Invariant measures of stochastic partial differential equations and conditioned diffusions

Reznikoff, Maria G.; Vanden‐Eijnden, Eric

doi:10.1016/j.crma.2004.12.025

Cited by 35 publications

(38 citation statements)

References 4 publications

(2 reference statements)

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“…The transition paths start at ∂A and terminate at ∂B, and hence they can be viewed as paths of a bridge process between s A and s B. In this perspective, our work is related to the conditional path sampling for SDEs studied in [SVW04,RVE05,HSVW05,HSV07]. In those works, stochastic partial differential equations were proposed to sample SDE paths with fixed end points.…”

Section: Introductionmentioning

confidence: 99%

Reactive trajectories and the transition path process

Nolen

2014

Probab. Theory Relat. Fields

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Abstract. We study the trajectories of a solution Xt to an Itô stochastic differential equation in R d , as the process passes between two disjoint open sets, A and B. These segments of the trajectory are called transition paths or reactive trajectories, and they are of interest in the study of chemical reactions and thermally activated processes. In that context, the sets A and B represent reactant and product states. Our main results describe the probability law of these transition paths in terms of a transition path process Yt, which is a strong solution to an auxiliary SDE having a singular drift term. We also show that statistics of the transition path process may be recovered by empirical sampling of the original process Xt. As an application of these ideas, we prove various representation formulas for statistics of the transition paths. We also identify the density and current of transition paths. Our results fit into the framework of the transition path theory by E and Vanden-Eijnden.

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Section: Introductionmentioning

confidence: 99%

Reactive trajectories and the transition path process

Nolen

2014

Probab. Theory Relat. Fields

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“…In that paper bridge processes are considered for SDEs whose drift is of gradient form. This idea is used to study the connection between invariant measures of SPDEs and bridge processes in [16] and its use in the context of sampling is studied further in [11] [11]. Many open problems remain, in particular for nonlinear drift fields.…”

Section: Discussionmentioning

confidence: 99%

“…For the bridge path problem (2.1) the choice of sign in the f �� term in the SPDE can be justified rigorously by appealing to the Girsanov Formula, and writing the density with respect to Brownian bridge (see [2], [11] and [16]). However we pursue a selfcontained numerical justification of the choice of sign.…”

Section: Resolving the Ambiguitymentioning

confidence: 99%

Conditional Path Sampling of SDEs and the Langevin MCMC Method

Stuart¹,

Voß²,

Wilberg³

2004

Communications in Mathematical Sciences

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“…In this sense our work complements existing work on the behaviour of MCMC methods in high dimensions (see [21] for a review). The methodology builds on recent results on the derivation of infinite-dimensional Langevin SDEs with a pre-specified equilibrium law corresponding to a conditioned diffusion process [11,12,13,18,24]. The Langevin equation is discretised to provide a proposal for a Metropolis-Hastings algorithm on the pathspace.…”

Section: Introductionmentioning

confidence: 99%

McMc Methods for Diffusion Bridges

et al. 2008

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We present and study a Langevin MCMC approach for sampling nonlinear diffusion bridges. The method is based on recent theory concerning stochastic partial differential equations (SPDEs) reversible with respect to the target bridge, derived by applying the Langevin idea on the bridge pathspace. In the process, a Random-Walk Metropolis algorithm and an Independence Sampler are also obtained. The novel algorithmic idea of the paper is that proposed moves for the MCMC algorithm are determined by discretising the SPDEs in the time direction using an implicit scheme, parameterised by θ ∈ [0, 1]. We show that the resulting infinite-dimensional MCMC sampler is well defined only if θ = 1/2, when the MCMC proposals have the correct quadratic variation. Previous Langevin-based MCMC methods used explicit schemes, corresponding to θ = 0. The significance of the choice θ = 1/2 is inherited by the finite-dimensional approximation of the algorithm used in practice. We present numerical results illustrating the phenomenon and the theory that explains it. Diffusion bridges (with additive noise) are representative of the family of laws defined as a change of measure from Gaussian distributions on arbitrary separable Hilbert spaces; the analysis in this paper can be readily extended to target laws from this family and an example from signal processing illustrates this fact.

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Invariant measures of stochastic partial differential equations and conditioned diffusions

Cited by 35 publications

References 4 publications

Reactive trajectories and the transition path process

Reactive trajectories and the transition path process

Conditional Path Sampling of SDEs and the Langevin MCMC Method

McMc Methods for Diffusion Bridges

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