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Eugene Lytvynov

Publications and source records attributed to Eugene Lytvynov.

At least 37 records · Page 2Linked to original sources
arXiv2016

Meixner class of orthogonal polynomials of a non-commutative monotone Levy noise

Let $(X_t)_{t\ge0}$ denote a non-commutative monotone Lévy process. Let $ω=(ω(t))_{t\ge0}$ denote the corresponding monotone Lévy noise.. A continuous polynomial of $ω$ is an element of the corresponding non-commutative $L^2$-space $L^2(τ)$ that has the form $\sum_{i=0}^n\langle ω^{\otimes i},f^{(i)}\rangle$, where $f^{(i)}\in C_0(\mathbb R_+^i)$. We denote by $\mathbf{CP}$ the space of all continuous polynomials of $ω$. For $f^{(n)}\in C_0(\mathbb R_+^n)$, the orthogonal polynomial $\langle P^{(n)}(ω),f^{(n)}\rangle$ is defined as the orthogonal projection of the monomial $\langleω^{\otimes n},f^{(n)}\rangle$ onto the subspace of $L^2(τ)$ that is orthogonal to all continuous polynomials of $ω$ of order $\le n-1$. We denote by $\mathbf{OCP}$ the linear span of the orthogonal polynomials. Each orthogonal polynomial $\langle P^{(n)}(ω),f^{(n)}\rangle$ depends only on the restriction of the function $f^{(n)}$ to the set $\{(t_1,\dots,t_n)\in\mathbb R_+^n\mid t_1\ge t_2\ge\dots\ge t_n\}$. The orthogonal polynomials allow us to construct a unitary operator $J:L^2(τ)\to\mathbb F$, where $\mathbb F$ is an extended monotone Fock space. Thus, we may think of the monotone noise $ω$ as a distribution of linear operators acting in $\mathbb F$. We say that the orthogonal polynomials belong to the Meixner class if $\mathbf{CP}=\mathbf{OCP}$. We prove that each system of orthogonal polynomials from the Meixner class is characterized by two parameters: $λ\in\mathbb R$ and $η\ge0$. In this case, the monotone Lévy noise has the representation $ω(t)=\partial_t^†+λ\partial_t^†\partial_t+\partial_t+η\partial_t^†\partial_t\partial_t$. Here, $\partial_t^†$ and $\partial_t$ are the (formal) creation and annihilation operators at $t\in\mathbb R_+$ acting in $\mathbb F$.

math.PR↗
arXiv2016

Gauge-invriant quasi-free states on the algebra of the anyon commutation relations

Let $X=\mathbb R^2$ and let $q\in\mathbb C$, $|q|=1$. For $x=(x^1,x^2)$ and $y=(y^1,y^2)$ from $X^2$, we define a function $Q(x,y)$ to be equal to $q$ if $x^1 y^1$, and to $\Re q$ if $x^1=y^1$. Let $\partial_x^+$, $\partial_x^-$ ($x\in X$) be operator-valued distributions such that $\partial_x^+$ is the adjoint of $\partial_x^-$. We say that $\partial_x^+$, $\partial_x^-$ satisfy the anyon commutation relations (ACR) if $\partial^+_x\partial_y^+=Q(y,x)\partial_y^+\partial_x^+$ for $x\ne y$ and $\partial^-_x\partial_y^+=δ(x-y)+Q(x,y)\partial_y^+\partial^-_x$ for $(x,y)\in X^2$. In particular, for $q=1$, the ACR become the canonical commutation relations and for $q=-1$, the ACR become the canonical anticommutation relations. We define the ACR algebra as the algebra generated by operator-valued integrals of $\partial_x^+$, $\partial_x^-$. We construct a class of gauge-invariant quasi-free states on the ACR algebra. Each state from this class is completely determined by a positive self-adjoint operator $T$ on the real space $L^2(X,dx)$ which commutes with any operator of multiplication by a bounded function $ψ(x^1)$. In the case $\Re q<0$, the operator $T$ additionally satisfies $0\le T\le -1/\Re q$. Further, for $T=κ^2\mathbf 1$ ($κ>0$), we discuss the corresponding particle density $ρ(x):=\partial_x^+\partial_x^-$. For $\Re q\in(0,1]$, using a renormalization, we rigorously define a vacuum state on the commutative algebra generated by operator-valued integrals of $ρ(x)$. This state is given by a negative binomial point process. A scaling limit of these states as $κ\to\infty$ gives the gamma random measure, depending on parameter $\Re q$.

