Change point estimation for a stochastic heat equation
We study a change point model based on a stochastic partial differential equation (SPDE) corresponding to the heat equation governed by the weighted Laplacian Δϑ=∇ϑ∇, where ϑ=ϑ(x) is a space dependent diffusivity. As a basic problem, the domain (0,1) is considered with a piecewise constant diffusivi...
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| Main Authors: | , , |
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| Format: | Article (Journal) |
| Language: | English |
| Published: |
February 2026
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| In: |
The annals of statistics
Year: 2026, Volume: 54, Issue: 1, Pages: 277-299 |
| ISSN: | 2168-8966 |
| DOI: | 10.1214/25-AOS2567 |
| Online Access: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.1214/25-AOS2567 Verlag, lizenzpflichtig, Volltext: https://projecteuclid.org/journals/annals-of-statistics/volume-54/issue-1/Change-point-estimation-for-a-stochastic-heat-equation/10.1214/25-AOS2567.full 
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| Author Notes: | by Markus Reiß, Claudia Strauch and Lukas Trottner |
| Summary: | We study a change point model based on a stochastic partial differential equation (SPDE) corresponding to the heat equation governed by the weighted Laplacian Δϑ=∇ϑ∇, where ϑ=ϑ(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 with rate δ, while the diffusivity constants can be recovered with convergence rate δ3/2. Furthermore, when the diffusivity parameters are known and the jump height vanishes as the spatial resolution tends 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 ϑ, tight concentration bounds for quadratic functionals in the solution and a generalization of classical M-estimators are developed. |
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| Item Description: | Gesehen am 18.05.2026 |
| Physical Description: | Online Resource |
| ISSN: | 2168-8966 |
| DOI: | 10.1214/25-AOS2567 |