  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.
