  We analyse the linear kinetic transport equation with a BGK relaxation
operator. We study the large scale hyperbolic limit $(t,x)\to (t/\eps,x/\eps)$.
We derive a new type of limiting Hamilton-Jacobi equation, which is analogous
to the classical eikonal equation derived from the heat equation with small
diffusivity. We prove well-posedness of the phase problem and convergence
towards the viscosity solution of the Hamilton-Jacobi equation. This is a
preliminary work before analysing the propagation of reaction fronts in kinetic
equations.
