  The purpose of this paper is to study the dynamical behavior of the sequence
produced by a forward-backward algorithm involving two random maximal monotone
operators and a sequence of decreasing step sizes. Defining a mean monotone
operator as an Aumann integral, and assuming that the sum of the two mean
operators is maximal (sufficient maximality conditions are provided), it is
shown that with probability one, the interpolated process obtained from the
iterates is an asymptotic pseudo trajectory in the sense of Bena\"{\i}m and
Hirsch of the differential inclusion involving the sum of the mean operators.
The convergence of the empirical means of the iterates towards a zero of the
sum of the mean operators is shown, as well as the convergence of the sequence
itself to such a zero under a demipositivity assumption. These results find
applications in a wide range of optimization or variational inequality problems
in random environments.
