We discuss general concept of Markov statistical dynamics in the continuum. For a class of spatial birth-and-death models, we develop a perturbative technique for the construction of statistical ...dynamics. Particular examples of such systems are considered. For the case of Glauber type dynamics in the continuum we describe a Markov chain approximation approach that gives more detailed information about statistical evolution in this model.
We consider the non-equilibrium dynamics for the Widom–Rowlinson model (without hard-core) in the continuum. The Lebowitz–Penrose-type scaling of the dynamics is studied and the system of the ...corresponding kinetic equations is derived. In the space-homogeneous case, the equilibrium points of this system are described. Their structure corresponds to the dynamical phase transition in the model. The bifurcation of the system is shown.
We consider the extinction regime in the spatial stochastic logistic model in \(\mathbb{R}^d\) (a.k.a. Bolker--Pacala--Dieckmann--Law model of spatial populations) using the first-order perturbation ...beyond the mean-field equation. In space homogeneous case (i.e. when the density is non-spatial and the covariance is translation invariant), we show that the perturbation converges as time tends to infinity; that yields the first-order approximation for the stationary density. Next, we study the critical mortality---the smallest constant death rate which ensures the extinction of the population---as a function of the mean-field scaling parameter \(\varepsilon>0\). We find the leading term of the asymptotic expansion (as \(\varepsilon\to0\)) of the critical mortality which is apparently different for the cases \(d\geq3\), \(d=2\), and \(d=1\).
A Markov evolution of a system of point particles in ℝ
d
is described at micro- and mesoscopic levels. The particles reproduce themselves at distant points (dispersal) and die, independently and ...under the effect of each other (competition). The microscopic description is based on an infinite chain of equations for correlation functions, similar to the BBGKY hierarchy used in the Hamiltonian dynamics of continuum particle systems. The mesoscopic description is based on a Vlasov-type kinetic equation for the particle’s density obtained from the mentioned chain via a scaling procedure. The main conclusion of the microscopic theory is that the competition can prevent the system from clustering, which makes its description in terms of densities reasonable. A possible homogenization of the solutions to the kinetic equation in the long-time limit is also discussed.
We construct a correlation functions evolution corresponding to the Glauber dynamics in continuum. Existence of the corresponding strongly continuous contraction semigroup in a proper Banach space is ...shown. Additionally we prove the existence of the evolution of states and study their ergodic properties.
We develop a new approach for the construction of the Glauber dynamics in continuum. Existence of the corresponding strongly continuous contraction semigroup in a proper Banach space is shown. ...Additionally we present the finite‐ and infinite‐volume approximations of the semigroup by families of bounded linear operators.
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