math.OA↗
arXiv2016

Approximation of a free Poisson process by systems of freely independent particles

Let $σ$ be a non-atomic, infinite Radon measure on $\mathbb R^d$, for example, $dσ(x)=z\,dx$ where $z>0$. We consider a system of freely independent particles $x_1,\dots,x_N$ in a bounded set $Λ\subset\mathbb R^d$, where each particle $x_i$ has distribution $\frac1{σ(Λ)}\,σ$ on $Λ$ and the number of particles, $N$, is random and has Poisson distribution with parameter $σ(Λ)$. If the particles were classically independent rather than freely independent, this particle system would be the restriction to $Λ$ of the Poisson point process on $\mathbb R^d$ with intensity measure $σ$. In the case of free independence, this particle system is not the restriction of the free Poisson process on $\mathbb R^d$ with intensity measure $σ$. Nevertheless, we prove that this is true in an approximative sense: if bounded sets $Λ^{(n)}$ ($n\in\mathbb N$) are such that $Λ^{(1)}\subsetΛ^{(2)}\subsetΛ^{(3)}\subset\dotsm$ and $\bigcup_{n=1}^\infty Λ^{(n)}=\mathbb R^d$, then the corresponding particle system in $Λ^{(n)}$ converges (as $n\to\infty$) to the free Poisson process on $\mathbb R^d$ with intensity measure $σ$. We also prove the following $N/V$-limit: Let $N^{(n)}$ be a determinstic sequence of natural numbers such that $\lim_{n\to\infty}N^{(n)}/σ(Λ^{(n)})=1$. Then the system of $N^{(n)}$ freely independent particles in $Λ^{(n)}$ converges (as $n\to\infty$) to the free Poisson process. We finally extend these results to the case of a free Lévy white noise (in particular, a free Lévy process) without free Gaussian part.

math.PR↗
arXiv2015

Laplace operators on the cone of Radon measures

We consider the infinite-dimensional Lie group $\mathfrak G$ which is the semidirect product of the group of compactly supported diffeomorphisms of a Riemannian manifold $X$ and the commutative multiplicative group of functions on $X$. The group $\mathfrak G$ naturally acts on the space $\mathbb M(X)$ of Radon measures on $X$. We would like to define a Laplace operator associated with a natural representation of $\mathfrak G$ in $L^2(\mathbb M(X),μ)$. Here $μ$ is assumed to be the law of a measure-valued Lévy process. A unitary representation of the group cannot be determined, since the measure $μ$ is not quasi-invariant with respect to the action of the group $\mathfrak G$. Consequently, operators of a representation of the Lie algebra and its universal enveloping algebra (in particular, a Laplace operator) are not defined. Nevertheless, we determine the Laplace operator by using a special property of the action of the group $\mathfrak G$ (a partial quasi-invariance). We further prove the essential self-adjointness of the Laplace operator. Finally, we explicitly construct a diffusion process on $\mathbb M(X)$ whose generator is the Laplace operator.

math.PR↗
arXiv2015

A moment problem for random discrete measures

Let $X$ be a locally compact Polish space. A random measure on $X$ is a probability measure on the space of all (nonnegative) Radon measures on $X$. Denote by $\mathbb K(X)$ the cone of all Radon measures $η$ on $X$ which are of the form $η=\sum_{i}s_iδ_{x_i}$, where, for each $i$, $s_i>0$ and $δ_{x_i}$ is the Dirac measure at $x_i\in X$. A random discrete measure on $X$ is a probability measure on $\mathbb K(X)$. The main result of the paper states a necessary and sufficient condition (conditional upon a mild a priori bound) when a random measure $μ$ is also a random discrete measure. This condition is formulated solely in terms of moments of the random measure $μ$. Classical examples of random discrete measures are completely random measures and additive subordinators, however, the main result holds independently of any independence property. As a corollary, a characterisation via a moments is given when a random measure is a point process.

