We obtain estimates for the weighted L^1-norm of the difference of two probability solutions to Kolmogorov equations in terms of the difference of the diffusion matrices and the drifts. Unlike the previously known results, our estimate does not involve Sobolev derivatives of solutions and coefficients. The diffusion matrices are supposed to be non-singular, bounded and satisfy the Dini mean oscillation condition.

This work presents a novel, fully Riemannian framework for Low-Rank Adaptation (LoRA) that geometrically treats low-rank adapters by optimizing them directly on the fixed-rank manifold. This formulation eliminates the parametrization ambiguity present in standard Euclidean optimizers. Our framework integrates three key components to achieve this: (1) we derive **Riemannion**, a new Riemannian optimizer on the fixed-rank matrix manifold that generalizes the recently proposed Muon optimizer; (2) we develop a Riemannian gradient-informed LoRA initialization, and (3) we provide an efficient implementation without prominent overhead that uses automatic differentiation to compute arising geometric operations while adhering to best practices in numerical linear algebra. Comprehensive experimental results on both LLM and diffusion model architectures demonstrate that our approach yields consistent and noticeable improvements in convergence speed and final task performance over both standard LoRA and its state-of-the-art modifications.

We study the asymptotic behaviour at infinity of solutions to the Cauchy problem for the Fokker–Planck–Kolmogorov equation. Our main result is an estimate for the difference of two solutions with different initial conditions in the weighted total variation norm in the case when the coefficients depend on time.

We consider Kolmogorov operators with constant diffusion matrices and linear drifts, i.e., Ornstein–Uhlenbeck operators, and show that all solutions to the corresponding stationary Fokker–Planck–Kolmogorov equations (including signed solutions) are invariant measures for the generated semigroups. This also gives a relatively explicit description of all solutions.

We obtain broad sufficient conditions for reconstructing the coefficients of a Kolmogorov operator by means of a solution to the Cauchy problem for the corresponding Fokker–Planck–Kolmogorov equation.

We discuss various definitions of Sobolev and Besov classes on infinite-dimensional spaces, give a survey of the results on coincidence of some of these classes, and obtain a number of new results.

. Nonlinear Fokker-Planck-Kolmogorov equations are investigated. Sufficient conditions are obtained for the existence and uniqueness of a nonnegative solution with a prescribed value of the integral. Convergence of solutions for the Cauchy problem to a solution of the stationary equation is shown. An important distinction from the known results is a very general form of the nonlinearity, which makes it possible to consider simultaneously a local and nonlocal dependence of coefficients on solutions.

We prove pseudocompactness of a Tychonoff space X and the space 𝒫(X) of Radon probability measures on it with the weak topology under the condition that the Stone–Čech compactification of the space 𝒫(X) is homeomorphic to the space 𝒫(β X) of Radon probability measures on the Stone–Čech compactification of the space X.

This paper gives a survey of recent investigations on nonlinear Fokker-Planck-Kolmogorov equations of elliptic and parabolic types and contains a number of new results. We discuss in detail the problems of existence and uniqueness of solutions, various estimates of solutions, connections with linear equations, and the convergence of solutions of parabolic equations to stationary solutions. Bibliography: 116 items.

В работе сравнивается стоун-чеховская компактификация $\beta \mathcal{P}(X)$ пространства радоновских вероятностных мер $\mathcal{P}(X)$ на тихоновском пространстве $X$, наделенного слабой топологией, с пространством $\mathcal{P}(\beta X)$ радоновских вероятностных мер на стоун-чеховской компактификации $\beta X$ самого пространства $X$. Показано, что для некомпактного метрического пространства $X$ компактификация $\beta \mathcal{P}(X)$ не совпадает с $\mathcal{P}(\beta X)$. Обсуждается случай более общих тихоновских пространств, а также случай компактификации Самюэля, для которой совпадение имеет место.

We consider optimal transportation of measures on metric and topological spaces in the case where the cost function and marginal distributions depend on a parameter with values in a metric space. The Hausdorff distance between the sets of probability measures with prescribed marginals is estimated in terms of the distances between the marginals themselves. This estimate is used to prove the continuity of the cost of optimal transportation with respect to the parameter in the case of the continuous dependence of the cost function and marginal distributions on this parameter. Existence of approximate optimal plans continuous with respect to the parameter is established. It is shown that the optimal plan is continuous with respect to the parameter in the case of uniqueness. However, examples are constructed when there is no continuous selection of optimal plans. Another application of the estimate for the Hausdorff distance concerns discrete approximations of the transportation problem. Finally, a general result on the convergence of Monge optimal mappings is proved. Bibliography: 46 titles.

