### Abstract

We explore Itô stochastic differential equations where the drift term possibly depends on the infinite past. Assuming the existence of a Lyapunov function, we prove the existence of a stationary solution assuming only minimal continuity of the coefficients. Uniqueness of the stationary solution is proven if the dependence on the past decays sufficiently fast. The results of this paper are then applied to stochastically forced dissipative partial differential equations such as the stochastic Navier-Stokes equation and stochastic Ginsburg-Landau equation.

Original language | English (US) |
---|---|

Pages (from-to) | 553-582 |

Number of pages | 30 |

Journal | Communications in Contemporary Mathematics |

Volume | 7 |

Issue number | 5 |

DOIs | |

State | Published - Oct 2005 |

### Fingerprint

### Keywords

- Ergodicity
- Lyapunov functions
- Memory
- Stationary solutions
- Stochastic differential equations
- Stochastic Ginsburg-Landau equation
- Stochastic Navier-Stokes equation

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**Stationary solutions of stochastic differential equations with memory and stochastic partial differential equations.** / Bakhtin, Yuri; Mattingly, Jonathan C.

Research output: Contribution to journal › Article

*Communications in Contemporary Mathematics*, vol. 7, no. 5, pp. 553-582. https://doi.org/10.1142/S0219199705001878

}

TY - JOUR

T1 - Stationary solutions of stochastic differential equations with memory and stochastic partial differential equations

AU - Bakhtin, Yuri

AU - Mattingly, Jonathan C.

PY - 2005/10

Y1 - 2005/10

N2 - We explore Itô stochastic differential equations where the drift term possibly depends on the infinite past. Assuming the existence of a Lyapunov function, we prove the existence of a stationary solution assuming only minimal continuity of the coefficients. Uniqueness of the stationary solution is proven if the dependence on the past decays sufficiently fast. The results of this paper are then applied to stochastically forced dissipative partial differential equations such as the stochastic Navier-Stokes equation and stochastic Ginsburg-Landau equation.

AB - We explore Itô stochastic differential equations where the drift term possibly depends on the infinite past. Assuming the existence of a Lyapunov function, we prove the existence of a stationary solution assuming only minimal continuity of the coefficients. Uniqueness of the stationary solution is proven if the dependence on the past decays sufficiently fast. The results of this paper are then applied to stochastically forced dissipative partial differential equations such as the stochastic Navier-Stokes equation and stochastic Ginsburg-Landau equation.

KW - Ergodicity

KW - Lyapunov functions

KW - Memory

KW - Stationary solutions

KW - Stochastic differential equations

KW - Stochastic Ginsburg-Landau equation

KW - Stochastic Navier-Stokes equation

UR - http://www.scopus.com/inward/record.url?scp=26444510187&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=26444510187&partnerID=8YFLogxK

U2 - 10.1142/S0219199705001878

DO - 10.1142/S0219199705001878

M3 - Article

AN - SCOPUS:26444510187

VL - 7

SP - 553

EP - 582

JO - Communications in Contemporary Mathematics

JF - Communications in Contemporary Mathematics

SN - 0219-1997

IS - 5

ER -