arXiv subjects

E. Yu. Shchetinin

Publications and source records attributed to E. Yu. Shchetinin.

3 recordsLinked to original sources

arXiv2026

Noncommutative Anisotropic Diffusion in Hilbert Space. II. Global Closure of the Logarithmic Gradient, Lower Bounds, and Nanosystem Applications

This second part of the series develops the statistical and applied layer of the theory built in Part I [1]. Unlike current Hilbert diffusion models [2-5], we focus not only on well-posedness of the infinite-dimensional generative dynamics, but on the noncommutative $A$-geometry, explicit entropy constants, and checkable lower bounds. The analytic estimate of Part I reduces stability of the backward evolution to control of a validation error $\mathcal{E}_{val}$. We prove three results. First, we construct a cylindrically weak denoising label for the infinite-dimensional score field, consistent with the exact logarithmic gradient. Second, the local parametric closure is replaced by a global nonparametric score closure, its complexity controlled via the Dudley entropy integral and uniform empirical bounds in $L^2(ν_*;A)$. Third, for the trace-smoothed nonlinear class we construct minimax bounds by the Le Cam-Assouad method, showing the root statistical rate cannot be improved without additional quadratic structure. The final section applies the theory to anisotropic diffusion in a nanosystem model and checks the constants $c_A,C_A,C_{LSI}^A$ independently of any smallness condition, establishing explicit accuracy orders: the parametric trace-smoothed minimax score risk is $p/M$, while the uniform error of the risk functional is $\sqrt{p/M}$, showing the statistical plateau of order $p/M$ is not a proof artifact. The applied layer closes with an independent analytic benchmark for the isotropic case, a comparison with classical cell homogenization, and an approximation theorem for smooth logarithmic gradients by $A$-adapted spectral networks.

math.AP↗

arXiv2026

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

This first part of the series builds the analytic layer of noncommutative anisotropic diffusion in a separable Hilbert space. Let $μ_0=\mathcal{N}(0,Q)$ 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 $σ(x)=D(x)^{1/2}Q^{1/2}$, the correct energy form is given not by the expression $\langle D\nabla u,\nabla v\rangle$ but by the consistent form $Γ_A(u,v)= \langle Q^{1/2}D(x)^{1/2}\nabla u, Q^{1/2}D(x)^{1/2}\nabla v\rangle$. 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 $v=\mathsf{A}\nablaΦ$ 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.

math.AP↗

arXiv2026

Entropy Geometry and Augmented Mobility for Reactive Maxwell--Stefan Membrane Transport with Finite Occupancy

We study a reactive Maxwell--Stefan-type membrane transport system under a finite-occupancy constraint with explicit vacancies. The admissible state space is $\mathcal{D} = \{u \in (0,1)^n : \varrho(u) := \sum_{i=1}^n u_i < 1\}$, where the vacancy fraction $1-\varrho(u)$ represents the local free volume. This bounded-occupancy geometry induces a Boltzmann--Fermi entropy and a global parametrization by entropy variables. The associated mobility is assumed to split into a composition channel, corresponding to redistribution at fixed total occupancy, and a mass channel, corresponding to variation of the filling fraction. The main structural difficulty is that the unaugmented mobility may lose coercivity in the mass channel. We show that a single rank-one augmentation of the form $γ_0 \mathbf{1} \otimes \mathbf{1}$ restores full coercivity while leaving the composition block unchanged. On this basis, we prove four results: a quantitative channel-wise coercivity estimate for the augmented mobility; global existence of entropy weak solutions via an implicit Rothe scheme in entropy variables; a weak--strong stability estimate in relative entropy with uniqueness in the strong class; and convergence of a fully implicit finite-volume approximation that preserves the bounded-occupancy structure and satisfies a discrete entropy inequality.

math.AP↗