  We consider a random walk on a second countable locally compact topological
space endowed with an invariant Radon measure. We show that if the walk is
symmetric and if every subset which is invariant by the walk has zero or
infinite measure, then one has escape of mass for almost every starting point.
We then apply this result in the context of homogeneous random walks on
infinite volume spaces, and deduce a converse to Eskin-Margulis recurrence
theorem.
