arXiv subjects

Alexander Bendikov

Publications and source records attributed to Alexander Bendikov.

At least 19 recordsLinked to original sources

arXiv2025

Riesz transform, function spaces and their applications on infinite dimensional compact groups

On a compact connected group $G$, consider the infinitesimal generator $-L$ of a central symmetric Gaussian convolution semigroup $(\mu_t)_{t>0}$. We establish several regularity results of the solution to the Poisson equation $LU=F$, both in strong and weak senses. To this end, we introduce two classes of Lipschitz spaces for $1\le p\le \infty$: $\Lambda_{\theta}^p$, defined via the associated Markov semigroup, and $\mathrm L_{\theta}^p$, defined via the intrinsic distance. In the strong sense, we prove a priori Sobolev regularity and Lipschitz regularity in the class of $\Lambda_{\theta}^p$ space. In the distributional sense, we further show local regularity in the class of $\mathrm L_{\theta}^{\infty}$ space. These results require some strong assumptions on $-L$. Our main techniques build on the differentiability of the associated semigroup, explicit dimension-free $L^p$ ($1<p<\infty$) boundedness of first and second order Riesz transforms, and a comparison between the two Lipschitz norms.

math.AP↗

arXiv2023

Random walks on a finite group and the Frobenius-Schur theorem

We consider random walk on a finite group $G$ as follows. We can consider $G$ as a group of substitutions. Randomly (i.e. with probability $U(g)=|G|^{-1}$ ) we choose a substitution $g \in G$ and execute it twice in a row, i.e. execute a substitution $g^2 \in G$ . Then the set of squares of elements of the group $G$ be a carrier of a probability $P(g)=\frac{r(g)}{|G|}\ (g \in G)$ , where $r(g)$ is a number of elements $h \in G$ such that $h^2 = g$ . Using well-known Frobenius-Schur theorem we find speed of convergence of $n$-fold convolution of $P$ to the uniform probability $U$ and conditions for the convergence.

math.RT↗

arXiv2020

Hierarchical Schrödinger-type operators: the case of potentials with local singularities

The goal of this paper is twofold. We prove that the operator $H=L+V$ , a perturbation of the Taibleson-Vladimirov multiplier $L=\mathfrak{D}^α$ by a potential $V(x)=b\left\Vert x\right\Vert ^{-α},$ $b\geq b_{\ast},$ is essentially self-adjoint and non-negative definite (the critical value $b_{\ast}$ depends on $α$ and will be specified later). While the operator $H$ is non-negative definite the potential $V(x)$ may well take negative values, e.g. $b_{\ast}<0$ for all $0<α<1$. The equation $Hu=v$ admiits a Green function $g_{H}(x,y)$, the integral kernel of the operator $H^{-1}$. We obtain sharp lower- and upper bounds on the ratio of the functions $g_{H}(x,y)$ and $g_{L}(x,y)$. Examples illustrate our exposition.

math.SP↗

arXiv2020

On the spectrum of the hierarchical Schrödinger type operators

The goal of this paper is the spectral analysis of the Schrödinger type operator $H=L+V$, the perturbation of the Taibleson-Vladimirov multiplier $L=\mathfrak{D}^α$ by a potential $V$. Assuming that $V$ belongs to a certain class of potentials we show that the discrete part of the spectrum of $H$ may contain negative energies, it also appears in the spectral gaps of $L$. We will split the spectrum of $H$ in two parts: high energy part containing eigenvalues which correspond to the eigenfunctions located on the support of the potential $V,$ and low energy part which lies in the spectrum of certain bounded Schrödinger-type operator acting on the Dyson hierarchical lattice. We pay special attention to the class of sparse potentials. In this case we obtain precise spectral asymptotics for $H$ provided the sequence of distances between locations tends to infinity fast enough. We also obtain certain results concerning localization theory for $H$ subject to (non-ergodic) random potential $V$. Examples illustrate our approach.

