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Sergio Albeverio

Publications and source records attributed to Sergio Albeverio.

At least 55 records · Page 3Linked to original sources
arXiv2015

On metric, dynamical, probabilistic and fractal phenomena connected with the second Ostrogradsky expansion

We consider the second Ostrogradsky expansion from the number theory, probability theory, dynamical systems and fractal geometry points of view, and establish several new phenomena connected with this expansion. First of all we prove the singularity of the random second Ostrogradsky expansion. Secondly we study properties of the symbolic dynamical system generated by the natural one-sided shift-transformation $T$ on the second Ostrogradsky expansion. It is shown, in particular, that there are no probability measures which are simultaneously invariant and ergodic (w.r.t. $T$) and absolutely continuous (w.r.t. Lebesgue measure). So, the classical ergodic approach to the development of the metric theory is not applicable for the second Ostrogradsky expansion. We develop instead the metric and dimensional theories for this expansion using probabilistic methods. We show, in particular, that for Lebesgue almost all real numbers any digit $i$ from the alphabet $A= \mathbb{N} $ appears only finitely many times in the difference-version of the second Ostrogradsky expansion, and the set of all reals with bounded digits of this expansion is of zero Hausdorff dimension. Finally, we compare metric, probabilistic and dimensional theories for the second Ostrogradsky expansions with the corresponding theories for the Ostrogradsky-Pierce expansion and for the continued fractions expansion.

math.PR↗
arXiv2015

Ostrogradsky-Sierpiński-Pierce expansion: dynamical systems, probability theory and fractal geometry points of view

We establish several new probabilistic, dynamical, dimensional and number theoretical phenomena connected with Ostrogradsky-Sierpiński-Pierce expansion. First of all, we develop metric, ergodic and dimensional theories of the Ostrogradsky-Sierpiński-Pierce expansion. In particular, it is proven that for Lebesgue almost all real numbers any digit $i$ from the alphabet $A= \mathbb{N} $ appears only finitely many times in the difference-version of the Ostrogradsky-Sierpiński-Pierce expansion. Properties of the symbolic dynamical system generated by a shift-transformation $T$ on the difference-version of the Ostrogradsky-Sierpiński-Pierce expansion are also studied in details. It is shown that there are no probability measures which are invariant and ergodic (w.r.t. $T$) and absolutely continuous (w.r.t. Lebesgue measure). Thirdly, we study properties of random variables $η$ with independent identically distributed differences of the Ostrogradsky-Sierpiński-Pierce expansion. Necessary and sufficient conditions for $η$ to be discrete resp. singularly continuous are found. We prove that $η$ can not be absolutely continuously distributed.

math.PR↗
arXiv2015

On Bernoulli convolutions generated by second Ostrogradsky series and their fine fractal properties

We study properties of Bernoulli convolutions generated by the second Ostrogradsky series, i.e., probability distributions of random variables \begin{equation} ξ= \sum_{k=1}^\infty \frac{(-1)^{k+1}ξ_k}{q_k}, \end{equation} where $q_k$ is a sequence of positive integers with $q_{k+1}\geq q_k(q_k+1)$, and $\{ξ_k\}$ are independent random variables taking the values $0$ and $1$ with probabilities $p_{0k}$ and $p_{1k}$ respectively. We prove that $ξ$ has an anomalously fractal Cantor type singular distribution ($\dim_H (S_ξ)=0$) whose Fourier-Stieltjes transform does not tend to zero at infinity. We also develop different approaches how to estimate a level of "irregularity" of probability distributions whose spectra are of zero Hausdorff dimension. Using generalizations of the Hausdorff measures and dimensions, fine fractal properties of the probability measure $μ_ξ$ are studied in details. Conditions for the Hausdorff--Billingsley dimension preservation on the spectrum by its probability distribution function are also obtained.

math.PR↗
arXiv2014

A unified approach to infinite dimensional integration

An approach to infinite dimensional integration which unifies the case of oscillatory integrals and the case of probabilistic type integrals is presented. It provides a truly infinite dimensional construction of integrals as linear functionals, as much as possible independent of the underlying topological and measure theoretical structure. Various applications are given, including, next to Schrödinger and diffusion equations, also higher order hyperbolic and parabolic equations.

math.PR↗
arXiv2014

Bounds on variation of the spectrum and spectral subspaces of a few-body Hamiltonian

We overview the recent results on the shift of the spectrum and norm bounds for variation of spectral subspaces of a Hermitian operator under an additive Hermitian perturbation. Along with the known results, we present a new subspace variation bound for the generic off-diagonal subspace perturbation problem. We also demonstrate how some of the abstract results may work for few-body Hamiltonians.

math-ph↗
arXiv2014

A class of Lévy driven SDEs and their explicit invariant measures

We describe a class of explicit invariant measures for both finite and infinite dimensional Stochastic Differential Equations (SDE) driven by Lévy noise. We first discuss in details the finite dimensional case with a linear, resp. non linear, drift. In particular, we exhibit a class of such SDEs for which the invariant measures are given in explicit form, coherently in all dimensions. We then indicate how to relate them to invariant measures for SDEs on separable Hilbert spaces.

