arXiv subjects

Dmitri Finkelshtein

Publications and source records attributed to Dmitri Finkelshtein.

At least 19 recordsLinked to original sources

arXiv2025

Lie structures of the group of Sheffer operators

Let $\Phi$ be an (LB)-space over $\mathbb F=\mathbb R$ or $\mathbb C$, and let $\Phi'$ be the dual space of~$\Phi$. We study the set $\mathbb S(\Phi)$ of Sheffer operators acting in polynomials on $\Phi'$. We prove that $\mathbb S(\Phi)$ is a group for the usual product of operators. We equip $\mathbb S(\Phi)$ with a natural topology which makes $\mathbb S(\Phi)$ into an infinite-dimensional manifold with a global parametrization. We show that $\mathbb S(\Phi)$ is an infinite-dimensional, regular Lie group, and provide an explicit description of the Lie algebra of $\mathbb S(\Phi)$, including an explicit form of the Lie bracket on it. Our main results are new even in the one-dimensional case, $\Phi=\mathbb{F}$. Furthermore, our results lead to improved understanding of the Lie algebra of the Riordan group, cf.\ Cheon, Luz\'on, Mor\'on, Prieto-Martinez, {\it Adv. Math.} 319 (2017) 522--566.

math.FA↗

arXiv2025

Stochastic dynamics on evolving geometric graphs

We consider an infinite locally finite system (configuration) $\gamma$ of particles distributed over a Euclidean space $X$. Each particle located at $x\in X$ carries an internal parameter (mark, or ``spin'') $\sigma_{x}\in S=\mathbb{R}.$ Such collections of particles form the space of marked configurations $\Gamma(X,S)$. We construct the following stochastic dynamics in $\Gamma(X,S)$: while the configuration $\gamma$ of particle positions performs a random evolution, the corresponding marks interact with each other and perform a coupled infinite-dimensional diffusion. The study of a spin dynamics on a fixed configuration $\gamma$ was initiated by Daletskii and Finkelshtein, J. Stat. Phys. 122 ( 2018), and continued by Chargaziya and Daletskii, J. Math. Phys. 66 (2025), and is based on the generalisation of the Ovsjannikov method. In the present paper, the underlying configuration evolves according to a Birth-and-Death process. We prove the existence and uniqueness of such dynamics and show that it forms a c\`adl\`ag process in $\Gamma(X,S)$.

math.PR↗

arXiv2024

Entropies of the Poisson distribution as functions of intensity: "normal" and "anomalous" behavior

The paper extends the analysis of the entropies of the Poisson distribution with parameter $\lambda$. It demonstrates that the Tsallis and Sharma-Mittal entropies exhibit monotonic behavior with respect to $\lambda$, whereas two generalized forms of the R\'enyi entropy may exhibit "anomalous" (non-monotonic) behavior. Additionally, we examine the asymptotic behavior of the entropies as $\lambda \to \infty$ and provide both lower and upper bounds for them.

math.PR↗

arXiv2024

Finite difference calculus in the continuum

The paper describes known and new results about finite difference calculus on configuration spaces. We describe finite difference geometry on configuration spaces, connect finite difference operators with cannonical commutation relations, find explicit form for certain finite difference Markov generators on configuration spaces, and describe spaces of Newton series defined over the configuration spaces.

math-ph↗

arXiv2023

Analysis on the cone of discrete Radon measures

We study analysis on the cone of discrete Radon measures over a locally compact Polish space $X$. We discuss probability measures on the cone and the corresponding correlation measures and correlation functions on the sub-cone of finite discrete Radon measures over $X$. For this, we consider on the cone an analogue of the harmonic analysis on the configuration space developed in [12]. We also study elements of the difference calculus on the cone: we introduce discrete birth-and-death gradients and study the corresponding Dirichlet forms; finally, we discuss a system of polynomial functions on the cone which satisfy the binomial identity.

