We study three stochastic processes on the group of Sobolev diffeomorphisms of the flat n-dimensional torus \(\mathcal {T}^n\) that satisfy a second-order stochastic differential equation with backward mean derivatives and a certain relation between their first-order backward mean derivatives and symmetric mean derivatives. From the relation, it follows that those processes are martingales and their probabilistic densities and current velocities satisfy analogs of the continuity equations. It follows from the second-order equation that they generate flows of non-Newtonian fluids on the torus, satisfying analogs of Burgers equations.
