Change point estimation for a stochastic heat equation

TL;DR

This paper develops a statistical method to detect and estimate change points in the diffusivity of a stochastic heat equation, providing convergence rates and limit distributions under various conditions.

Contribution

It introduces a novel M-estimator for change points in a SPDE with space-dependent diffusivity and analyzes its asymptotic properties.

Findings

Change point estimator converges at rate δ.

Diffusivity constants can be estimated at rate δ^{3/2}.

Limit theorem derived for known diffusivity and vanishing jump height.

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 $\Delta_\vartheta = \nabla\vartheta\nabla$, 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 $\tau$. Based on local measurements of the solution in space with resolution $\delta$ 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 $\delta$, while the diffusivity constants can be recovered with convergence rate $\delta^{3/2}$. 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.