ON STOCHASTIC DEFORMATIONS OF DYNAMICAL SYSTEMS

I discuss the connection of the three different questions: The existence of the Gibbs steady state distributions for the stochastic differential equations, the notion and the existence of the conservation laws for such equations, and the convergence of the smooth random perturbations of dynamical systems to stochastic differential equations in the Ito sense. I show that in all cases one needs to include some additional term in the standard form of stochastic equation. I call such approach to describing the influence of the noise on the dynamical systems the stochastic deformation to distinguish it from the conventional stochastic perturbation. I also discuss some consequences of this approach, in particular, a connection between the noise intensity and the temperature. This connection is known in physics (for the case of linear system of differential equations) as fluctuation-dissipation theorem (L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 9, Statistical Physics. Part 2). In conclusion, I present an interesting physical example of the dynamics of magnetic dipole in a random magnetic field.