math.PR↗
arXiv2015

Equilibrium diffusion on the cone of discrete Radon measures

Let $\mathbb K(\mathbb R^d)$ denote the cone of discrete Radon measures on $\mathbb R^d$. There is a natural differentiation on $\mathbb K(\mathbb R^d)$: for a differentiable function $F:\mathbb K(\mathbb R^d)\to\mathbb R$, one defines its gradient $\nabla^{\mathbb K} F $ as a vector field which assigns to each $η\in \mathbb K(\mathbb R^d)$ an element of a tangent space $T_η(\mathbb K(\mathbb R^d))$ to $\mathbb K(\mathbb R^d)$ at point $η$. Let $ϕ:\mathbb R^d\times\mathbb R^d\to\mathbb R$ be a potential of pair interaction, and let $μ$ be a corresponding Gibbs perturbation of (the distribution of) a completely random measure on $\mathbb R^d$. In particular, $μ$ is a probability measure on $\mathbb K(\mathbb R^d)$ such that the set of atoms of a discrete measure $η\in\mathbb K(\mathbb R^d)$ is $μ$-a.s.\ dense in $\mathbb R^d$. We consider the corresponding Dirichlet form $$ \mathscr E^{\mathbb K}(F,G)=\int_{\mathbb K(\mathbb R^d)}\langle\nabla^{\mathbb K} F(η), \nabla^{\mathbb K} G(η)\rangle_{T_η(\mathbb K)}\,dμ(η). $$ Integrating by parts with respect to the measure $μ$, we explicitly find the generator of this Dirichlet form. By using the theory of Dirichlet forms, we prove the main result of the paper: If $d\ge2$, there exists a conservative diffusion process on $\mathbb K(\mathbb R^d)$ which is properly associated with the Dirichlet form $\mathscr E^{\mathbb K}$.

math.PR↗
arXiv2015

An extended anyon Fock space and noncommutative Meixner-type orthogonal polynomials in infinite dimensions

Let $ν$ be a finite measure on $\mathbb R$ whose Laplace transform is analytic in a neighborhood of zero. An anyon Lévy white noise on $(\mathbb R^d,dx)$ is a certain family of noncommuting operators $\langleω,φ\rangle$ in the anyon Fock space over $L^2(\mathbb R^d\times\mathbb R,dx\otimesν)$. Here $φ=φ(x)$ runs over a space of test functions on $\mathbb R^d$, while $ω=ω(x)$ is interpreted as an operator-valued distribution on $\mathbb R^d$. Let $L^2(τ)$ be the noncommutative $L^2$-space generated by the algebra of polynomials in variables $\langle ω,φ\rangle$, where $τ$ is the vacuum expectation state. We construct noncommutative orthogonal polynomials in $L^2(τ)$ of the form $\langle P_n(ω),f^{(n)}\rangle$, where $f^{(n)}$ is a test function on $(\mathbb R^d)^n$. Using these orthogonal polynomials, we derive a unitary isomorphism $U$ between $L^2(τ)$ and an extended anyon Fock space over $L^2(\mathbb R^d,dx)$, denoted by $\mathbf F(L^2(\mathbb R^d,dx))$. The usual anyon Fock space over $L^2(\mathbb R^d,dx)$, denoted by $\mathcal F(L^2(\mathbb R^d,dx))$, is a subspace of $\mathbf F(L^2(\mathbb R^d,dx))$. Furthermore, we have the equality $\mathbf F(L^2(\mathbb R^d,dx))=\mathcal F(L^2(\mathbb R^d,dx))$ if and only if the measure $ν$ is concentrated at one point, i.e., in the Gaussian/Poisson case. Using the unitary isomorphism $U$, we realize the operators $\langle ω,φ\rangle$ as a Jacobi (i.e., tridiagonal) field in $\mathbf F(L^2(\mathbb R^d,dx))$. We derive a Meixner-type class of anyon Lévy white noise for which the respective Jacobi field in $\mathbf F(L^2(\mathbb R^d,dx))$ has a relatively simple structure.

