Higher-dimensional change-set (interface) estimation for discontinuous diffusivity

Let $d\ge 2$, let $D\subset\mathbb R^d$ be a bounded domain with $C^2$ boundary, fix $T>0$, and fix known constants $\vartheta_1,\vartheta_2$ with $0<\vartheta_1\neq \vartheta_2<\infty$. For an unknown measurable set $G\subset D$, define the diffusivity

Consider the stochastic parabolic equation (in weak/mild sense)

with boundary condition $u_t(x)=0$ for $x\in\partial D$, initial condition $u_0\in L^2(D)$ known, $W$ a cylindrical Wiener process on $L^2(D)$, and $B$ a known linear noise operator chosen so the equation is well posed.

This setup follows Reiß et al. (2025).

For each spatial resolution $\delta>0$, assume one observes the locally averaged field

where $K_\delta(z)=\delta^{-d}K(z/\delta)$ for a known compactly supported kernel $K$ with $\int_{\mathbb R^d}K=1$, and $D_\delta=\{x\in D:\operatorname{dist}(x,\partial D)>\delta\,\operatorname{rad}(\operatorname{supp}K)\}$. Thus $\delta\to 0$ corresponds to increasingly local spatial measurements over fixed time horizon $[0,T]$.