The problem of data assimilation can be formulated as the sampling of solutions of time-dependent equations given noisy observations. We introduce a stochastic PDE based approach to sampling paths of differential equations, conditional on observations. The stochastic SPDEs are derived by generalising the Langevin MCMC method to infinite dimensions. Various applications are described, including sampling paths subject to two end-point conditions (bridges), nonlinear filter/smoothers and a toy model for data assimilation.
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