Multiple change-point inference for piecewise constant diffusivity

Consider a one-dimensional stochastic heat equation with multiple change points in piecewise-constant diffusivity and local spatial measurements at resolution $\delta\to 0$. Let $(\Omega,\mathcal F,(\mathcal F_t)_{0\le t\le T},\mathbb P)$ be a filtered probability space, $T\in(0,\infty)$ fixed, and let $W$ be a cylindrical Wiener process on $L^2(0,1)$. Consider

with piecewise-constant diffusivity

where $m\in\mathbb N_0$, jump locations $(\tau_j)_{j=1}^m$, and levels $(\vartheta_j)_{j=1}^{m+1}$ are unknown, subject to $0<\underline\vartheta\le\vartheta_j\le\overline\vartheta<\infty$.

For $\delta=1/n\to0$, let $x_k=k\delta$ and $K\in H^2(\mathbb R)$ be compactly supported, with $K_{\delta,k}(x)=\delta^{-1/2}K((x-x_k)/\delta)$. In line with the single-jump setup, use both local measurements

for admissible $k$ (with support inside $(0,1)$), where in 1D $\Delta=\partial_{xx}$.