Ergodicity of Stochastic Differential Equations

Instead of a deterministic system of ordinary differential equations, we now have a stochastic system. Given a set of points Mo in the phase space, we consider the stochastic paths emanating from these points, with each point from the distribution viewed as an initial condition for the evolving SDE. At any time t we take a snapshot M, of the resulting set of random variables. Even if Mq is bounded, the evolving set of points would be expected to expand and fill in the phase space accessible at the temperature T eventually we hope that the points will be distributed in the entire phase space in accordance with the target measure (usually the Gibbs distribution 

Equivalently we may study a single solution trajectory eminating from almost any initial point on Afo, and ask that in the long-time limit the entirety of the phase space is covered according to the desired measure. The discussion we present here is necessarily condensed we direct the motivated reader to [269] for a more complete discussion. 

---