math.SP↗

arXiv2019

Oscillating heat kernels on ultrametric spaces

Let $(X,d)$ be a proper ultrametric space. Given a measure $m$ on $X$ and a function $B \mapsto C(B)$ defined on the collection of all non-singleton balls $B$ of $X$, we consider the associated hierarchical Laplacian $L=L_{C}\,$. The operator $L$ acts in $\mathcal{L}^{2}(X,m),$ is essentially self-adjoint and has a pure point spectrum. It admits a continuous heat kernel $\mathfrak{p}(t,x,y)$ with respect to $m$. We consider the case when $X$ has a transitive group of isometries under which the operator $L$ is invariant and study the asymptotic behaviour of the function $t\mapsto \mathfrak{p}(t,x,x)=\mathfrak{p}(t)$. It is completely monotone, but does not vary regularly. When $X=\mathbb{Q}_{p}\,$, the ring of $p$-adic numbers, and $L=\mathcal{D}^α $, the operator of \ fractional derivative of order $α,$ we show that $\mathfrak{p}(t)=t^{-1/α}\mathcal{A}% (\log_{p}t)$, where $\mathcal{A}(τ)$ is a continuous non-constant $α$-periodic function. We also study asymptotic behaviour of $\min\mathcal{A}$ and $\max\mathcal{A}$ as the space parameter $p$ tends to $\infty$. When $X=S_{\infty}\,$, the infinite symmetric group, and $L$ is a hierarchical Laplacian with metric structure analogous to $\mathcal{D}^α,$ we show that, contrary to the previous case, the completely monotone function $\mathfrak{p}(t)$ oscillates between two functions $ψ(t)$ and $Ψ(t)$ such that $ψ(t)/Ψ(t)\to 0$ as $t \to \infty\,$.

math.PR↗

arXiv2018

On the spectrum of the hierarchical Schrödinger operator

The goal of this paper is the spectral analysis of the Schrödinger operator $H=L+V$ , the perturbation of the Taibleson-Vladimirov multiplier $L=\mathcal{D}^α$ by a potential $V$. Assuming that $V$ belonges to a class of fast decreasing potentials we show that the discrete part of the spectrum of $H$ may contain negative energies, it also appears in the spectral gaps of $L$. We will split the spectrum of $H$ in two parts: high energy part containing eigenvalues which correspond to the eigenfunctions located on the support of the potential $V,$ and low energy part which lies in the spectrum of certain bounded Schrödinger operator acting on the Dyson hierarchical lattice. The spectral asymptotics \ strictly depend on the transience versus recurrence properties of the underlying hierarchical random walk. In the transient case we will prove results in spirit of CLR theory, for the recurrent case we will provide Bargmann's type asymptotics.

math.FA↗

arXiv2017

Limit theorems for random walks

We consider a random walk $S_τ$ which is obtained from the simple random walk $S$ by a discrete time version of Bochner's subordination. We prove that under certain conditions on the subordinator $τ$ appropriately scaled random walk $S_τ$ converges in the Skorohod space to the symmetric $α$-stable process $B^α$. We also prove asymptotic formula for the transition function of $S_τ$ similar to the Pólya's asymptotic formula for $B^α$.

math.PR↗

arXiv2017

On the rate of convergence in the central limit theorem for hierarchical Laplacian

Let $(X,d)$ be a proper ultrametric space. Given a measure $m$ on $X$ and a function $C(B)$ defined on the set of all non-singleton balls $B$ we consider the hierarchical Laplacian $L=L_{C}$. Choosing a sequence $\{\varepsilon (B)\}$ of i.i.d. random variables we define the perturbed function $C(B,ω)$ and the perturbed hierarchical Laplacian $L^{ω}=L_{C(ω)}.$ We study the arithmetic means $\overline{λ}(ω)$ of the $L^{ω}$-eigenvalues. Under some mild assumptions the normalized arithmetic means $\big( \overline{λ}-\mathbb{E}\overline{λ}\big) /σ\big( \overline{λ}\big) $ converge in law to the standard normal distribution. In this note we study convergence in the total variation distance and estimate the rate of convergence.

math.PR↗

arXiv2016

On massive sets for subordinated random walks

We study massive (reccurent) sets with respect to a certain random walk $S_α$ defined on the integer lattice $\mathbb{Z} ^d$, $d=1,2$. Our random walk $S_α$ is obtained from the simple random walk $S$ on $\mathbb{Z} ^d$ by the procedure of discrete subordination. $S_α$ can be regarded as a discrete space and time counterpart of the symmetric $α$-stable Lévy process in $\mathbb{R}^d$. In the case $d=1$ we show that some remarkable proper subsets of $\mathbb{Z}$ , e.g. the set $\mathcal{P}$ of primes, are massive whereas some proper subsets of $\mathcal{P}$ such as Leitmann primes $\mathcal{P}_h$ are massive/non-massive depending on the function $h$. Our results can be regarded as an extension of the results of McKean (1961) about massiveness of the set of primes for the simple random walk in $\mathbb{Z}^3$. In the case $d=2$ we study massiveness of thorns and their proper subsets.

math.PR↗

arXiv2016

Alpha-stable random walk has massive thorns

We introduce and study a class of random walks defined on the integer lattice $ \mathbb{Z} ^d$ -- a discrete space and time counterpart of the symmetric $α$-stable process in $\mathbb{R} ^d$. When $0< α<2$ any coordinate axis in $\mathbb{Z} ^d$, $d\geq 3$, is a non-massive set whereas any cone is massive. We provide a necessary and sufficient condition for the thorn to be a massive set.

