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Analysis of spdes arising in path sampling part II: The nonlinear case
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Hairer, Martin, Stuart, A. M. and Voss, J. (2007) Analysis of spdes arising in path sampling part II: The nonlinear case. Annals of Applied Probability, Vol.17 (No.5-6). pp. 1657-1706. doi:10.1214/07-AAP441 ISSN 1050-5164.
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Official URL: http://dx.doi.org/10.1214/07-AAP441
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: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | Annals of Applied Probability | ||||
Publisher: | Institute of Mathematical Statistics | ||||
ISSN: | 1050-5164 | ||||
Official Date: | October 2007 | ||||
Dates: |
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Volume: | Vol.17 | ||||
Number: | No.5-6 | ||||
Number of Pages: | 50 | ||||
Page Range: | pp. 1657-1706 | ||||
DOI: | 10.1214/07-AAP441 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Open Access (Creative Commons) |
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