Noncommutative Anisotropic Diffusion in Hilbert Space. I. The Consistent A-Geometry, Mosco Stability, and the Weak Bridge

Abstract

This first part of the series builds the analytic layer of noncommutative anisotropic diffusion in a separable Hilbert space. Let $\hat{I}{¼}_0=\mathcal{N}(0,Q)$ be the reference Gaussian measure, with $Q\in L^1(\mathcal{H})$, and let $D(x)$ be a positive, state-dependent anisotropy. We do not assume that $[D(x),Q]=0$. Consequently, for the forward SDE with $\ddot{I}ƒ(x)=D(x{)}^{1/2}{Q}^{1/2}$, the correct energy form is given not by the expression $⟨D\nabla u,\nabla v⟩$ but by the consistent form $\hat{I}{“}_A(u,v)=⟨Q^{1/2}D(x{)}^{1/2}\nabla u,Q^{1/2}D(x{)}^{1/2}\nabla v⟩$. We prove closability of the form, well-posedness of the forward dynamics, Galerkin convergence, stability of the $A$-LSI under a Mosco limit, the chain rule for relative entropy, and a general weak-bridge theorem. The main result of Part~I is a functional-analytic theorem: if $A$-consistency, a uniform $A$-LSI, and representability of the right-hand side of the backward weak form in the negative energy space all hold, then a backward weak drift $v=\mathsf{A}\nabla \hat{I}¦$ exists and the basic entropy dissipation estimate holds. In addition, we single out a three-dimensional tensor class of anisotropies, formulate a condition for the absence of diffusion degeneracy, and obtain a rate estimate for the homogenization limit, first on cylindrical subspaces and then on compact-tail classes, which yields strong resolvent convergence and convergence of the forward SDEs. The statistical closure, an independent isotropic benchmark, and an approximation theorem for $A$-adapted networks are treated in Part~II.