An estimate for the average spectral measure of random band matrices

For a class of random band matrices of band width $W$, we prove regularity of the average spectral measure at scales $\epsilon \geq W^{-0.99}$, and find its asymptotics at these scales.Comment: 19 pp., revised versio

Identifying the factors that contribute to sustainable development of the national economy

The structural imbalance is the main problem hindering the development of the Russian national economy. It leads to significant difference in economic efficiency of various industrial sectors. Moreover, the structural imbalance adversely affects the interaction between industries and hampers to foster an enabling environment that would accelerate economic growth consistent with the principles of sustainable development. The right balance between economic sectors provides favorable conditions for a successful interaction between industries. The article suggests the methodology intended to identify the factors contributing to sustainable development of the national economy, to assess the status of the economy as well as to estimate the dynamics of economic growth. The methodology is a promising approach building a network of interactions between different industries to deepen the diversification of economic sectors. The authors propose a set of indicators - indicators of economic imbalances - that allow, based on primary statistical data, to quantitatively determine the degree of difference and the changing dynamics in economic, financial, technological and social characteristics of several economic sectors. The paper details the developed system of monitoring and multi-criteria evaluation of growth in several economic sectors. The system makes it possible to estimate key factors affecting sustainable development of the economy as well as to get the right diagnosis of economic processes that shape the sectoral structure of the Russian economy.peer-reviewe

Kernel estimates for nonautonomous Kolmogorov equations with potential term

Using time dependent Lyapunov functions, we prove pointwise upper bounds for the heat kernels of some nonautonomous Kolmogorov operators with possibly unbounded drift and diffusion coefficients and a possibly unbounded potential term

Construction of Lp\mathcal L^p-strong Feller Processes via Dirichlet Forms and Applications to Elliptic Diffusions

We provide a general construction scheme for $\mathcal L^p$-strong Feller processes on locally compact separable metric spaces. Starting from a regular Dirichlet form and specified regularity assumptions, we construct an associated semigroup and resolvents of kernels having the $\mathcal L^p$-strong Feller property. They allow us to construct a process which solves the corresponding martingale problem for all starting points from a known set, namely the set where the regularity assumptions hold. We apply this result to construct elliptic diffusions having locally Lipschitz matrix coefficients and singular drifts on general open sets with absorption at the boundary. In this application elliptic regularity results imply the desired regularity assumptions

Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below

This paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces (X,d,m). Our main results are: - A general study of the relations between the Hopf-Lax semigroup and Hamilton-Jacobi equation in metric spaces (X,d). - The equivalence of the heat flow in L^2(X,m) generated by a suitable Dirichlet energy and the Wasserstein gradient flow of the relative entropy functional in the space of probability measures P(X). - The proof of density in energy of Lipschitz functions in the Sobolev space W^{1,2}(X,d,m). - A fine and very general analysis of the differentiability properties of a large class of Kantorovich potentials, in connection with the optimal transport problem. Our results apply in particular to spaces satisfying Ricci curvature bounds in the sense of Lott & Villani [30] and Sturm [39,40], and require neither the doubling property nor the validity of the local Poincar\'e inequality.Comment: Minor typos corrected and many small improvements added. Lemma 2.4, Lemma 2.10, Prop. 5.7, Rem. 5.8, Thm. 6.3 added. Rem. 4.7, Prop. 4.8, Prop. 4.15 and Thm 4.16 augmented/reenforced. Proof of Thm. 4.16 and Lemma 9.6 simplified. Thm. 8.6 corrected. A simpler axiomatization of weak gradients, still equivalent to all other ones, has been propose