We study metrizability of compact sets in spaces of Radon measures with the weak topology. It is shown that if all compacta in a given completely regular topological space are metrizable, then every uniformly tight compact set in the space of Radon measures on this space is also metrizable. It is proved that the property that compact sets of measures on a given space are metrizable is preserved for products of this space with spaces that can be embedded into separable metric spaces. In addition, we construct a Radon probability measure on the space of Radon probability measures on a completely regular space such that its barycenter is not a Radon measure.

В статье дан обзор нескольких направлений исследований, связанных с работами А. Н. Колмогорова о параболических и эллиптических уравнениях Фоккера-Планка-Колмогорова для переходных и стационарных вероятностей диффузионных процессов. Приведены основные результаты о существовании решений, единственности, свойствах плотностей решений. Упомянуты открытые вопросы в этой области.

In this note, we study the Kantorovich problem of optimal transportation of measures on metric spaces in the case where the cost function and marginal distributions depend on a parameter from a metric space. It is shown that the Hausdorff distance between the sets of probability measures with given marginals can be estimated by the distances between the marginals. As a corollary, it is proved that the cost of optimal transportation is continuous with respect to the parameter if the cost function and marginal distributions are continuous in this parameter.

Рассматривается оптимальная транспортировка мер на метрических и топологических пространствах в случае, когда функция стоимости и маргинальные распределения зависят от параметра со значениями в метрическом пространстве. Расстояние Хаусдорфа между множествами вероятностных мер с заданными проекциями оценивается через расстояния между самими проекциями. Эта оценка используется для доказательства непрерывности стоимости оптимальной транспортировки относительно параметра в случае непрерывной зависимости функции стоимости и маргинальных распределений от этого параметра. Установлено существование приближенных оптимальных планов, непрерывных относительно параметра. Показано, что оптимальный план непрерывен по параметру в случае единственности. Однако построены примеры, когда не существует непрерывного выбора оптимальных планов. Другое применение оценки для расстояния Хаусдорфа связано с дискретными приближениями транспортных задач. Наконец, доказан общий результат о сходимости оптимальных отображений Монжа. Библиография: 46 названий.

We study Kantorovich type optimal transportation problems that combine two new generalizations introduced and investigated recently by several authors: nonlinearity of cost functions, including dependence on conditional measures of transport plans, and constraints on densities of transport plans. New existence results are established and some counter-examples are constructed.

The paper gives a survey of several directions of research connected with the works of A.N. Kolmogorov on parabolic and elliptic Fokker--Planck--Kolmogorov equations for transition and stationary probabilities of diffusion processes. We present the fundamental results on existence of solutions, their uniqueness, and the properties of solution densities. Open questions in this area are mentioned.

We study qualitative properties of solutions to double divergence form elliptic equations (or stationary Kolmogorov equations) on R-d: It is shown that the Harnack inequality holds for nonnegative solutions if the diffusion matrix A is nondegenerate and satisfies the Dini mean oscillation condition and the drift coefficient b is locally integrable to some power p > d. We establish new estimates for the L-p-norms of solutions and obtain a generalization of the known theorem of Hasminskii on the existence of a probability solution to the stationary Kolmogorov equation to the case where the matrix A satisfies Dini's condition or belongs to the class VMO. These results are based on a new analytic version of Zvonkin's transform of the drift coefficient.

— In this note we develop a new analytic version of Zvonkin’s transform of the drift coefficient of a stationary Kolmogorov equation and apply this transform to derive the Harnack inequality for nonnegative solutions in the case where the diffusion matrix is not locally Sobolev. We also obtain a generalization of the known theorem of Hasminskii on existence of a probability solution to the stationary Kolmogorov equation.

We study two topologies $\tau_{KR}$ and $\tau_K$ on the space of measures on a completely regular space generated by Kantorovich--Rubinshtein and Kantorovich seminorms analogous to their classical norms in the case of a metric space. The Kantorovich--Rubinshtein topology $\tau_{KR}$ coincides with the weak topology on nonnegative measures and on bounded uniformly tight sets of measures. A~sufficient condition is given for the compactness in the Kantorovich topology. We show that for logarithmically concave measures and stable measures weak convergence implies convergence in the Kantorovich topology. We also obtain an efficiently verified condition for convergence of the barycenters of Radon measures from a sequence or net weakly converging on a locally convex space. As an application it is shown that for weakly convergent logarithmically concave measures and stable measures convergence of their barycenters holds without additional conditions. The same is true for measures given by polynomial densities of a fixed degree with respect to logarithmically concave measures.