General birth-and-death as well as hopping stochastic dynamics of infinite particle systems in the continuum are considered. We derive corresponding evolution equations for correlation functions and ...generating functionals. General considerations are illustrated in a number of concrete examples of Markov evolutions appearing in applications.
We construct the time evolution for states of Glauber 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, leading to a local (in time) solution which, under certain initial conditions, might be extended to a global one. An application of this approach to Vlasov-type scaling in terms of generating functionals is considered as well.
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.
We study existence, uniqueness, and a limiting behaviour 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 Math. Models Methods Appl. Sci., 25, 2, 2015, pp.343-370 for a particular equation. An application to a birth-and-death stochastic dynamics in the continuum is considered.
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.
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.
We consider two types of convolutions (\(\ast\) and \(\star\)) of functions on spaces of finite configurations (finite subsets of a phase space), and some their properties are studied. A connection ...of the \(\ast\)-convolution with the convolution of measures on spaces of finite configurations is shown. Properties of multiplication and derivative operators with respect to the \(\ast\)-convolution are discovered. We present also conditions when the \(\ast\)-convolution will be positive definite with respect to the \(\star\)-convolution.
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