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 $${\mathbbm {R}}^d$$Rd 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 $$\mu _0 \mapsto \mu _t$$μ0↦μt such that the moments $$\mu _t(N_\Lambda ^n)$$μt(NΛn), $$n\in {\mathbbm {N}}$$n∈N, of the number of entities in compact $$\Lambda \subset {\mathbbm {R}}^d$$Λ⊂Rd 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.