# Evolution of states in a continuum migration model

Evolution of states in a continuum migration model The Markov evolution of states of a continuum migration model is studied. The model describes an infinite system of entities placed in $${\mathbbm {R}}^d$$ 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 $$\mu _0 \mapsto \mu _t$$ μ 0 ↦ μ t such that the moments $$\mu _t(N_\Lambda ^n)$$ μ t ( N Λ n ) , $$n\in {\mathbbm {N}}$$ n ∈ N , of the number of entities in compact $$\Lambda \subset {\mathbbm {R}}^d$$ Λ ⊂ R d remain bounded for all $$t>0$$ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

# Evolution of states in a continuum migration model

, Volume 8 (1) – Mar 1, 2017
29 pages

/lp/springer-journals/evolution-of-states-in-a-continuum-migration-model-erzbFU7tRf
Publisher
Springer Journals
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-017-0166-8
Publisher site
See Article on Publisher Site

### Abstract

The Markov evolution of states of a continuum migration model is studied. The model describes an infinite system of entities placed in $${\mathbbm {R}}^d$$ 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 $$\mu _0 \mapsto \mu _t$$ μ 0 ↦ μ t such that the moments $$\mu _t(N_\Lambda ^n)$$ μ t ( N Λ n ) , $$n\in {\mathbbm {N}}$$ n ∈ N , of the number of entities in compact $$\Lambda \subset {\mathbbm {R}}^d$$ Λ ⊂ R d remain bounded for all $$t>0$$ 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.

### Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Mar 1, 2017