There is a gap in the proof of the main theorem in the article [5] on optimal bounds for the Morse lemma in Gromov-hyperbolic spaces. We correct this gap, showing that the main theorem of [5] is true. We also describe a computer certification of this result.

The Morse lemma is fundamental in hyperbolic group theory. Using exponential contraction, we establish an upper bound for the Morse lemma that is optimal up to multiplicative constants, which we demonstrate by presenting a concrete example. We also prove an “anti” version of the Morse lemma. We introduce the notion of a geodesically rich space and consider applications of these results to the displacement of points under quasi-isometries that fix the ideal boundary.

The present work stemmed from the study of the problem of harmonic analysis on the infinite-dimensional unitary group U(∞). That problem consisted in the decomposition of a certain 4-parameter family of unitary representations, which replace the nonexisting two-sided regular representation (Olshanski [31]). The required decomposition is governed by certain probability measures on an infinite-dimensional space Ω, which is a dual object to U(∞). A way to describe those measures is to convert them into determinantal point processes on the real line; it turned out that their correlation kernels are computable in explicit form - they admit a closed expression in terms of the Gauss hypergeometric function F12 (Borodin and Olshanski [8]).In the present work we describe a (nonevident) q-discretization of the whole construction. This leads us to a new family of determinantal point processes. We reveal its connection with an exotic finite system of q-discrete orthogonal polynomials - the so-called pseudo big q-Jacobi polynomials. The new point processes live on a double q-lattice and we show that their correlation kernels are expressed through the basic hypergeometric function ϕ12.A crucial novel ingredient of our approach is an extended version G of the Gelfand-Tsetlin graph (the conventional graph describes the Gelfand-Tsetlin branching rule for irreducible representations of unitary groups). We find the q-boundary of G, thus extending previously known results (Gorin [17]). © 2015 Elsevier Inc.

We solve a problem about the orthogonal complement of the space spanned by restricted shifts of functions in L^2 [0,1] posed by M.Carlsson and C.Sundberg.

Convergence in variation of solutions of nonlinear Fokker-Planck-Kolmogorov equations to stationary measures has been studied.

We estimate the total variation and Kantorovich distances between transition probabilities of two diffusions with different diffusion matrices and drifts via a natural quadratic distance between the drifts and diffusion matrices. Applications to nonlinear Fokker–Planck–Kolmogorov equations, optimal control and mean field games are given.

Let X be a Stein manifold, and let Y be a closed complex submanifold of X. Denote by O(X) the algebra of holomorphic functions on X. We show that the weak (i.e., flat) homological dimension of O(Y) as a Fr'echet O(X)-module equals the codimension of Y in X. In the case where X and Y are of Liouville type, the same formula is proved for the projective homological dimension of O(Y) over O(X). On the other hand, we show that if X is of Liouville type and Y is hyperconvex, then the projective homological dimension of O(Y) over O(X) equals the dimension of X.

In this paper, we prove the well-posedness of a conditional McKean Lagrangian stochastic model, endowed with the specular boundary condition, and further the mean no-permeability condition, in a smooth bounded confinement domain . This result extends our previous work Bossy and Jabir 2011, where the confinement domain was the upper-half plane and where the specular boundary condition has been constructed owing to some well known results on the law of the passage times at zero of the Brownian primitive. The extension of the construction to more general confinement domain exhibits difficulties that we handle by combining stochastic calculus and the analysis of kinetic equations. As a prerequisite for the study of the nonlinear case, we construct a Langevin process confined in D and satisfying the specular boundary condition. We then use PDE techniques to construct the time-marginal densities of the nonlinear process from which we are able to exhibit the conditional McKean Lagrangian stochastic model.

Let γ be a Gaussian measure on a locally convex space X and H be the corresponding Cameron-Martin space. It has been recently shown by L. Ambrosio and A. Figalli that the linear first-order transportational PDE on X admits a weak solution under broad assumptions. Applying transportation of measures via triangular maps we prove a similar result for a large class of non-Gaussian probability measures ν on $\R^{\infty}$, under the main assumption of integrability of logarithmic derivativesof v. We also show uniqueness of the solution for a wide class of measures. This class includes uniformly log-concave Gibbs measures and certain product measures. measures.

We consider a quantitative form of the quasi-isometry problem. We discuss several arguments which lead us to a number of results and bounds of quasi-isometric distortion: comparison of volumes, connectivity, etc. Then we study the transport of Poincaré constants by quasi-isometries and we give sharp lower and upper bounds for the homotopy distortion growth for a certain class of hyperbolic metric spaces, a quotient of a Heintze group by . We also prove the linear distortion growth between hyperbolic space and a tree.

For a densely defined self-adjoint operator $\mathcal{H}$ in Hilbert space $\mathcal{F}$ the operator $\exp(-it\mathcal{H})$ is the evolution operator for the Schr\"odinger equation $i\psi'_t=\mathcal{H}\psi$, i.e. if $\psi(0,x)=\psi_0(x)$ then $\psi(t,x)=(\exp(-it\mathcal{H})\psi_0)(x)$ for $x\in Q.$ The space $\mathcal{F}$ here is the space of wave functions $\psi$ defined on an abstract space $Q$, the configuration space of a quantum system, and $\mathcal{H}$ is the Hamiltonian of the system. In this paper the operator $\exp(-it\mathcal{H})$ for all real values of $t$ is expressed in terms of the family of self-adjoint bounded operators $S(t), t\geq 0$, which is Chernoff-tangent to the operator $-\mathcal{H}$. One can take $S(t)=\exp(-t\mathcal{H})$, or use other, simple families $S$ that are listed in the paper. The main theorem is proven on the level of semigroups of bounded operators in $\mathcal{F}$ so it can be used in a wider context due to its generality. Two examples of application are provided.

We investigate the quasi-potential problem for the entropy cost functionals of non-entropic solutions to scalar conservation laws with smooth fluxes. We prove that the quasi-potentials coincide with the integral of a suitable Einstein entropy.

We consider a non-local operator Lα which is the sum of a fractional Laplacian △ α/2 , α ∈ (0, 1), plus a first order term which is measurable in the time variable and locally β-Hölder continuous in the space variables. Importantly, the fractional Laplacian ∆ α/2 does not dominate the first order term. We show that global parabolic Schauder estimates hold even in this case under the natural condition α + β > 1. Thus, the constant appearing in the Schauder estimates is in fact independent of the L ∞-norm of the first order term. In our approach we do not use the so-called extension property and we can replace △ α/2 with other operators of α-stable type which are somehow close, including the relativistic α-stable operator. Moreover, when α ∈ (1/2, 1), we can prove Schauder estimates for more general α-stable type operators like the singular cylindrical one, i.e., when △ α/2 is replaced by a sum of one dimensional fractional Laplacians d k=1 (∂ 2 x k x k) α/2.