On solvability of stochastic differential equations with osmotic velocities

We give conditions such that solutions exist of the stochastic equation with the so-called osmotic velocities (the so-called Nelson's antisymmetric mean derivatives). We show that the solutions may not exist. Specifically, initial conditions are determined by the right-hand sides; however, some right-hand sides do not determine initial conditions, and in this case the solutions do not exist. Moreover, under the “correct” initial condition the solution is shown to be nonunique.