  Stochastic differential equations of Langevin-diffusion form have received
significant attention, thanks to their foundational role in both Bayesian
sampling algorithms and optimization in machine learning. In the latter, they
serve as a conceptual model of the stochastic gradient flow in training
over-parameterized models. However, the literature typically assumes smoothness
of the potential, whose gradient is the drift term. Nevertheless, there are
many problems for which the potential function is not continuously
differentiable, and hence the drift is not Lipschitz continuous everywhere.
This is exemplified by robust losses and Rectified Linear Units in regression
problems. In this paper, we show some foundational results regarding the flow
and asymptotic properties of Langevin-type Stochastic Differential Inclusions
under assumptions appropriate to the machine-learning settings. In particular,
we show strong existence of the solution, as well as an asymptotic minimization
of the canonical free-energy functional.
