Ergodicity of Brownian Dynamics

Imagine a distribution po(X) which we may take to be an initial macroscopic state of a stochastic differential equation system. This might be a smooth probability density such as a Gaussian, the indicator function for a small disk D in the phase space, or, in the extreme case a Dirac delta distribution (indicating that all initial conditions are clustered at a single point in phase space). The density evolves according to the partial differential equation 

Under what conditions (and in what sense) can we say that the distribution obtained in this way converges to a known invariant distribution  

For example, let us return to Brownian dynamics (or overdamped Langevin dynamics) and set Af = / and y = 1 in (6.36), giving an equation of motion 

We demonstrate a very useful property of this operator. 

Combining this with the first part of the inner product, we see that 

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