Séminaire Orléans
Gaussian approximations to diffusion bridgesHendrik Weber (Warwick)
Thursday 29 June 2017 14:00 - Orléans - Salle de Séminaire
Résumé :
Complex measures on infinite dimensional spaces arise in many applications, e.g. the Bayesian approach to inverse problems or as laws of diffusion bridges for models of molecular dynamics. We are concerned with the question of how to extract useful information from such a measure and propose to approximate it with a Gaussian measure viewing the mean of this Gaussian as the "most likely point" and the covariance operator as means to measure the fluctuations around this most likely point. We propose to measure the quality of the approximation in terms of the Kullback-Leibler divergence or relative entropy. After briefly discussing the existence of optimal Gaussian approximations I will focus the example of diffusion bridges for the over-damped Langevin model of molecular dynamics. We give an explicit expression for the Kullback-Leibler divergence in this case and study the low temperature limit via Gamma-convergence of the associated variational problem. We show that in this limit the optimal Gaussian means converge to the paths predicted by large deviation theory. The fluctuations have a natural interpretation as coming from a time-inhomogeneous Ornstein-Uhlenbeck process found by linearisation of the Langevin dynamics around the mean of the Gaussian.
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