### Abstract

Original language | English |
---|---|

Pages (from-to) | 185-232 |

Number of pages | 47 |

Journal | Stochastic Processes and their Applications |

Volume | 101 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2002 |

### Fingerprint

### Keywords

- geometric ergodicity
- stochastic equations
- Langevin equation
- gradient systems
- additive noise
- time-discretization
- computer science
- applied mathematics

### Cite this

*Stochastic Processes and their Applications*,

*101*(2), 185-232. https://doi.org/10.1016/S0304-4149(02)00150-3

}

*Stochastic Processes and their Applications*, vol. 101, no. 2, pp. 185-232. https://doi.org/10.1016/S0304-4149(02)00150-3

**Ergodicity for SDEs and approximations: Locally Lipschitz vector fields and degenerate noise.** / Mattingly, J.; Stuart, A.M.; Higham, D.J.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Ergodicity for SDEs and approximations: Locally Lipschitz vector fields and degenerate noise

AU - Mattingly, J.

AU - Stuart, A.M.

AU - Higham, D.J.

PY - 2002

Y1 - 2002

N2 - 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, such as that expounded by Meyn-Tweedie. Application of these Markov chain results leads to straightforward proofs of geometric ergodicity for a variety of SDEs, including problems with degenerate noise and for problems with locally Lipschitz vector fields. Applications where this theory can be usefully applied include damped-driven Hamiltonian problems (the Langevin equation), the Lorenz equation with degenerate noise and gradient systems. The same Markov chain theory is then used to study time-discrete approximations of these SDEs. The two primary ingredients for ergodicity are a minorization condition and a Lyapunov condition. 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; for pathwise approximations 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.

AB - 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, such as that expounded by Meyn-Tweedie. Application of these Markov chain results leads to straightforward proofs of geometric ergodicity for a variety of SDEs, including problems with degenerate noise and for problems with locally Lipschitz vector fields. Applications where this theory can be usefully applied include damped-driven Hamiltonian problems (the Langevin equation), the Lorenz equation with degenerate noise and gradient systems. The same Markov chain theory is then used to study time-discrete approximations of these SDEs. The two primary ingredients for ergodicity are a minorization condition and a Lyapunov condition. 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; for pathwise approximations 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.

KW - geometric ergodicity

KW - stochastic equations

KW - Langevin equation

KW - gradient systems

KW - additive noise

KW - time-discretization

KW - computer science

KW - applied mathematics

U2 - 10.1016/S0304-4149(02)00150-3

DO - 10.1016/S0304-4149(02)00150-3

M3 - Article

VL - 101

SP - 185

EP - 232

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

SN - 0304-4149

IS - 2

ER -