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### Sampling conditioned hypoelliptic diffusions

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Hairer, Martin, Stuart, A. M. and Voss, Jochen.
(2011)
*Sampling conditioned hypoelliptic diffusions.*
The Annals of Applied Probability, Volume 21
(Number 2).
pp. 669-698.
ISSN 1050-5164

**Full text not available from this repository.**

Official URL: http://dx.doi.org/10.1214/10-AAP708

## Abstract

A series of recent articles introduced a method to construct stochastic partial differential equations (SPDEs) which are invariant with respect to the distribution of a given conditioned diffusion. These works are restricted to the case of elliptic diffusions where the drift has a gradient structure and the resulting SPDE is of second-order parabolic type. The present article extends this methodology to allow the construction of SPDEs which are invariant with respect to the distribution of a class of hypoelliptic diffusion processes, subject to a bridge conditioning, leading to SPDEs which are of fourth-order parabolic type. This allows the treatment of more realistic physical models, for example, one can use the resulting SPDE to study transitions between meta-stable states in mechanical systems with friction and noise. In this situation the restriction of the drift being a gradient can also be lifted.

[error in script] [error in script]Item Type: | Journal Article |
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Subjects: | Q Science > QA Mathematics |

Divisions: | Faculty of Science > Mathematics |

Library of Congress Subject Headings (LCSH): | Stochastic partial differential equations, Diffusion processes |

Journal or Publication Title: | The Annals of Applied Probability |

Publisher: | Institute of Mathematical Statistics |

ISSN: | 1050-5164 |

Date: | 2011 |

Volume: | Volume 21 |

Number: | Number 2 |

Page Range: | pp. 669-698 |

Identification Number: | 10.1214/10-AAP708 |

Status: | Peer Reviewed |

Publication Status: | Published |

Funder: | Engineering and Physical Sciences Research Council (EPSRC), European Research Council (ERC) |

Grant number: | EP/E002269/1 (EPSRC) |

References: | [1] APTE, A., HAIRER, M., STUART, A. M. and VOSS, J. (2007). Sampling the posterior: An approach to non-Gaussian data assimilation. Phys. D 230 50–64. MR2345202 [2] BLÖMKER, D.,MAIER-PAAPE, S. andWANNER, T. (2001). Spinodal decomposition for the Cahn–Hilliard–Cook equation. Comm. Math. Phys. 223 553–582. MR1866167 [3] DA PRATO, G. and DEBUSSCHE, A. (1996). Stochastic Cahn–Hilliard equation. Nonlinear Anal. 26 241–263. MR1359472 [4] DA PRATO, G. and ZABCZYK, J. (1992). Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44. Cambridge Univ. Press, Cambridge. MR1207136 [5] HAIRER, M. (2009). An introduction to stochastic PDEs. Available at http://arxiv.org/abs/0907. 4178. [6] HAIRER, M. (2010). Singular perturbations to semilinear stochastic heat equations. Preprint. Available at http://arxiv.org/abs/1002.3722. [7] HAIRER, M., STUART, A. M. and VOSS, J. (2007). Analysis of SPDEs arising in path sampling. II. The nonlinear case. Ann. Appl. Probab. 17 1657–1706. MR2358638 [8] HAIRER, M., STUART, A. M., VOSS, J. andWIBERG, P. (2005). Analysis of SPDEs arising in path sampling. I. The Gaussian case. Commun. Math. Sci. 3 587–603. MR2188686 [9] HAIRER, M., STUART, A. and VOSS, J. (2009). Sampling conditioned diffusions. In Trends in Stochastic Analysis. London Mathematical Society Lecture Note Series 353 159–185. Cambridge Univ. Press, Cambridge. MR2562154 [10] KATO, T. (1966). Perturbation Theory for Linear Operators. Springer, Berlin. |

URI: | http://wrap.warwick.ac.uk/id/eprint/41615 |

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