math.PR↗
arXiv2014

Quasi regular Dirichlet forms and the stochastic quantization problem

After recalling basic features of the theory of symmetric quasi regular Dirichlet forms we show how by applying it to the stochastic quantization equation, with Gaussian space-time noise, one obtains weak solutions in a large invariant set. Subsequently, we discuss non symmetric quasi regular Dirichlet forms and show in particular by two simple examples in infinite dimensions that infinitesimal invariance, does not imply global invariance. We also present a simple example of non-Markov uniqueness in infinite dimensions.

math.PR↗
arXiv2013

Explicit invariant measures for infinite dimensional SDE driven by Lévy noise with dissipative nonlinear drift I

We stu\dd y a class of nonlinear stochastic partial differential equations with dissipative nonlinear drift, driven by Lévy noise. Our work is divided in two parts. In the present part I we first define a Hilbert-Banach setting in which we can prove existence and uniqueness of solutions under general assumptions on the drift and the Lévy noise. We then prove a decomposition of the solution process in a stationary component and a component which vanishes asymptotically for large times in the $L^p-$sense, $p\geq1$. The law of the stationary component is identified with the unique invariant probability measure of the process. In part II we will exhibit the invariant measure as the limit of explicit invariant measures for finite dimensional approximants.

math.PR↗
arXiv2013

Klein-Dirac Quadric and Multidimensional Toda Lattice via Pseudo-positive Moment Problem

In 1974 Jürgen Moser has shown that the classical Moment Problem plays a fundamental role for the theory of completely integrable systems, by proving that the simplest case of the finite Toda lattice is described exhaustively in its terms. In particular, the Jacobi matrix defined by the Flaschka variables corresponds to the Schrödinger operator related to the Korteweg-de Vries operator. In the present paper we use a recent breakthrough in the area of the multidimensional Moment Problem in order to develop a multidimensional generalization of the finite Toda lattice. Our construction is based on the notion of pseudo-positive measure and the notion of multidimensional Markov-Stieltjes transform naturally defined on the Klein-Dirac quadric. An important consequence of the properties of the Markov-Stieltjes transform is a new method for summation of multidimensional divergent series on the Klein-Dirac quadric.

math.AP↗
arXiv2013

On the Hausdorff dimension faithfulness and the Cantor series expansion

We study families $Φ$ of coverings which are faithful for the Hausdorff dimension calculation on a given set $E$ (i. e., special relatively narrow families of coverings leading to the classical Hausdorff dimension of an arbitrary subset of $E$) and which are natural generalizations of comparable net-coverings. They are shown to be very useful for the determination or estimation of the Hausdorff dimension of sets and probability measures. We give general necessary and sufficient conditions for a covering family to be faithful and new techniques for proving faithfulness/non-faithfulness for the family of cylinders generated by expansions of real numbers. Motivated by applications in the multifractal analysis of infinite Bernoulli convolutions, we study in details the Cantor series expansion and prove necessary and sufficient conditions for the corresponding net-coverings to be faithful. To the best of our knowledge this is the first known sharp condition of the faithfulness for a class of covering families containing both faithful and non-faithful ones. Applying our results, we characterize fine fractal properties of probability measures with independent digits of the Cantor series expansion and show that a class of faithful net-coverings essentially wider that the class of comparable ones. We construct, in particular, rather simple examples of faithful families $\mathcal{A}$ of net-coverings which are "extremely non-comparable" to the Hausdorff measure.

math.PR↗
arXiv2013

On Titchmarsh-Weyl functions and eigenfunction expansions of first-order symmetric systems

We study general (not necessarily Hamiltonian) first-order symmetric systems $J y'(t)-B(t)y(t)=\D(t) f(t)$ on an interval $\cI=[a,b> $ with the regular endpoint $a$. It is assumed that the deficiency indices $n_\pm(\Tmi)$ of the minimal relation $\Tmi$ in $\LI$ satisfy $n_-(\Tmi)\leq n_+(\Tmi)$. By using a Nevanlinna boundary parameter $τ=τ(ł)$ at the singular endpoint $b$ we define self-adjoint and $ł$-depending Nevanlinna boundary conditions which are analogs of separated self-adjoint boundary conditions for Hamiltonian systems. With a boundary value problem involving such conditions we associate the $m$-function $m(\cd)$, which is an analog of the Titchmarsh-Weyl coefficient for the Hamiltonian system. By using $m$-function we obtain the Fourier transform $V:\LI\to L^2(\Si)$ with the spectral function $\Si(\cd)$ of the minimally possible dimension. If $V$ is an isometry, then the (exit space) self-adjoint extension $\wt T$ of $\Tmi$ induced by the boundary problem is unitarily equivalent to the multiplication operator in $L^2(\Si)$; hence the spectrum of $\wt T$ is defined by the spectral function $\Si(\cd)$. We show that all the objects of the boundary problem are determined by the parameter $τ$, which enables us to parametrize all spectral function $\Si(\cd) $ immediately in terms of $τ$. Similar results for various classes of boundary problems were obtained by Kac and Krein, Fulton, Hinton and Shaw and other authors.

