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

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) → t ε , 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 → 0 of the potential towards the viscosity solution of a Hamilton-Jacobi equation ∂tϕ + H (∇xϕ) = 0 where the hamiltonian may lack C 1 regularity, which is quite unseen in this type of studies. Résumé