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arXiv · 2307.10960

Change point estimation for a stochastic heat equation

Markus Reiß·Claudia Strauch·Lukas Trottner
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Abstract

We study a change point model based on a stochastic partial differential equation (SPDE) corresponding to the heat equation governed by the weighted Laplacian $Δ_\vartheta = \nabla\vartheta\nabla$, where $\vartheta=\vartheta(x)$ is a space-dependent diffusivity. As a basic problem the domain $(0,1)$ is considered with a piecewise constant diffusivity with a jump at an unknown point $τ$. Based on local measurements of the solution in space with resolution $δ$ over a finite time horizon, we construct a simultaneous M-estimator for the diffusivity values and the change point. The change point estimator converges at rate $δ$, while the diffusivity constants can be recovered with convergence rate $δ^{3/2}$. Moreover, when the diffusivity parameters are known and the jump height vanishes with the spatial resolution tending to zero, we derive a limit theorem for the change point estimator and identify the limiting distribution. For the mathematical analysis, a precise understanding of the SPDE with discontinuous $\vartheta$, tight concentration bounds for quadratic functionals in the solution, and a generalisation of classical M-estimators are developed.

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Markus Reiß, Claudia Strauch, Lukas Trottner. 2024-10-27. Change point estimation for a stochastic heat equation. https://doi.org/10.1214/25-aos2567

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