math-ph↗

arXiv2021

Stirling operators in spatial combinatorics

We define and study a spatial (infinite-dimensional) counterpart of Stirling numbers. In classical combinatorics, the Pochhammer symbol $(m)_n$ can be extended from a natural number $m\in\mathbb N$ to the falling factorials $(z)_n=z(z-1)\dotsm (z-n+1)$ of an argument $z$ from $\mathbb F=\mathbb R\text{ or }\mathbb C$, and Stirling numbers of the first and second kinds are the coefficients of the expansions of $(z)_n$ through $z^k$, $k\leq n$ and vice versa. When taking into account spatial positions of elements in a locally compact Polish space $X$, we replace $\mathbb N$ by the space of configurations -- discrete Radon measures $γ=\sum_iδ_{x_i}$ on $X$, where $δ_{x_i}$ is the Dirac measure with mass at $x_i$.The spatial falling factorials $(γ)_n:=\sum_{i_1}\sum_{i_2\ne i_1}\dotsm\sum_{i_n\ne i_1,\dots, i_n\ne i_{n-1}}δ_{(x_{i_1},x_{i_2},\dots,x_{i_n})}$ can be naturally extended to mappings $M^{(1)}(X)\niω\mapsto (ω)_n\in M^{(n)}(X)$, where $M^{(n)}(X)$ denotes the space of $\mathbb F$-valued, symmetric (for $n\ge2$) Radon measures on $X^n$. There is a natural duality between $M^{(n)}(X)$ and the space $\mathcal {CF}^{(n)}(X)$ of $\mathbb F$-valued, symmetric continuous functions on $X^n$ with compact support. The Stirling operators of the first and second kind, $\mathbf{s}(n,k)$ and $\mathbf{S}(n,k)$, are linear operators, acting between spaces $\mathcal {CF}^{(n)}(X)$ and $\mathcal {CF}^{(k)}(X)$ such that their dual operators, acting from $M^{(k)}(X)$ into $M^{(n)}(X)$, satisfy $(ω)_n=\sum_{k=1}^n\mathbf{s}(n,k)^*ω^{\otimes k}$ and $ω^{\otimes n}=\sum_{k=1}^n\mathbf{S}(n,k)^*(ω)_k$, respectively. We derive combinatorial properties of the Stirling operators, present their connections with a generalization of the Poisson point process and with the Wick ordering under the canonical commutation relations.

math.CO↗

arXiv2021

Markov dynamics on the cone of discrete Radon measures

We start with a brief overview of the known facts about the spaces of discrete Radon measures those may be considered as generalizations of configuration spaces. Then we study three Markov dynamics on the spaces of discrete Radon measures: analogues of the contact model, of the Bolker--Dieckmann--Law--Pacala model, and of the Glauber-type dynamics. We show how the results obtained previously for the configuration spaces can be modified for the case of the spaces of discrete Radon measures.

math-ph↗

arXiv2020

Extinction threshold in spatial stochastic logistic model: Space homogeneous case

We consider the extinction regime in the spatial stochastic logistic model in $\mathbb{R}^d$ (a.k.a. Bolker--Pacala--Dieckmann--Law model of spatial populations) using the first-order perturbation beyond the mean-field equation. In space homogeneous case (i.e. when the density is non-spatial and the covariance is translation invariant), we show that the perturbation converges as time tends to infinity; that yields the first-order approximation for the stationary density. Next, we study the critical mortality---the smallest constant death rate which ensures the extinction of the population---as a function of the mean-field scaling parameter $\varepsilon>0$. We find the leading term of the asymptotic expansion (as $\varepsilon\to0$) of the critical mortality which is apparently different for the cases $d\geq3$, $d=2$, and $d=1$.

math.PR↗

arXiv2019

Sheffer homeomorphisms of spaces of entire functions in infinite dimensional analysis

For certain Sheffer sequences $(s_n)_{n=0}^\infty$ on $\mathbb C$, Grabiner (1988) proved that, for each $α\in[0,1]$, the corresponding Sheffer operator $z^n\mapsto s_n(z)$ extends to a linear self-homeomorphism of $\mathcal E^α_{\mathrm{min}}(\mathbb C)$, the Fréchet topological space of entire functions of order at most $α$ and minimal type (when the order is equal to $α>0$). In particular, every function $f\in \mathcal E^α_{\mathrm{min}}(\mathbb C)$ admits a unique decomposition $f(z)=\sum_{n=0}^\infty c_n s_n(z)$, and the series converges in the topology of $\mathcal E^α_{\mathrm{min}}(\mathbb C)$. Within the context of a complex nuclear space $Φ$ and its dual space $Φ'$, in this work we generalize Grabiner's result to the case of Sheffer operators corresponding to Sheffer sequences on $Φ'$. In particular, for $Φ=Φ'=\mathbb C^n$ with $n\ge2$, we obtain the multivariate extension of Grabiner's theorem. Furthermore, for an Appell sequence on a general co-nuclear space $Φ'$, we find a sufficient condition for the corresponding Sheffer operator to extend to a linear self-homeomorphism of $\mathcal E^α_{\mathrm{min}}(Φ')$ when $α>1$. The latter result is new even in the one-dimensional case.

math.FA↗

arXiv2019

Fecundity regulation in a spatial birth-and-death process

We study a Markov birth-and-death process on a space of locally finite configurations, which describes an ecological model with a density dependent fecundity regulation mechanism. We establish existence and uniqueness of this process and analyze its properties. In particular, we show global time-space boundedness of the population density and, using a constructed Foster-Lyapunov-type function, we study return times to certain level sets of tempered configurations. We find also sufficient conditions that the degenerate invariant distribution is unique for the considered process.

math.PR↗

arXiv2018

Accelerated front propagation for monostable equations with nonlocal diffusion: Multidimensional case

