Noise can create periodic behavior and stabilize nonlinear diffusions

It is shown that unlike nondegenerate (linear) diffusion processes, nonlinear diffusion processes can have a periodic law. We provide an example of a nonlinear diffusion for which periodic behavior is even created by the noise, i.e. no periodicity occurs when the noise is turned off. In the second part of the paper we give an example of a one-dimensional nonlinear diffusion which can be stabilized by noise. Finally we show also that the N-dimensional (N ≥ 2) ‘linear’ diffusion approximations of that system are stabilized by noise.