math.PR↗

arXiv2016

Brownian motion on treebolic space: positive harmonic functions

Treebolic space HT(q,p) is a key example of a strip complex in the sense of Bendikov, Saloff-Coste, Salvatori, and Woess [Adv. Math. 226 (2011), 992-1055]. It is an analog of the Sol geometry, namely, it is a horocylic product of the hyperbolic upper half plane with a "stretching" parameter q and the homogeneous tree T with vertex degree p+1 < 2, the latter seen as a one-complex. In a previous paper [arXiv:1212.6151, Rev. Mat. Iberoamericana, in print] we have explored the metric structure and isometry group of that space. Relying on the analysis on strip complexes, a family of natural Laplacians with "vertical drift" and the escape to infinity of the associated Brownian motion were considered. Here, we undertake a potential theoretic study, investigating the positive harmonic functions associated with those Laplacians. The methodological subtleties stem from the singularites of treebolic space at its bifurcation lines. We first study harmonic functions on simply connected sets with "rectangular" shape that are unions of strips. We derive a Poisson representation and obtain a solution of the Dirichlet problem on sets of that type. This provides properties of the density of the induced random walk on the collection of all bifuraction lines. Subsequently, we prove that each positive harmonic function with respect to that random walk has a unique extension which is harmonic with respect to the Laplacian on treebolic space. Finally, we derive a decomposition theorem for positive harmonic functions on the entire space that leads to a characterisation of the weak Liouville property. We determine all minimal harmonic functions in those cases where our Laplacian arises from lifting a (smooth) hyperbolic Laplacian with drift from the hyperbolic plane to treebolic space.

math.PR↗

arXiv2015

Poisson statistics of eigenvalues in the hierarchical Dyson model

Let $(X,d)$ be a locally compact separable ultrametric space. Given a measure $m$ on $X$ and a function $C$ defined on the set $\mathcal{B}$ of all balls $B\subset X$ we consider the hierarchical Laplacian $L=L_{C}$. The operator $L$ acts in $L^{2}(X,m)$, is essentially self-adjoint, and has a purely point spectrum. Choosing a family $\{\varepsilon(B)\}_{B\in \mathcal{B}}$ of i.i.d. random variables, we define the perturbed function $\mathcal{C}(B)=C(B)(1+\varepsilon(B))$ and the perturbed hierarchical Laplacian $\mathcal{L}=L_{\mathcal{C}}$. All outcomes of the perturbed operator $\mathcal{L}$ are hierarchical Laplacians. In particular they all have purely point spectrum. We study the empirical point process $M$ defined in terms of $\mathcal{L}$-eigenvalues. Under some natural assumptions $M$ can be approximated by a Poisson point process. Using a result of Arratia, Goldstein, and Gordon based on the Chen-Stein method, we provide total variation convergence rates for the Poisson approximation. We apply our theory to random perturbations of the operator $\mathfrak{D}^{α}$, the $p$-adic fractional derivative of order $α>0$. This operator, related to the concept of $p$-adic Quantum Mechanics, is a hierarchical Laplacian which acts in $L^{2}(X,m)$ where $X=\mathbb{Q}_{p}$ is the field of $p$-adic numbers and $m$ is Haar measure. It is translation invariant and the set $\mathsf{Spec}(\mathfrak{D}^{α})$ consists of eigenvalues $p^{αk}$, $k\in \mathbb{Z}$, each of which has infinite multiplicity.

math.PR↗

arXiv2014

Brownian motion on treebolic space: escape to infinity

Treebolic space is an analog of the Sol geometry, namely, it is the horocylic product of the hyperbolic upper half plane H and the homogeneous tree T with degree p+1 > 2, the latter seen as a one-complex. Let h be the Busemann function of T with respect to a fixed boundary point. Then for real q > 1 and integer p > 1, treebolic space HT(q,p) consists of all pairs (z=x+i y,w) in H x T with h(w) = log_{q} y. It can also be obtained by glueing together horziontal strips of H in a tree-like fashion. We explain the geometry and metric of HT and exhibit a locally compact group of isometries (a horocyclic product of affine groups) that acts with compact quotient. When q=p, that group contains the amenable Baumslag-Solitar group BS(p)$ as a co-compact lattice, while when q and p are distinct, it is amenable, but non-unimodular. HT(q,p) is a key example of a strip complex in the sense of our previous paper in Advances in Mathematics 226 (2011) 992-1055. Relying on the analysis of strip complexes developed in that paper, we consider a family of natural Laplacians with "vertical drift" and describe the associated Brownian motion. The main difficulties come from the singularites which treebolic space (as any strip complex) has along its bifurcation lines. In this first part, we obtain the rate of escape and a central limit theorem, and describe how Brownian motion converges to the natural geometric boundary at infinity. Forthcoming work will be dedicated to positive harmonic functions.

