Long-Time Influence of Small Perturbations: Systems with Conservation Laws

 I will consider long-time effects caused by perturbations of diffusion processes, in particular of dynamical systems. If the system has some conservation laws, they can be broken by the perturbations. Long- time  motion of the perturbed system can be described, under certain assumptions, by the evolution, in an appropriate time scale, of smooth and discrete first integrals. This evolution is defined, as a rule, by laws of large numbers, CLT-type results, limit theorems for large deviations. More or less all classical  and recent results of this type can be described in this way. I will consider perturbations of diffusion processes with sufficiently rich collections of invariant sets in the phase space as well as the case when some of conservation laws are defined by the Martin boundary of the non-perturbed system.