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

Caillerie, Nils

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 $\left(t,x,v\right)\to \left(t/\epsilon ,x/\epsilon ,v\right)$, puis nous effectuons une transformée de Hopf–Cole qui nous donne une équation cinétique suivie par un potentiel. Nous montrons la convergence pour $\epsilon \to 0$ 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.

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 $(t,x,v)\to (t/\epsilon ,x/\epsilon ,v)$ and then, perform a Hopf–Cole transform which gives us a kinetic equation on a potential. We prove the convergence of this potential as $\epsilon \to 0$ 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.

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.

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