Publications [#10352] of Jonathan C. Mattingly

Papers Published

1. Mattingly, J. C. and Stuart, A. M. and Higham, D. J., Ergodicity for {SDE}s and approximations: locally Lipschitz vector fields and degenerate noise, Stochastic Process. Appl., vol. 101 no. 2 (2002), pp. 185--232 [MR2003i:60103], [pdf]
   (last updated on 2004/07/13)

   Abstract:
   The ergodic properties of SDEs, and various time discretizations for SDEs, are studied. The ergodicity of SDEs is established by using techniques from the theory of Markov chains on general state spaces. Application of these Markov chain results leads to straightforward proofs of ergodicity for a variety of SDEs, in particular for problems with degenerate noise and for problems with locally Lipschitz vector fields. The key points which need to be verified are the existence of a Lyapunov function inducing returns to a compact set, a uniformly reachable point from within that set, and some smoothness of the probability densities; the last two points imply a minorization condition. Together the minorization condition and Lyapunov structure give geometric ergodicity. Applications include the Langevin equation, the Lorenz equation with degenerate noise and gradient systems. The ergodic theorems proved are strong, yielding exponential convergence of expectations for classes of measurable functions restricted only by the condition that they grow no faster than the Lyapunov function. The same Markov chain theory is then used to study time-discrete approximations of these SDEs. It is shown that the minorization condition is robust under approximation. For globally Lipschitz vector fields this is also true of the Lyapunov condition. However in the locally Lipschitz case the Lyapunov condition fails for explicit methods such as Euler-Maruyama; it is, in general, only inherited by specially constructed implicit discretizations. Examples of such discretization based on backward Euler methods are given, and approximation of the Langevin equation studied in some detail.