We describe acceleration of the front propagation for solutions to a class of monostable nonlinear equations with a nonlocal diffusion in $\mathbb{R}^d$, $d\geq1$. We show that the acceleration takes place if either the diffusion kernel or the initial condition has 'regular' heavy tails in $\mathbb{R}^d$ (in particular, decays slower than exponentially). Under general assumptions which can be verified for particular models, we present sharp estimates for the time-space zone which separates the region of convergence to the unstable zero solution with the region of convergence to the stable positive constant solution. We show the variety of different possible rates of the propagation starting from a little bit faster than a linear one up to the exponential rate. The paper generalizes to the case $d>1$ our results for the case $d=1$ obtained early in https://dx.doi.org/10.1080/00036811.2017.1400537 .

math.AP↗

arXiv2018

Doubly nonlocal Fisher-KPP equation: Existence and properties of traveling waves

We consider a reaction-diffusion equation with nonlocal anisotropic diffusion and a linear combination of local and nonlocal monostable-type reactions in a space of bounded functions on $\mathbb{R}^d$. Using the properties of the corresponding semiflow, we prove the existence of monotone traveling waves along those directions where the diffusion kernel is exponentially integrable. Among other properties, we prove continuity, strict monotonicity and exponential integrability of the traveling wave profiles.

math.AP↗

arXiv2018

Doubly nonlocal Fisher-KPP equation: Speeds and uniqueness of traveling waves

We study traveling waves for a reaction-diffusion equation with nonlocal anisotropic diffusion and a linear combination of local and nonlocal monostable-type reactions. We describe relations between speeds and asymptotic of profiles of traveling waves, and prove the uniqueness of the profiles up to shifts.

math.AP↗

arXiv2018

Doubly nonlocal Fisher-KPP equation: Front propagation

We study propagation over $\mathbb{R}^d$ of the solution to a nonlocal nonlinear equation with anisotropic kernels, which can be interpretted as a doubly nonlocal reaction-diffusion equation of the Fisher--KPP-type. We prove that if the kernel of the nonlocal diffusion is exponentially integrable in a direction and if the initial condition decays in this direction faster than any exponential function, then the solution propagates at most linearly in time in that direction. Moreover, if both the kernel and the initial condition have the above properties in any direction (being, in general, anisotropic), then we prove linear in time propagation of the corresponding solution over $\mathbb{R}^d$.

math.AP↗

arXiv2018

An Ikehara-type theorem for functions convergent to zero

We prove an analogue of the Ikehara theorem for positive non-increasing functions convergent to zero, generalising the results postulated in Diekmann, Kaper (1978, Nonlinear Anal. 2(6), 721--737) and Carr, Chmaj (2004, Proc. AMS 132(8), 2433--2439).

math.CV↗

arXiv2018

Row-finite systems of ordinary differential equations in a scale of Banach spaces

Motivated by the study of dynamics of interacting spins for infinite particle systems, we consider an infinite family of first order differential equations in a Euclidean space, parameterized by elements $x$ of a fixed countable set. We suppose that the system is row-finite, that is, the right-hand side of the $x$-equation depends on a finite but in general unbounded number $n_x$ of variables. Under certain dissipativity-type conditions on the right-hand side and a bound on the growth of $n_x$, we show the existence of the solutions with infinite life-time, and prove that they live in an increasing scale of Banach spaces. For this, we obtain uniform estimates for solutions to approximating finite systems using a version of Ovsyannikov's method for linear systems in a scale of Banach spaces. As a by-product, we develop an infinite-time generalization of the Ovsyannikov method.

math.FA↗

arXiv2018

An infinite dimensional umbral calculus

The aim of this paper is to develop foundations of umbral calculus on the space $\mathcal D'$ of distributions on $\mathbb R^d$, which leads to a general theory of Sheffer polynomial sequences on $\mathcal D'$. We define a sequence of monic polynomials on $\mathcal D'$, a polynomial sequence of binomial type, and a Sheffer sequence. We present equivalent conditions for a sequence of monic polynomials on $\mathcal D'$ to be of binomial type or a Sheffer sequence, respectively. We also construct a lifting of a sequence of monic polynomials on $\mathbb R$ of binomial type to a polynomial sequence of binomial type on $\mathcal D'$, and a lifting of a Sheffer sequence on $\mathbb R$ to a Sheffer sequence on $\mathcal D'$. Examples of lifted polynomial sequences include the falling and rising factorials on $\mathcal D'$, Abel, Hermite, Charlier, and Laguerre polynomials on $\mathcal D'$. Some of these polynomials have already appeared in different branches of infinite dimensional (stochastic) analysis and played there a fundamental role.

math.FA↗

arXiv2018

Kesten's bound for sub-exponential densities on the real line and its multi-dimensional analogues

We study the tail asymptotic of sub-exponential probability densities on the real line. Namely, we show that the n-fold convolution of a sub-exponential probability density on the real line is asymptotically equivalent to this density times n. We prove Kesten's bound, which gives a uniform in n estimate of the n-fold convolution by the tail of the density. We also introduce a class of regular sub-exponential functions and use it to find an analogue of Kesten's bound for functions on $\mathbb{R}^d$. The results are applied for the study of the fundamental solution to a nonlocal heat-equation.

math.PR↗