Change point estimation for a stochastic heat equation
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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_ϑ = ∇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 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 ϑ, tight concentration bounds for quadratic functionals in the solution, and a generalisation of classical M-estimators are developed.
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