The pioneering work on stochastic stabilization was done by Has'minskii in 1969. In this paper we shall use stochastic perturbation described by linear Ito integration w.r.t. Brownian motions to stabilize or destabilize the given non-linear system. On the other hand, this system can also be destabilzed if the dimension of the state is more than 1. It should be stressed that in this paper we use Ito integral rather than Stratonovich's.
Analogous results exist for other particular models.
A unifying framework is given for such fixed-r results, and it is shown that if the site density is finite and the labeled graph has positive finite energy, then with probability one, uniqueness holds simultaneously for all values of r. An example is given to show that when the site density is infinite, positive finite energy does not ensure uniqueness, even for fixed r. There are extensive studies on heat conduction in a composite medium. In one dimensional case, there is a general method by solving the boundary problem when medium is composed of several pieces of diffrent materials. However, in the general case, there is no unified method.