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 $\partial$tÏ + H ($\nabla$xÏ) = 0 where the hamiltonian may lack C 1 regularity, which is quite unseen in this type of studies. R{é}sum{é}

References in corpus (2)

• A Hamilton-Jacobi method to describe the evolutionary equilibria in heterogeneous environments and with non-vanishing effects of mutations
• Spreading in kinetic reaction-transport equations in higher velocity dimensions