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

TL;DR

This paper investigates the large deviations of a velocity jump process using a Hamilton-Jacobi framework, revealing convergence to a viscosity solution even with non-smooth Hamiltonians.

Contribution

It introduces a Hamilton-Jacobi approach to analyze large deviations in velocity jump processes, including cases with non-C1 Hamiltonians, which is novel.

Findings

Convergence of the potential to a viscosity solution of a Hamilton-Jacobi equation.

Extension of large deviation analysis to Hamiltonians lacking C1 regularity.

Application of Hopf-Cole transform to kinetic equations in this context.

Abstract

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) $\to$ t $\epsilon$, x $\epsilon$, v, with $\epsilon$ \textgreater{} 0, before proceeding to a Hopf-Cole transform, which gives a kinetic equation on a potential. We show convergence as $\epsilon$ $\to$ 0 of the potential towards the viscosity solution of a Hamilton-Jacobi equation $\partial$t\"I + H ($\nabla$x\"I) = 0 where the hamiltonian may lack C 1 regularity, which is quite unseen in this type of studies. R{\'e}sum{\'e}