arXiv subjects

Yuri Kondratiev

Publications and source records attributed to Yuri Kondratiev.

At least 55 records · Page 3Linked to original sources

arXiv2017

Stochastic averaging for a spatial population model in random environment

In this work we study the non-equilibrium Markov state evolution for a spatial population model on the space of locally finite configurations $Γ^2 = Γ^+ \times Γ^-$ over $\mathbb{R}^d$ where particles are marked by spins $\pm$. Particles of type '+' reproduce themselves independently of each other and, moreover, die due to competition either among particles of the same type or particles of different type. Particles of type '-' evolve according to a non-equilibrium Glauber-type dynamics with activity $z$ and potential $ψ$. Let $L^S$ be the Markov operator for '+' -particles and $L^E$ the Markov operator for '-' -particles. The non-equilibrium state evolution $(μ_t^{\varepsilon})_{t \geq 0}$ is obtained from the Fokker-Planck equation with Markov operator $L^S + \frac{1}{\varepsilon}L^E$, $\varepsilon > 0$, which itself is studied in terms of correlation function evolution on a suitable chosen scale of Banach spaces. We prove that in the limiting regime $\varepsilon \to 0$ the state evolution $μ_t^{\varepsilon}$ converges weakly to some state evolution $\overlineμ_t$ associated to the Fokker-Planck equation with (heuristic) Markov operator obtained from $L^S$ by averaging the interactions of the system with the environment with respect to the unique invariant Gibbs measure of the environment.

math-ph↗

arXiv2017

Stochastic averaging principle for spatial Markov evolutions in the continuum

We study a spatial birth-and-death process on the phase space of locally finite configurations $Γ^+ \times Γ^-$ over $\mathbb{R}^d$. Dynamics is described by an non-equilibrium evolution of states obtained from the Fokker-Planck equation and associated with the Markov operator $L^+(γ^-) + \frac{1}{\varepsilon}L^-$, $\varepsilon > 0$. Here $L^-$ describes the environment process on $Γ^-$ and $L^+(γ^-)$ describes the system process on $Γ^+$, where $γ^-$ indicates that the corresponding birth-and-death rates depend on another locally finite configuration $γ^- \in Γ^-$. We prove that, for a certain class of birth-and-death rates, the corresponding Fokker-Planck equation is well-posed, i.e. there exists a unique evolution of states $μ_t^{\varepsilon}$ on $Γ^+ \times Γ^-$. Moreover, we give a sufficient condition such that the environment is ergodic with exponential rate. Let $μ_{\mathrm{inv}}$ be the invariant measure for the environment process on $Γ^-$. In the main part of this work we establish the stochastic averaging principle, i.e. we prove that the marginal of $μ_t^{\varepsilon}$ onto $Γ^+$ converges weakly to an evolution of states on $Γ^+$ associated with the averaged Markov birth-and-death operator $\overline{L} = \int_{Γ^-}L^+(γ^-)d μ_{\mathrm{inv}}(γ^-)$.

math-ph↗

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↗

arXiv2017

Global stability in a nonlocal reaction-diffusion equation

We study stability of stationary solutions for a class of non-local semilinear parabolic equations. To this end, we prove the Feynman--Kac formula for a Lévy processes with time-dependent potentials and arbitrary initial condition. We propose sufficient conditions for asymptotic stability of the zero solution, and use them to the study of the spatial logistic equation arising in population ecology. For this equation, we find conditions which imply that its positive stationary solution is asymptotically stable. We consider also the case when the initial condition is given by a random field.

math.AP↗

arXiv2017

Self-regulation in the Bolker-Pacala model

The Markov dynamics is studied of an infinite system of point entities placed in $\mathds{R}^d$, in which the constituents disperse and die, also due to competition. Assuming that the dispersal and competition kernels are continuous and integrable we show that the evolution of states of this model preserves their sub-Poissonicity, and hence the local self-regulation (suppression of clustering) takes place. Upper bounds for the correlation functions of all orders are also obtained for both long and short dispersals, and for all values of the intrinsic mortality rate.

math.DS↗

arXiv2017

Self-regulation in continuum population models

We study the Markov dynamics of an infinite birth-and-death system of point entities placed in $\mathbb{R}^d$, in which the constituents disperse and die, also due to competition. Assuming that the dispersal and competition kernels are just continuous and integrable we prove that the evolution of states of this model preserves their sub-Poissonicity, and hence the local self-regulation (suppression of clustering) takes place. Upper bounds for the correlation functions of all orders are also obtained for both long and short dispersals, and for all values of the intrinsic mortality rate $m\geq 0$.

math.DS↗

arXiv2016

Fractional statistical dynamics and fractional kinetics

We apply the subordination principle to construct kinetic fractional statistical dynamics in the continuum in terms of solutions to Vlasov-type hierarchies. As a by-product we obtain the evolution of the density of particles in the fractional kinetics in terms of a non-linear Vlasov-type kinetic equation. As an application we study the intermittency of the fractional mesoscopic dynamics.

math-ph↗

arXiv2016

Fractional kinetic hierarchies and intermittency

We consider general convolutional derivatives and related fractional statistical dynamics of continuous interacting particle systems. We apply the subordination principle to construct kinetic fractional statistical dynamics in the continuum in terms of solutions to Vlasov-type hierarchies. Conditions for the intermittency property of fractional kinetic dynamics are obtained.

