  Our work is part of the close link between continuous-time dissipative
dynamical systems and optimization algorithms, and more precisely here, in the
stochastic setting. We aim to study stochastic convex minimization problems
through the lens of stochastic inertial differential inclusions that are driven
by the subgradient of a convex objective function. This will provide a general
mathematical framework for analyzing the convergence properties of stochastic
second-order inertial continuous-time dynamics involving vanishing viscous
damping and measurable stochastic subgradient selections. Our chief goal in
this paper is to develop a systematic and unified way that transfers the
properties recently studied for first-order stochastic differential equations
to second-order ones involving even subgradients in lieu of gradients. This
program will rely on two tenets: time scaling and averaging, following an
approach recently developed in the literature by one of the co-authors in the
deterministic case.
  Under a mild integrability assumption involving the diffusion term and the
viscous damping, our first main result shows that almost surely, there is weak
convergence of the trajectory towards a minimizer of the objective function and
fast convergence of the values and gradients. We also provide a comprehensive
complexity analysis by establishing several new pointwise and ergodic
convergence rates in expectation for the convex, strongly convex, and (local)
Polyak-Lojasiewicz case. Finally, using Tikhonov regularization with a properly
tuned vanishing parameter, we can obtain almost sure strong convergence of the
trajectory towards the minimum norm solution.
