  This paper is concerned with stochastic processes that model multiple (or
iterated) scattering in classical mechanical systems of billiard type, defined
below. From a given (deterministic) system of billiard type, a random process
with transition probabilities operator P is introduced by assuming that some of
the dynamical variables are random with prescribed probability distributions.
Of particular interest are systems with weak scattering, which are associated
to parametric families of operators P_h, depending on a geometric or mechanical
parameter h, that approaches the identity as h goes to 0. It is shown that (P_h
-I)/h converges for small h to a second order elliptic differential operator L
on compactly supported functions and that the Markov chain process associated
to P_h converges to a diffusion with infinitesimal generator L. Both P_h and L
are selfadjoint (densely) defined on the space L2(H,{\eta}) of
square-integrable functions over the (lower) half-space H in R^m, where {\eta}
is a stationary measure. This measure's density is either (post-collision)
Maxwell-Boltzmann distribution or Knudsen cosine law, and the random processes
with infinitesimal generator L respectively correspond to what we call MB
diffusion and (generalized) Legendre diffusion. Concrete examples of simple
mechanical systems are given and illustrated by numerically simulating the
random processes.
