  A stochastic Forward-Backward algorithm with a constant step is studied. At
each time step, this algorithm involves an independent copy of a couple of
random maximal monotone operators. Defining a mean operator as a selection
integral, the differential inclusion built from the sum of the two mean
operators is considered. As a first result, it is shown that the interpolated
process obtained from the iterates converges narrowly in the small step regime
to the solution of this differential inclusion. In order to control the long
term behavior of the iterates, a stability result is needed in addition. To
this end, the sequence of the iterates is seen as a homogeneous Feller Markov
chain whose transition kernel is parameterized by the algorithm step size. The
cluster points of the Markov chains invariant measures in the small step regime
are invariant for the semiflow induced by the differential inclusion.
Conclusions regarding the long run behavior of the iterates for small steps are
drawn. It is shown that when the sum of the mean operators is demipositive, the
probabilities that the iterates are away from the set of zeros of this sum are
small in Ces\`aro mean. The ergodic behavior of these iterates is studied as
well. Applications of the proposed algorithm are considered. In particular, a
detailed analysis of the random proximal gradient algorithm with constant step
is performed.
