Some Estimates for 2-Dimensional Infinite and Bounded Dilute Random Lorentz Gases

We present a (mostly) rigorous approach to unbounded and bounded (open) dilute random Lorentz gases. Relying on previous rigorous results on the dilute (Boltzmann–Grad) limit we compute the asymptotics of the Lyapunov exponent in the unbounded case. For the bounded open case in a circular region we give here an incomplete rigorous analysis which gives the asymptotics for large radius of the escape rate and of the rescaled quasi-invariant (q.i., or quasi-stationary) measure. We finally give a complete proof on existence and asymptotic properties of the q.i. measure in a one-dimensional caricature.