Ergodicity of Langevin Dynamics

The key feature of Brownian dynamics that makes it possible to verify the ergodic property is the fact that each variable of the system is directly in contact with an independent stochastic Wiener process. This ensures that at each point of phase space, all possible directions are sampled and the paths will have freedom to move in any direction. In Langevin dynamics, taking M = / for simplicity, 

In these more complicated examples, we are able to demonstrate ergodicity without finding an explicit solution (as in the Ornstein-Uhlenbeck example) or study of the dynamics generator (as in Brownian dynamics), given some assumptions on the behavior of solutions. We state (without proof) a powerful theorem on the ergodicity of degenerate stochastic diffusions, whose proof is essentially contained in [257, Theorem 2.5] (see also [44, 160, 161, 253, 266]). We denote by Hfix) the open ball in D centered on the points of radius while B(D) is the Borel a-algebra on T) (see Sect. 5.2.1). 

Assumption 1 The Markov process generated by the SDE (6.43) satisfies, for some fixed compact set C e B D), the conditions  

If these two assumptions hold, then we are able to state the following theorem [165]  

Theorem 6.2 (Geometric Ergodicity) Let the solution to (6.43) be denoted X(f). If Assumption 1 and Assumption 2 are satisfied for 

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