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

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}