arXiv subjects

Alexei Daletskii

Publications and source records attributed to Alexei Daletskii.

15 recordsLinked to original sources

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↗

arXiv2023

Stochastic dynamics of particle systems on unbounded degree graphs

We consider an infinite system of coupled stochastic differential equations (SDE) describing dynamics of the following infinite particle system. Each partricle is characterised by its position $x\in \mathbb{R}^{d}$ and internal parameter (spin) $\sigma _{x}\in \mathbb{R}$. While the positions of particles form a fixed ("quenched") locally-finite set (configuration) $ \gamma \subset $ $\mathbb{R}^{d}$, the spins $\sigma _{x}$ and $\sigma _{y}$ interact via a pair potential whenever $\left\vert x-y\right\vert <\rho $, where $\rho >0$ is a fixed interaction radius. The number $n_{x}$ of particles interacting with a particle in positionn $x$ is finite but unbounded in $x$. The growth of $n_{x}$ as $x\rightarrow \infty $ creates a major technical problem for solving our SDE system. To overcome this problem, we use a finite volume approximation combined with a version of the Ovsjannikov method, and prove the existence and uniqueness of the solution in a scale of Banach spaces of weighted sequences. As an application example, we construct stochastic dynamics associated with Gibbs states of our particle system.

math.FA↗

arXiv2021

Stochastic differential equations in a scale of Hilbert spaces 2. Global solutions

A stochastic differential equation with coefficients defined in a scale of Hilbert spaces is considered. The existence, uniqueness and path-continuity of infinite-time solutions is proved by an extension of the Ovsyannikov method. This result is applied to a system of equations describing non-equilibrium stochastic dynamics of (real-valued) spins of an infinite particle system on a typical realization of a Poisson or Gibbs point process in ${\mathbb{R}}^{n}$. The paper improves the results of the work by the second named author "Stochastic differential equations in a scale of Hilbert spaces", Electron. J. Probab. 23, where finite-time solutions were constructed.

math.FA↗

arXiv2019

Stochastic Camassa-Holm equation with convection type noise

We consider a stochastic Camassa-Holm equation driven by a one-dimensional Wiener process with a first order differential operator as diffusion coefficient. We prove the existence and uniqueness of local strong solutions of this equation. In order to do so, we transform it into a random quasi-linear partial differential equation and apply Kato's operator theory methods. Some of the results have potential to find applications to other nonlinear stochastic partial differential equations.

math.FA↗

arXiv2019

Fock representations of multicomponent (particularly non-Abelian anyon) commutation relations

Let $H$ be a separable Hilbert space and $T$ be a self-adjoint bounded linear operator on $H^{\otimes 2}$ with norm $\le1$, satisfying the Yang--Baxter equation. Bożejko and Speicher (1994) proved that the operator $T$ determines a $T$-deformed Fock space $\mathcal F(H)=\bigoplus_{n=0}^\infty\mathcal F_n(H)$. We start with reviewing and extending the known results about the structure of the $n$-particle spaces $\mathcal F_n(H)$ and the commutation relations satisfied by the corresponding creation and annihilation operators acting on $\mathcal F(H)$. We then choose $H=L^2(X\to V)$, the $L^2$-space of $V$-valued functions on $X$. Here $X:=\mathbb R^d$ and $V:=\mathbb C^m$ with $m\ge2$. Furthermore, we assume that the operator $T$ acting on $H^{\otimes 2}=L^2(X^2\to V^{\otimes 2})$ is given by $(Tf^{(2)})(x,y)=C_{x,y}f^{(2)}(y,x)$. Here, for a.a.\ $(x,y)\in X^2$, $C_{x,y}$ is a linear operator on $V^{\otimes 2}$ with norm $\le1$ that satisfies $C_{x,y}^*=C_{y,x}$ and the spectral quantum Yang--Baxter equation. The corresponding creation and annihilation operators describe a multicomponent quantum system. A special choice of the operator-valued function $C_{xy}$ in the case $d=2$ determines non-Abelian anyons (also called plektons). For a multicomponent system, we describe its $T$-deformed Fock space and the available commutation relations satisfied by the corresponding creation and annihilation operators. Finally, we consider several examples of multicomponent quantum systems.

math-ph↗

arXiv2018

Stochastic differential equations in a scale of Hilbert spaces

A stochastic differential equation with coefficients defined in a scale of Hilbert spaces is considered. The existence and uniqueness of finite time solutions is proved by an extension of the Ovsyannikov method. This result is applied to a system of equations describing non-equilibrium stochastic dynamics of (real-valued) spins of an infinite particle system on a typical realization of a Poisson or Gibbs point process in a Euclidean space.

math.FA↗

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↗

arXiv2017

Gibbs Measures on Marked Configuration Spaces: Existence and Uniqueness

We study equilibrium states of an infinite system of interacting particles in a Euclidean space. The particles bear `unbounded' spins with a given symmetric a priori distribution. The interaction between the particles is pairwise and splits into position-position and spin-spin parts. The position-position part is described by a superstable potential, and the spin-spin part is attractive and of finite range. Thermodynamic states of the system are defined as tempered Gibbs measures on the space of marked configurations. We derive sufficient conditions of the existence and uniqueness of these Gibbs measures.

