There is studied an infinite system of point entities in R d which reproduce themselves and die, also due to competition. The system's states are probability measures on the space of configurations of entities. Their evolution is described by means of a BBGKY-type equation for the corresponding correlation (moment) functions. It is proved that: (a) these functions evolve on a bounded time interval and remain sub-Poissonian due to the competition; (b) in the Vlasov scaling limit they converge to the correlation functions of the time-dependent Poisson point field the density of which solves the kinetic equation obtained in the scaling limit from the equation for the correlation functions. A number of properties of the solutions of the kinetic equation are also established.

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A rigorous description of the equilibrium thermodynamic properties of an infinite system of interacting ν-dimensional quantum anharmonic oscillators is given. The oscillators are indexed by the elements of a countable set L ⊂ R d , possibly irregular; the anharmonic potentials vary from site to site. The description is based on the representation of the Gibbs states in terms of path measures -the so called Euclidean Gibbs measures. It is proven that: (a) the set of such measures G t is non-void and compact; (b) every µ ∈ G t obeys an exponential integrability estimate, the same for the whole set G t ; (c) every µ ∈ G t has a Lebowitz-Presutti type support; (d) G t is a singleton at high temperatures. In the case of attractive interaction and ν = 1 we prove that |G t | > 1 at low temperatures. The uniqueness of Gibbs measures due to quantum effects and at a nonzero external field are also proven in this case. Thereby, a qualitative theory of phase transitions and quantum effects, which interprets most important experimental data known for the corresponding physical objects, is developed. The mathematical result of the paper is a complete description of the set G t , which refines and extends the results known for models of this type.

The evolution of states in a spatial population model is studied. The model describes an infinite system of point entities in 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, and their evolution is obtained from a hierarchical chain of differential equations for the corresponding correlation functions derived from the Fokker-Planck equation for the states. 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 at each moment of time the correlation function corresponds to a unique sub-Poissonian state. Some further properties of the evolution of states constructed in this way are described.

The dynamics of an infinite system of point particles in R d , which hop and interact with each other, is described at both micro-and mesoscopic levels. The states of the system are probability measures on the space of configurations of particles. For a bounded time interval [0, T ), the evolution of states μ 0 → μ t is shown to hold in a space of sub-Poissonian measures. This result is obtained by: (a) solving equations for correlation functions, which yields the evolution k 0 → k t , t ∈ [0, T ), in a scale of Banach spaces; (b) proving that each k t is a correlation function for a unique measure μ t . The mesoscopic theory is based on a Vlasov-type scaling, that yields a mean-field-like approximate description in terms of the particles' density which obeys a kinetic equation. The latter equation is rigorously derived from that for the correlation functions by the scaling procedure. We prove that the kinetic equation has a unique solution t , t ∈ [0, +∞).