Analysis of SPDEs arising in path sampling. Part I: The Gaussian case

By Martin Hairer, Andrew M. Stuart, Jochen Voss, P. Wiberg. Communications in Mathematical Sciences, 2005

99 citations recorded by the source corpus.

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 SDE used in finite dimensional sampling. Here the theory is developed for conditioned Gaussian processes for which the resulting SPDE is linear. Applications include the Kalman-Bucy filter/smoother. A companion paper studies the nonlinear case, building on the linear analysis provided here.

Subfields: Artificial Intelligence, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/be013fe58bc7466b0ba373ae6005c8e74ba0e0b6

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Citations in the corpus

Conditioning Gaussian measure on Hilbert space (2015, background): “3 of Hairer, Stuart, Voss, and Wiber, see also Stuart, characterizes the conditional measure through a measurable extension result of Dalecky and Fomin of an operator defined on the CameronMartin reproducing kernel Hilbert space [3-5].”
Pointwise eigenfunction estimates and intrinsic ultracontractivity-type properties of Feynman-Kac semigroups for a class of L\'{e}vy processes (2012, unknown): “The Annals of Probability 2015, Vol. 43, No. 3, 1350–1398 DOI: 10.1214/13-AOP897 © Institute of Mathematical Statistics, 2015”
The Bayesian Approach to Inverse Problems (2013, background): “tain (3.12) for the posterior measure, now with the new deﬁnition of G, and hence Φ. The Bayesian Approach to Inverse Problems 35 3.5. Bibliographic Notes. • Subsection 3.1. Theorem 3.1 is taken from [44] where it is used to compute expressions for the measure induced by various conditionings applie”
Inverse problems: A Bayesian perspective (2010, background): “…this invariance under µ. Applying this idea in the infinite-dimensional context is described in Apte, Hairer, Stuart and Voss (2007) and Beskos and Stuart (2009), based on the idea of Langevin equations in infinite dimensions (Hairer et al. 2005, Hairer et al. 2007, Hairer, Stuart and Voss 2009).”
Nonparametric estimation of diffusions: a differential equations approach (2012, background): “This is demonstrated in Hairer et al. (2005) where various conditionings are discussed including fixing the value of the process at the endpoints (bridge constraints) and conditioning upon another linear process whose drift is a linear function of the given SDE (the KalmanBucy filter).”
The Jain–Monrad criterion for rough paths and applications to random Fourier series and non-Markovian Hörmander theory (2013, background): “…been constructed one can view (4.1) as an evolution in a rough path space, a point of view which has proven extremely fruitful in solving new classes of until now ill-posed stochastic PDE [Hai11, HW11, Hai13], arising for example in path sampling problems for R d -valued SDE [HSVW05, HSV07, Hai11].”
Bayesian inverse problems for functions and applications to fluid mechanics (2009, methodology): “As stated here it is taken from [ 33 ], where it is used to prove a specific instance of”
Sampling the posterior: An approach to non-Gaussian data assimilation (2007, methodology): “The SPDE based approach to sampling continuous time paths was introduced in [38] and is subsequently analyzed in [15] and [16], building on analysis in [40].”