Ergodic Stochastic Differential Equations

Working in flatL space, we have that the infinitesmal generator for (6.43) acting on a smooth test function p(X) is 

In the case of Langevin dynamics, the matrix B is diagonal, simplifying the form of the operators by reducing the term involving second derivatives to a Laplacian. 

We have seen that invariant distributions correspond in general to solutions of 

Solutions to this equation are of central importance to the probabilistic behavior of solutions to (6.43). Such solutions give distributions of points in phase space which will remain constant as solutions evolve in time. We may ask a number of questions related to these distributions  

The property of ergodicity combined with associated mixing properties similar to those mentioned in Chap. 5, provide a framework to give precise answers to some of these questions 

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