  We analyze the stochastic proximal subgradient descent in the case where the
objective functions are path differentiable and verify a Sard-type condition.
While the accumulation set may not be reduced to unique point, we show that the
time spent by the iterates to move from one accumulation point to another goes
to infinity. An oscillation-type behavior of the drift is established. These
results show a strong stability property of the proximal subgradient descent.
Using the theory of closed measures, Bolte, Pauwels and R\'ios-Zertuche
established this type of behavior for the deterministic subgradient descent.
Our technique of proof relies on the classical work of Bena\"im, Hofbauer and
Sorin, which allows us to extend results in the deterministic case to a
stochastic and proximal setting, as well as to treat these different cases in a
unified manner.
