Evolution of states in a continuum migration model

Abstract

The Markov evolution of states of a continuum migration model is studied. The model describes an infinite system of entities placed in 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 → μ t such that the moments μ t (N n ), n ∈ N, of the number of entities in compact ⊂ 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 point-wise bounded globally in time.

Mutual service processes in Euclidean spaces: existence and ergodicity

“…They also develop conservation equations for higher order moment measures [6,4]. The model studied in our paper appears in, e.g., [3] and [13], and the latter also gives a proof of the existence of an evolution starting at time 0.…”

An interplay between attraction and repulsion in infinite populations

“…It is clear, cf. [22,Sect. 2.3], that without interaction the distribution of migrants should eventually reflect the heterogeneity of the habitat and be Poissonian-in view of the randomness mentioned above and the lack of interactions.…”

Correction to: Evolution of states in a continuum migration model

“…Thus, starting from the second line in (4.17), we have, see the beginning of Sect. 3.2.2, d dt q (1)…”

“…The proof of Lemma 4.1 of [1] has a certain inexactness which should be corrected. Namely, in proving the estimate in (4.18), one has to consider in (4.17) the case of l = 1 separately from all other cases as F (l−1) (∅) = 0 holds only for l ≥ 2.…”

“…where R μ t is the density of the projection of μ t with respect to the Lebesgue-Poisson measure λ. By (2.5) and (3.32), this can be rewritten d dt q (1) (1)…”