Identifiability and Stability of Generative Drifting with Companion-Elliptic Kernel Families

Abstract: This paper analyzes identifiability and stability for the drifting field underlying distributional matching in the Generative Drifting framework of Deng et al. First, we introduce the class of companion-elliptic kernels, which includes the Laplace kernel and is characterized by a second-order elliptic coupling between each kernel $κ$ in this class and its companion function $η$. For each kernel in this class and each pair of Borel probability measures, we prove that the drifting field vanishes if and only if the two probability measures are equal. We further show that this class consists precisely of Gaussian kernels and Matérn kernels with $ν\ge 1/2$. Second, by constructing counterexamples, we exhibit sequences for which mass escapes to infinity while the field tends to zero; in particular, control of the field norm alone does not guarantee weak convergence. Nevertheless, we prove that the only possible mode of failure is confined to the one-dimensional ray ${c\,p:0\le c\le 1}$. Consequently, weak convergence can be restored by imposing an asymptotic lower bound on the intrinsic overlap scalar, a linear observable defined by the kernel and the target measure.