Bernstein-von Mises theorems for time evolution equations
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866917414480904192 |
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| author | Nickl, Richard |
| author_facet | Nickl, Richard |
| contents | We consider a class of infinite-dimensional dynamical systems driven by non-linear parabolic partial differential equations with initial condition $θ$ 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 - Δu &= f(u) \\ u(0) &= θ\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 Schrödinger equation with `rough' Gaussian initial conditions whose covariance operator we describe. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_14781 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bernstein-von Mises theorems for time evolution equations Nickl, Richard Statistics Theory Numerical Analysis Analysis of PDEs Probability We consider a class of infinite-dimensional dynamical systems driven by non-linear parabolic partial differential equations with initial condition $θ$ 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 - Δu &= f(u) \\ u(0) &= θ\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 Schrödinger equation with `rough' Gaussian initial conditions whose covariance operator we describe. |
| title | Bernstein-von Mises theorems for time evolution equations |
| topic | Statistics Theory Numerical Analysis Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2407.14781 |