Stochastic Differential Inclusions driven by Maximal Monotone Operators with empty interiors

Abstract

This paper studies the long-time behavior of stochastic differential inclusions driven by maximal monotone operators, motivated by continuous-time models of first-order optimization methods under noisy or approximate operator information. We first address well-posedness and show that existence and uniqueness can be established without the customary requirement that the operator's domain has nonempty interior, by adopting an appropriate notion of solution. We then analyze asymptotic properties of the resulting stochastic dynamics, extending convergence guarantees beyond previously studied settings that rely on smooth potentials, full-domain subdifferentials, or Lipschitz monotone operators. In addition, we consider a Tikhonov-type regularization of the stochastic inclusion and prove corresponding well-posedness and long-time convergence results.