math.PR↗
arXiv2013

Determinantal point processes with J-Hermitian correlation kernels

Let X be a locally compact Polish space and let m be a reference Radon measure on X. Let $Γ_X$ denote the configuration space over X, that is, the space of all locally finite subsets of X. A point process on X is a probability measure on $Γ_X$. A point process $μ$ is called determinantal if its correlation functions have the form $k^{(n)}(x_1,\ldots,x_n)=\det[K(x_i,x_j)]_{i,j=1,\ldots,n}$. The function K(x,y) is called the correlation kernel of the determinantal point process $μ$. Assume that the space X is split into two parts: $X=X_1\sqcup X_2$. A kernel K(x,y) is called J-Hermitian if it is Hermitian on $X_1\times X_1$ and $X_2\times X_2$, and $K(x,y)=-\overline{K(y,x)}$ for $x\in X_1$ and $y\in X_2$. We derive a necessary and sufficient condition of existence of a determinantal point process with a J-Hermitian correlation kernel K(x,y).

math.PR↗
arXiv2012

Equilibrium Kawasaki dynamics and determinantal point processes

Let "mu" be a point process on a countable discrete space "X". Under assumption that "mu" is quasi-invariant with respect to any finitary permutation of "X", we describe a general scheme for constructing an equilibrium Kawasaki dynamics for which "mu" is a symmetrizing (and hence invariant) measure. We also exhibit a two-parameter family of point processes "mu" possessing the needed quasi-invariance property. Each process of this family is determinantal, and its correlation kernel is the kernel of a projection operator in the Hilbert space of square-summable functions on "X".

math.PR↗
arXiv2011

Non-commutative Lévy processes for generalized (particularly anyon) statistics

Let $T=\mathbb R^d$. Let a function $Q:T^2\to\mathbb C$ satisfy $Q(s,t)=\bar{Q(t,s)}$ and $|Q(s,t)|=1$. A generalized statistics is described by creation operators $\partial_t^†$ and annihilation operators $\partial_t$, $t\in T$, which satisfy the $Q$-commutation relations. From the point of view of physics, the most important case of a generalized statistics is the anyon statistics, for which $Q(s,t)$ is equal to $q$ if $s t$. Here $q\in\mathbb C$, $|q|=1$. We start the paper with a detailed discussion of a $Q$-Fock space and operators $(\partial_t^†,\partial_t)_{t\in T}$ in it, which satisfy the $Q$-commutation relations. Next, we consider a noncommutative stochastic process (white noise) $ω(t)=\partial_t^†+\partial_t+λ\partial_t^†\partial_t$, $t\in T$. Here $λ\in\mathbb R$ is a fixed parameter. The case $λ=0$ corresponds to a $Q$-analog of Brownian motion, while $λ\ne0$ corresponds to a (centered) $Q$-Poisson process. We study $Q$-Hermite ($Q$-Charlier respectively) polynomials of infinitely many noncommutatative variables $(ω(t))_{t\in T}$. The main aim of the paper is to explain the notion of independence for a generalized statistics, and to derive corresponding Lévy processes. To this end, we recursively define $Q$-cumulants of a field $(ξ(t))_{t\in T}$. This allows us to define a $Q$-Lévy process as a field $(ξ(t))_{t\in T}$ whose values at different points of $T$ are $Q$-independent and which possesses a stationarity of increments (in a certain sense). We present an explicit construction of a $Q$-Lévy process, and derive a Nualart-Schoutens-type chaotic decomposition for such a process.

math.PR↗
arXiv2011

Binary jumps in continuum. II. Non-equilibrium process and a Vlasov-type scaling limit