math.PR↗

arXiv2014

Isotropic Markov semigroups on ultra-metric spaces

Let (X,d) be a locally compact separable ultra-metric space. Given a reference measure μ on X and a step length distribution on the non-negative reals, we construct a symmetric Markov semigroup P^t acting in L^2(X,μ). We study the corresponding Markov process. We obtain upper and lower bounds of its transition density and its Green function, give a transience criterion, estimate its moments and describe the Markov generator and its spectrum, which is pure point. In the particular case when X is the field of p-adic numbers, our construction recovers fractional derivative and the Taibleson Laplacian (spectral multiplier), and we can also apply our theory to the study of the Vladimirov Laplacian which is closely related to the concept of p-adic Quantum Mechanics. Even in this well established setting, several of our results are new. We also elaborate the relation between our processes and Kigami's jump processes on the boundary of a tree which are induced by a random walk. In conclusion, we provide examples illustrating the interplay between the fractional derivatives and random walks.

math.PR↗

arXiv2013

On the spectrum of the hierarchical Laplacian

Let $(X,d)$ be a locally compact separable ultrametric space. We assume that $(X,d)$ is proper, that is, any closed ball $B$ in $X$ is a compact set. Given a measure $m$ on $X$ and a function $C(B)$ defined on the set of balls (the choice function), we define the hierarchical Laplacian $L_C$ which is closely related to the concept of the hierarchical lattice of F.J. Dyson. $L_C$ is a non-negative definite, self-adjoint operator in $L^2(X,m)$. We address in this paper to the following question: How general can be the spectrum $\mathsf{Spec}(L_C)$ as a subset of the non-negative reals? When $(X,d)$ is compact, $\mathsf{Spec}(L_C)$ is an increasing sequence of eigenvalues of finite multiplicity which contains $0$. Assuming that $(X,d)$ is not compact we show that, under some natural conditions concerning the structure of the hierarchical lattice (= the tree of $d$-balls), any given closed subset $S$ of $[0,\infty)$, which contains $0$ as an accumulation point and is unbounded if $X$ is non-discrete, may appear as $\mathsf{Spec}(L_C)$ for some appropriately chosen function $C(B)$. The operator $-L_C$ extends to $L^q(X,m)$, $0 < q < \infty$, as Markov generator and its spectrum does not depend on $q$. As an example, we consider the operator $\mathfrak{D}^α$ of fractional derivative defined on the field $\mathbb{Q}_p$ of $p$-adic numbers.

math.PR↗

arXiv2012

Random walks driven by low moment measures

We study the decay of convolution powers of probability measures without second moment but satisfying some weaker finite moment condition. For any locally compact unimodular group G and any positive function $ρ:G \rightarrow [0,+\infty]$, we introduce a function $Φ_{G,ρ}$ which describes the fastest possible decay of $n \mapsto ϕ^{(2n)}(e)$ when ϕis a symmetric continuous probability density such that $\intρϕ$ is finite. We estimate $Φ_{G,ρ}$ for a variety of groups G and functions ρ. When ρis of the form $ρ=ρ\circ δ$ with $ρ:[0,+\infty) \rightarrow [0,+\infty)$, a fixed increasing function, and $δ:G \rightarrow [0,+\infty)$, a natural word length measuring the distance to the identity element in G, $Φ_{G,ρ}$ can be thought of as a group invariant.

math.PR↗

arXiv2011

Spectral properties of a class of random walks on locally finite groups

We study some spectral properties of random walks on infinite countable amenable groups with an emphasis on locally finite groups, e.g. the infinite symmetric group. On locally finite groups, the random walks under consideration are driven by infinite divisible distributions. This allows us to embed our random walks into continuous time Lévy processes whose heat kernels have shapes similar to the ones of alpha-stable processes. We obtain examples of fast/slow decays of return probabilities, a recurrence criterion, exact values and estimates of isospectral profiles and spectral distributions, formulae and estimates for the escape rates and for heat kernels.

math.SP↗

arXiv2009

Spectral distribution and $L^2$-isoperimetric profile of Laplace operators on groups

We give a formula relating the $L^2$-isoperimetric profile to the spectral distribution of the Laplace operator associated to a finitely generated group $Γ$ or a Riemannian manifold with a cocompact, isometric $Γ$-action. As a consequence, we can apply techniques from geometric group theory to estimate the spectral distribution of the Laplace operator in terms of the growth and the Følner's function of the group, generalizing previous estimates by Gromov and Shubin. This leads, in particular, to sharp estimates of the spectral distributions for several classes of solvable groups. Furthermore, we prove the asymptotic invariance of the spectral distribution under changes of measures with finite second moment.

math.GR↗