math.FA↗
arXiv2012

Small noise asymptotic expansions for stochastic PDE's driven by dissipative nonlinearity and Lévy noise

We study a reaction-diffusion evolution equation perturbed by a space-time Lévy noise. The associated Kolmogorov operator is the sum of the infinitesimal generator of a $C_0$-semigroup of strictly negative type acting in a Hilbert space and a nonlinear term which has at most polynomial growth, is non necessarily Lipschitz and is such that the whole system is dissipative. The corresponding Itô stochastic equation describes a process on a Hilbert space with dissi- pative nonlinear, non globally Lipschitz drift and a Lévy noise. Under smoothness assumptions on the non-linearity, asymptotics to all orders in a small parameter in front of the noise are given, with detailed estimates on the remainders. Applications to nonlinear SPDEs with a linear term in the drift given by a Laplacian in a bounded domain are included. As a particular case we provide the small noise asymptotic expansions for the SPDE equations of FitzHugh Nagumo type in neurobiology with external impulsive noise.

math.PR↗
arXiv2012

Schrödinger operators with concentric $δ$-shells

We investigate the spectral properties of the Schrödinger operators in $L^2(\mathbb{R}^n)$ with a singular interaction supported by an infinite family of concentric spheres $$ \mathbf{H}_{R,α}=-Δ+\sum_{k=1}^\inftyα_kδ(|x|-r_k). $$ We obtain necessary and sufficient conditions for the operator $\mathbf{H}_{R,α}$ to be self-adjoint, lower-semibounded. Also we investigate the spectral types of $\mathbf{H}_{R,α}$.

math-ph↗
arXiv2012

Diffusion processes in thin tubes and their limits on graphs

The present paper is concerned with diffusion processes running on tubular domains with conditions on nonreaching the boundary, respectively, reflecting at the boundary, and corresponding processes in the limit where the thin tubular domains are shrinking to graphs. The methods we use are probabilistic ones. For shrinking, we use big potentials, respectively, reflection on the boundary of tubes. We show that there exists a unique limit process, and we characterize the limit process by a second-order differential generator acting on functions defined on the limit graph, with Kirchhoff boundary conditions at the vertices.

math.PR↗
arXiv2012

Sharpening the norm bound in the subspace perturbation theory

Let A be a self-adjoint operator on a Hilbert space H. Assume that σ is an isolated component of the spectrum of A, i.e. dist(σ,Σ)=d>0 where Σ=spec(A)\σ. Suppose that V is a bounded self-adjoint operator on H such that ||V|| R^+, that is essentially stronger than the previously known estimates for ||P-Q||. In particular, the bound obtained ensures that ||P-Q||<1 and, thus, that the spectral subspaces Ran(P) and Ran(Q) are in the acute-angle case whenever ||V||<cd with c=0.454169... (the precise expression for c is also given). Our proof of the above results is based on using the triangle inequality for the maximal angle between subspaces and on employing the a priori generic \sin2θestimate for the variation of a spectral subspace. As an example, the boundedly perturbed quantum harmonic oscillator is discussed.

math.SP↗
arXiv2012

The Riemann zeta in terms of the dilogarithm

We give a representation of the classical Riemann $ζ$-function in the half plane $\Re s>0$ in terms of a Mellin transform involving the real part of the dilogarithm function with an argument on the unit circle (associated Clausen $Gl_2$-function). We also derive corresponding representations involving the derivatives of the $Gl_2$-function. A generalized symmetrized Müntz-type formula is also derived. For a special choice of test functions it connects to our integral representation of the $ζ$-function, providing also a computation of a concrete Mellin transform. Certain formulae involving series of zeta functions and gamma functions are also derived.

math.NT↗
arXiv2012

The a priori Tan Theta Theorem for spectral subspaces

Let A be a self-adjoint operator on a separable Hilbert space H. Assume that the spectrum of A consists of two disjoint components s_0 and s_1 such that the set s_0 lies in a finite gap of the set s_1. Let V be a bounded self-adjoint operator on H off-diagonal with respect to the partition spec(A)=s_0 \cup s_1. It is known that if ||V||<\sqrt{2}d, where d=\dist(s_0,s_1), then the perturbation V does not close the gaps between s_0 and s_1 and the spectrum of the perturbed operator L=A+V consists of two isolated components s'_0 and s'_1 grown from s_0 and s_1, respectively. Furthermore, it is known that if V satisfies the stronger bound ||V||< d then the following sharp norm estimate holds: ||E_L(s'_0)-E_A(s_0)|| \leq sin(arctan(||V||/d)), where E_A(s_0) and E_L(s'_0) are the spectral projections of A and L associated with the spectral sets s_0 and s'_0, respectively. In the present work we prove that this estimate remains valid and sharp also for d \leq ||V||< \sqrt{2}d, which completely settles the issue.

math.SP↗