The Annals of Applied Probability

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

M. Hairer, A. M. Stuart, and J. Voss

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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.

Article information

Source
Ann. Appl. Probab. Volume 17, Number 5/6 (2007), 1657-1706.

Dates
First available in Project Euclid: 3 October 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1191419180

Digital Object Identifier
doi:10.1214/07-AAP441

Mathematical Reviews number (MathSciNet)
MR2358638

Zentralblatt MATH identifier
1140.60329

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60] 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx]

Keywords
Path sampling stochastic PDEs ergodicity

Citation

Hairer, M.; Stuart, A. M.; Voss, J. Analysis of SPDEs arising in path sampling part II: The nonlinear case. Ann. Appl. Probab. 17 (2007), no. 5/6, 1657--1706. doi:10.1214/07-AAP441. https://projecteuclid.org/euclid.aoap/1191419180.


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