math-ph↗

arXiv2015

Phase Transitions in Continuum Ferromagnets with Unbounded Spins

States of thermal equilibrium of an infinite system of interacting particles in a Euclidean space are studied. The particles bear 'unbounded' spins with a given symmetric a priori distribution. The interaction between the particles is pairwise and splits into position-position and spin-spin parts. The position-position part is described by a superstable potential, and the spin-spin part is attractive and of finite range. Thermodynamic states of the system are defined as tempered Gibbs measures on the space of marked configurations. It is proved that the set of such measures contains at least two elements if the activity is big enough.

math-ph↗

arXiv2014

Gibbs States on Random Configurations

We study a class of Gibbs measures of classical particle spin systems with spin space $S=\mathbb{R}^{m}$ and unbounded pair interaction, living on a metric graph given by a typical realization $γ$ of a random point process in $\mathbb{R}^{n}$. Under certain conditions of growth of pair- and self-interaction potentials, we prove that the set $\mathcal{G}(S^γ)$ of all such Gibbs measures is not empty for almost all $γ$, and study support properties of $ν_γ\in \mathcal{G}(S^γ)$. Moreover we show the existence of measurable maps (selections) $γ\mapsto ν_γ$ and derive the corresponding averaged moment estimates.

math-ph↗

arXiv2014

A Phase Transition in a Quenched Amorphous Ferromagnet

Quenched thermodynamic states of an amorphous ferromagnet are studied. The magnet is a countable collection of point particles chaotically distributed over $\mathbb{R}^d$, $d\geq 2$. Each particle bears a real-valued spin with symmetric a priori distribution; the spin-spin interaction is pair-wise and attractive. Two spins are supposed to interact if they are neighbors in the graph defined by a homogeneous Poisson point process. For this model, we prove that with probability one: (a) quenched thermodynamic states exist; (b) they are multiple if the particle density (i.e., the intensity of the underlying point process) and the inverse temperature are big enough; (c) there exist multiple quenched thermodynamic states which depend on the realizations of the underlying point process in a measurable way.

math-ph↗

arXiv2011

L2 dimensions of spaces of braid-invariant harmonic forms

Let X be a Riemannian manifold endowed with a co-compact isometric action of an infinite discrete group. We consider L2 spaces of harmonic vector-valued forms on the product manifold X^N, which are invariant with respect to an action of the braid group B_N, and compute their von Neumann dimensions (the braided L2- Betti numbers)

math.FA↗

arXiv2011

Cluster point processes on manifolds

The probability distribution $μ_{cl}$ of a general cluster point process in a Riemannian manifold $X$ (with independent random clusters attached to points of a configuration with distribution $μ$) is studied via the projection of an auxiliary measure $\hatμ$ in the space of configurations $\hatγ=\{(x,\bar{y})\}\subset X\times\mathfrak{X}$, where $x\in X$ indicates a cluster "centre" and $\bar{y}\in\mathfrak{X}:=\bigsqcup_{n} X^n$ represents a corresponding cluster relative to $x$. We show that the measure $μ_{cl}$ is quasi-invariant with respect to the group $Diff_{0}(X)$ of compactly supported diffeomorphisms of $X$, and prove an integration-by-parts formula for $μ_{cl}$. The associated equilibrium stochastic dynamics is then constructed using the method of Dirichlet forms. General constructions are illustrated by examples including Euclidean spaces, Lie groups, homogeneous spaces, Riemannian manifolds of nonpositive curvature and metric spaces. The paper is an extension of our earlier results for Poisson cluster measures [J. Funct. Analysis 256 (2009) 432-478] and for Gibbs cluster measures [arxiv:1007.3148], where different projection constructions were utilised.

math.FA↗

arXiv2010

Gibbs cluster measures on configuration spaces

The distribution $g_{cl}$ of a Gibbs cluster point process in $X=\mathbb{R}^{d}$ (with i.i.d. random clusters attached to points of a Gibbs configuration with distribution $g$) is studied via the projection of an auxiliary Gibbs measure $\hat{g}$ in the space of configurations $\hat{gamma}=\{(x,\bar{y})\}\subset X\times\mathfrak{X}$, where $x\in X$ indicates a cluster "center" and $\bar{y}\in\mathfrak{X}:=\bigsqcup_{n} X^n$ represents a corresponding cluster relative to $x$. We show that the measure $g_{cl}$ is quasi-invariant with respect to the group $\mathrm{Diff}_{0}(X)$ of compactly supported diffeomorphisms of $X$, and prove an integration-by-parts formula for $g_{cl}$. The associated equilibrium stochastic dynamics is then constructed using the method of Dirichlet forms. These results are quite general; in particular, the uniqueness of the background Gibbs measure $g$ is not required. The paper is an extension of the earlier results for Poisson cluster measures %obtained by the authors [J. Funct. Analysis 256 (2009) 432-478], where a different projection construction was utilized specific to this "exactly soluble" case.

math.FA↗

arXiv2008

Poisson Cluster Measures: Quasi-invariance, Integration by Parts and Equilibrium Stochastic Dynamics

The distribution $μ_{cl}$ of a Poisson cluster process in $X=\mathbb{R}^{d}$ (with i.i.d. clusters) is studied via an auxiliary Poisson measure on the space of configurations in $\mathfrak{X}=\sqcup_{n} X^n$, with intensity measure defined as a convolution of the background intensity of cluster centres and the probability distribution of a generic cluster. We show that the measure $μ_{cl}$ is quasi-invariant with respect to the group of compactly supported diffeomorphisms of $X$ and prove an integration-by-parts formula for $μ_{cl}$. The corresponding equilibrium stochastic dynamics is then constructed using the method of Dirichlet forms.

math.FA↗