Nous étudions plus précisément l'équation de Kolmogorov après rééchelonnement hyperbolique $(t, x, v) \to (\frac{t}{ε}, \frac{x}{ε}, v)$ , $ε > 0$ , puis nous effectuons une transformée de Hopf–Cole qui nous donne une équation cinétique suivie par un potentiel. Nous montrons la convergence pour $ε \to 0$ de ce potentiel vers la solution de viscosités d'une équation de Hamilton–Jacobi $\partial_{t} φ + H (\nabla_{x} φ) = 0$ où le hamiltonien peut présenter une singularité $C^{1}$ , ce qui est assez inédit dans ce type d'études.

Article information
Cite this article

Received: 2016-02-15
Accepted: 2017-01-05
Published online: 2017-02-01

DOI: 10.1016/j.crma.2016.12.011

Author's affiliations:

Nils Caillerie ¹

¹ Université de Lyon, Université Claude-Bernard Lyon-1, CNRS UMR 5208, Institut Camille-Jordan, 43 bd du 11-Novembre-1918, 69622 Villeurbanne cedex, France

Nils Caillerie. Large deviations of a velocity jump process with a Hamilton–Jacobi approach. Comptes Rendus. Mathématique, Volume 355 (2017) no. 2, pp. 170-175. doi: 10.1016/j.crma.2016.12.011

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