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.

翻译：我们研究基于随机偏微分方程（SPDE）的变点模型，该模型对应于由加权拉普拉斯算子 $\Delta_\vartheta = \nabla\vartheta\nabla$ 控制的热方程，其中 $\vartheta=\vartheta(x)$ 为空间依赖的扩散系数。作为基本问题，考虑定义域 $(0,1)$ 内具有分片常数扩散系数（在未知点 $\tau$ 处存在跳跃）的情形。基于有限时间范围内空间分辨率为 $\delta$ 的局部观测解，我们构建了扩散系数值与变点的联合M估计量。变点估计量的收敛速度为 $\delta$，而扩散系数可达到 $\delta^{3/2}$ 的收敛速度。进一步地，当扩散参数已知且跳跃高度随空间分辨率趋零而消失时，我们推导了变点估计量的极限定理并识别其极限分布。在数学分析中，我们发展了具有不连续 $\vartheta$ 的SPDE的精确理解、解的二次泛函的紧浓度界以及经典M估计量的推广方法。