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Analysis of SPDEs arising in path sampling part II: The nonlinear case

Hairer, M. and Stuart, A. M. and Voss, J. (2007) Analysis of SPDEs arising in path sampling part II: The nonlinear case. Annals of Applied Probability, 17 (5/6). pp. 1657-1706. ISSN 1050-5164. doi:10.1214/07-AAP441. https://resolver.caltech.edu/CaltechAUTHORS:20170613-080132206

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Abstract

In many applications, it is important to be able to sample paths of SDEs conditional on observations of various kinds. This paper studies SPDEs which solve such sampling problems. The SPDE may be viewed as an infinite-dimensional analogue of the Langevin equation used in finite-dimensional sampling. In this paper, conditioned nonlinear SDEs, leading to nonlinear SPDEs for the sampling, are studied. In addition, a class of preconditioned SPDEs is studied, found by applying a Green’s operator to the SPDE in such a way that the invariant measure remains unchanged; such infinite dimensional evolution equations are important for the development of practical algorithms for sampling infinite dimensional problems. The resulting SPDEs provide several significant challenges in the theory of SPDEs. The two primary ones are the presence of nonlinear boundary conditions, involving first order derivatives, and a loss of the smoothing property in the case of the pre-conditioned SPDEs. These challenges are overcome and a theory of existence, uniqueness and ergodicity is developed in sufficient generality to subsume the sampling problems of interest to us. The Gaussian theory developed in Part I of this paper considers Gaussian SDEs, leading to linear Gaussian SPDEs for sampling. This Gaussian theory is used as the basis for deriving nonlinear SPDEs which affect the desired sampling in the nonlinear case, via a change of measure.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1214/07-AAP441DOIArticle
http://projecteuclid.org/euclid.aoap/1191419180PublisherArticle
ORCID:
AuthorORCID
Voss, J.0000-0001-7740-8811
Additional Information:© 2007 Institute of Mathematical Statistics. Received May 2006; revised March 2007. Supported by EPSRC Grant EP/E002269/1. Supported by EPSRC and ONR Grant N00014-05-1-0791.
Funders:
Funding AgencyGrant Number
Engineering and Physical Sciences Research Council (EPSRC)EP/E002269/1
Office of Naval Research (ONR)N00014-05-1-0791
Subject Keywords:Path sampling, stochastic PDEs, ergodicity
Other Numbering System:
Other Numbering System NameOther Numbering System ID
Andrew StuartJ70
Issue or Number:5/6
Classification Code:AMS 2000 subject classifications: 60H15, 60G35
DOI:10.1214/07-AAP441
Record Number:CaltechAUTHORS:20170613-080132206
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20170613-080132206
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:78140
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:13 Jun 2017 16:18
Last Modified:15 Nov 2021 17:37

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