Bernstein-von Mises theorems for time evolution equations
<|reference_start|>Bernstein-von Mises theorems for time evolution equations: We consider a class of infinite-dimensional dynamical systems driven by non-linear parabolic partial differential equations with initial condition $\theta$ modelled by a Gaussian process `prior' probability measure. Given discrete samples of the state of the system evolving in space-time, one obtains updated `posterior' measures on a function space containing all possible trajectories. We give a general set of conditions under which these non-Gaussian posterior distributions are approximated, in Wasserstein distance for the supremum-norm metric, by the law of a Gaussian random function. We demonstrate the applicability of our results to periodic non-linear reaction diffusion equations \begin{align*} \frac{\partial}{\partial t} u - \Delta u &= f(u) \\ u(0) &= \theta \end{align*} where $f$ is any smooth and compactly supported reaction function. In this case the limiting Gaussian measure can be characterised as the solution of a time-dependent Schr\"odinger equation with `rough' Gaussian initial conditions whose covariance operator we describe.<|reference_end|>
@article{nickl2024bernstein-von,
    title={Bernstein-von Mises theorems for time evolution equations},
    author={Richard Nickl},
    journal={arXiv preprint arXiv:2407.14781},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.14781},
    primaryClass={math.ST cs.NA math.AP math.NA math.PR stat.TH}
}