D. L. Finkelshtein

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Articles: 10

Markov dynamics on the cone of discrete Radon measures

Dmitri Finkelshtein, Yuri Kondratiev, Peter Kuchling

↓ Abstract   |   Article (.pdf)

MFAT 27 (2021), no. 2, 173-191


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.

Стаття розпочинається з короткого огляду відомих фактів про простори дискретних мір Радона, які можуть розглядатися як узагальнення просторів конфігурацій. Далі розглядаються три марківські динаміки на просторах дискретних мір Радона: аналоги моделі контактів та моделі Болкера--Дікмана--Лоу--Пакали та аналог динаміки типу Глаубера. Показано як результати, отримані для просторів конфігурацій, можуть бути узагальнені для випадки просторів дискретних мір Радона.

Around Ovsyannikov's method

Dmitri Finkelshtein

↓ Abstract   |   Article (.pdf)

MFAT 21 (2015), no. 2, 134–150


We study existence, uniqueness, and a limiting behavior of solutions to an abstract linear evolution equation in a scale of Banach spaces. The generator of the equation is a perturbation of the operator which satisfies the classical assumptions of Ovsyannikov's method by a generator of a $C_0$-semigroup acting in each of the spaces of the scale. The results are (slightly modified) abstract version of those considered in [10] for a particular equation. An application to a birth-and-death stochastic dynamics in the continuum is considered.

An operator approach to Vlasov scaling for some models of spatial ecology

D. Finkelshtein, Yu. Kondratiev, O. Kutoviy

↓ Abstract   |   Article (.pdf)

MFAT 19 (2013), no. 2, 108-126


We consider Vlasov-type scaling for Markov evolution of birth-and-death type in continuum, which is based on a proper scaling of corresponding Markov generators and has an algorithmic realization in terms of related hierarchical chains of correlation functions equations. The existence of rescaled and limiting evolutions of correlation functions and convergence to the limiting evolution are shown. The obtained results enable us to derive a non-linear Vlasov-type equation for the density of the limiting system.

Kawasaki dynamics in the continuum via generating functionals evolution

D. L. Finkelshtein, Yu. G. Kondratiev, M. J. Oliveira

↓ Abstract   |   Article (.pdf)

MFAT 18 (2012), no. 1, 55-67


We construct the time evolution of Kawasaki dynamics for a spatial infinite particle system in terms of generating functionals. This is carried out by an Ovsjannikov-type result in a scale of Banach spaces, which leads to a local (in time) solution. An application of this approach to Vlasov-type scaling in terms of generating functionals is considered as well.

Functional evolutions for homogeneous stationary death-immigration spatial dynamics

D. Finkelshtein

↓ Abstract   |   Article (.pdf)

MFAT 17 (2011), no. 4, 300-318


We discover death-immigration non-equilibrium stochastic dynamics in the continuum also known as the Surgailis process. Explicit expression for the correlation functions is presented. Dynamics of states and their generating functionals are studied. Ergodic properties for the evolutions are considered.

On two-component contact model in continuum with one independent component

D. O. Filonenko, D. L. Finkelshtein, Yu. G. Kondratiev

↓ Abstract   |   Article (.pdf)

MFAT 14 (2008), no. 3, 209-228


Properties of a contact process in continuum for a system of particles of two types, one which is independent of the other, are considered. We study dynamics of the first and the second order correlation functions, their asymptotics, and the dependence on parameters of the~system.

Measures on configuration spaces defined by relative energies

D. L. Finkelshtein, Yu. G. Kondratiev

MFAT 11 (2005), no. 2, 126-155


Spectral gap inequalities on configuration spaces

D. L. Finkelshtein

MFAT 9 (2003), no. 1, 1-8


Symmetric differential operators of the second order in Poisson spaces

D. L. Finkelshtein, Yu. G. Kondratiev, A. Yu. Konstantinov, M. Röckner

MFAT 6 (2000), no. 4, 14-25


On exponential model of Poisson spaces

D. L. Finkelshtein, G. F. Us

MFAT 4 (1998), no. 4, 5-21


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