In this paper we describe a class of nonlinear feedback systems perturbed by white noise for which explicit formulas for steady-state probability densities can be found, We show that this class includes what has been called monotemperaturic systems in earlier work and establish relationships with Lyapunov functions for the corresponding deterministic systems. We also treat a number of stochastic optimal control problems in the case of quantized feedback, with performance criteria formulated in terms of the steady-state probability density.