math-ph↗

arXiv2016

Evolution of states in a continuum migration model

The Markov evolution of states of a continuum migration model is studied. The model describes an infinite system of entities placed in $\mathds{R}^d$ in which the constituents appear (immigrate) with rate $b(x)$ and disappear, also due to competition. For this model, we prove the existence of the evolution of states $μ_0 \mapsto μ_t$ such that the moments $μ_t(N_Λ^n)$, $n\in \mathds{N}$, of the number of entities in compact $Λ\subset \mathds{R}^d$ remain bounded for all $t>0$. Under an additional condition, we prove that the density of entities and the second correlation function remain bounded globally in time.

math.DS↗

arXiv2016

Resolvent bounds for jump generators and ground state asymptotics for nonlocal Schrödinger operators

The paper deals with jump generators with a convolution kernel. Assuming that the kernel decays either exponentially or polynomially we prove a number of lower and upper bounds for the resolvent of such operators. We consider two applications of these results. First we obtain pointwise estimates for principal eigenfunction of jump generators perturbed by a compactly supported potential (so-called nonlocal Schrödinger operators). Then we consider the Cauchy problem for the corresponding inhomogeneous evolution equations and study the behaviour of its solutions.

math-ph↗

arXiv2016

A quasispecies continuous contact model in a critical regime

We study a new non-equilibrium dynamical model: a marked continuous contact model in $d$-dimensional space ($d \ge 3$). We prove that for certain values of rates (the critical regime) this system has the one-parameter family of invariant measures labelled by the spatial density of particles. Then we prove that the process starting from the marked Poisson measure converges to one of these invariant measures. In contrast with the continuous contact model studied earlier in \cite{KKP}, now the spatial particle density is not a conserved quantity.

math-ph↗

arXiv2016

On ground state of non local Schrodinger operators

We study a ground state of a non local Schrodinger operator associated with an evolution equation for the density of population in the stochastic contact model in continuum with inhomogeneous mortality rates. We found a new effect in this model, when even in the high dimensional case the existence of a small positive perturbation of a special form (so-called, small paradise) implies the appearance of the ground state. We consider the problem in the Banach space of bounded continuous functions $C_b (R^d)$ and in the Hilbert space $L^2(R^d)$.

math-ph↗

arXiv2015

The Microscopic Dynamics of a Spatial Ecological Model

The evolution of states of a spatial ecological model is studied. The model describes an infinite population of point entities placed in $\mathbb{R}^d$ which reproduce themselves at distant points (disperse) and die with rate that includes a competition term. The system's states are probability measures on the space of configurations of entities, and their evolution is described by means of a hierarchical chain of equations for the corresponding correlation functions derived from the Fokker-Planck equation for measures. Under natural conditions imposed on the model parameters it is proved that the correlation functions evolve in a scale of Banach spaces in such a way that each correlation function corresponds to a unique sub-Poissonian state. Some further properties of the evolution of states constructed in this way are also described.

math.DS↗

arXiv2015

Traveling waves and long-time behavior in a doubly nonlocal Fisher-KPP equation

We consider a Fisher-KPP-type equation, where both diffusion and nonlinear part are nonlocal, with anisotropic probability kernels. Under minimal conditions on the coefficients, we prove existence, uniqueness, and uniform space-time boundedness of the positive solution. We investigate existence, uniqueness, and asymptotic behavior of monotone traveling waves for the equation. We also describe the existence and main properties of the front of propagation.

math.AP↗

arXiv2015

Cantor series expansions and packing dimension faithfulness

The paper is devoted to the development of general theory of packing measures and dimensions via introducing the notion of < > and the packing analogues of the Billingsley dimension. To this aim we study equivalent definitions of packing dimension and prove theorems which can be considered as packing analogues of the famous Billingsley's theorems. The main result of the paper gives necessary and sufficient condition for the packing dimension faithfulness of the family of cylinders generated by the Cantor series expansion. To the best of our knowledge this is the first known sharp condition of the packing dimension faithfulness for a class of packing families containing both faithful and non-faithful ones.

math.MG↗

arXiv2015

On new fractal phenomena connected with infinite linear IFS

We establish several new fractal and number theoretical phenomena connected with expansions which are generated by infinite linear iterated function systems. First of all we show that the systems $Φ$ of cylinders of generalized Lüroth expansions are, generally speaking, not faithful for the Hausdorff dimension calculation. Using Yuval Peres' approach, we prove sufficient conditions for the non-faithfulness of such families of cylinders. On the other hand, rather general sufficient conditions for the faithfulness of such covering systems are also found. As a corollary of our main results, we obtain the non-faithfullness of the family of cylinders generated by the classical Lüroth expansion. Possible infinite entropy of the stochastic vector $Q_\infty$ which determines the metric relations for partitions of the generalized Lüroth expansions, possible non-faithfulness of the family $Φ(Q_\infty)$ and the absence of general formulae for the calculation of the Hausdorff dimension even for probability measures with independent identically distributed symbols of generalized Lüroth expansions are all facts that do not allow to apply usual probabilistic methods, as well as methods of dynamical systems to study fractal properties of corresponding subsets of non-normal numbers. Despite of the above mentioned difficulties, we develop new approach to the study of subsets of $Q_\infty$-essentially non-normal numbers and prove (without any additional restrictions on the stochastic vector $Q_{\infty}$) that this set is superfractal. This result answers the open problem mentioned in \cite{AKNT2} and completes the metric, dimensional and topological classification of real numbers via the asymptotic behaviour of frequencies their digits in the generalized Lüroth expansion.

math.MG↗

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

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↗