Let $Γ$ denote the space of all locally finite subsets (configurations) in $\mathbb R^d$. A stochastic dynamics of binary jumps in continuum is a Markov process on $Γ$ in which pairs of particles simultaneously hop over $\mathbb R^d$. We discuss a non-equilibrium dynamics of binary jumps. We prove the existence of an evolution of correlation functions on a finite time interval. We also show that a Vlasov-type mesoscopic scaling for such a dynamics leads to a generalized Boltzmann non-linear equation for the particle density.

math.PR↗
arXiv2011

Binary jumps in continuum. I. Equilibrium processes and their scaling limits

Let $Γ$ denote the space of all locally finite subsets (configurations) in $R^d$. A stochastic dynamics of binary jumps in continuum is a Markov process on $Γ$ in which pairs of particles simultaneously hop over $R^d$. In this paper, we study an equilibrium dynamics of binary jumps for which a Poisson measure is a symmetrizing (and hence invariant) measure. The existence and uniqueness of the corresponding stochastic dynamics are shown. We next prove the main result of this paper: a big class of dynamics of binary jumps converge, in a diffusive scaling limit, to a dynamics of interacting Brownian particles. We also study another scaling limit, which leads us to a spatial birth-and-death process in continuum. A remarkable property of the limiting dynamics is that its generator possesses a spectral gap, a property which is hopeless to expect from the initial dynamics of binary jumps.

math.PR↗
arXiv2010

A note on equilibrium Glauber and Kawasaki dynamics for permanental point processes

We construct two types of equilibrium dynamics of an infinite particle system in a locally compact metric space $X$ for which a permanental point process is a symmetrizing, and hence invariant measure. The Glauber dynamics is a birth-and-death process in $X$, while in the Kawasaki dynamics interacting particles randomly hop over $X$. In the case $X=\mathbb R^d$, we consider a diffusion approximation for the Kawasaki dynamics at the level of Dirichlet forms. This leads us to an equilibrium dynamics of interacting Brownian particles for which a permanental point process is a symmetrizing measure.

math.PR↗
arXiv2009

Meixner class of non-commutative generalized stochastic processes with freely independent values I. A characterization

Let $T$ be an underlying space with a non-atomic measure $σ$ on it (e.g. $T=\mathbb R^d$ and $σ$ is the Lebesgue measure). We introduce and study a class of non-commutative generalized stochastic processes, indexed by points of $T$, with freely independent values. Such a process (field), $ω=ω(t)$, $t\in T$, is given a rigorous meaning through smearing out with test functions on $T$, with $\int_T σ(dt)f(t)ω(t)$ being a (bounded) linear operator in a full Fock space. We define a set $\mathbf{CP}$ of all continuous polynomials of $ω$, and then define a con-commutative $L^2$-space $L^2(τ)$ by taking the closure of $\mathbf{CP}$ in the norm $\|P\|_{L^2(τ)}:=\|PΩ\|$, where $Ω$ is the vacuum in the Fock space. Through procedure of orthogonalization of polynomials, we construct a unitary isomorphism between $L^2(τ)$ and a (Fock-space-type) Hilbert space $\mathbb F=\mathbb R\oplus\bigoplus_{n=1}^\infty L^2(T^n,γ_n)$, with explicitly given measures $γ_n$. We identify the Meixner class as those processes for which the procedure of orthogonalization leaves the set $\mathbf {CP}$ invariant. (Note that, in the general case, the projection of a continuous monomial of oder $n$ onto the $n$-th chaos need not remain a continuous polynomial.) Each element of the Meixner class is characterized by two continuous functions $λ$ and $η\ge0$ on $T$, such that, in the $\mathbb F$ space, $ω$ has representation $ω(t)=\di_t^†+λ(t)\di_t^†\di_t+\di_t+η(t)\di_t^†\di^2_t$, where $\di_t^†$ and $\di_t$ are the usual creation and annihilation operators at point $t$.

math.PR↗