In this paper it is shown that contrary to "linear" homogeneous nondegenerate diffusion processes, "nonlinear" diffusion processes can have a time-periodic law. We study four qualitatively different simple examples: a law-dependent Ornstein-Uhlenbeck process, a deterministic example, a one-dimensional example with a "semilinear" drift and a two-dimensional example with a linear drift.
Michael Scheutzow. "Some Examples of Nonlinear Diffusion Processes Having a Time-Periodic Law." Ann. Probab. 13 (2) 379 - 384, May, 1985. https://doi.org/10.1214/aop/1176992997