Evolution of states in a continuum migration model
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Abstract
The Markov evolution of states of a continuum migration model is studied. The model describes an infinite system of entities placed in \({\mathbbm {R}}^d\) 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 \(\mu _0 \mapsto \mu _t\) such that the moments \(\mu _t(N_\Lambda ^n)\), \(n\in {\mathbbm {N}}\), of the number of entities in compact \(\Lambda \subset {\mathbbm {R}}^d\) 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.
Keywords
Markov evolution Competition kernel Poisson random fieldMathematics Subject Classification
60J80 92D25 82C221 Introduction
Assumption 1.1
The competition kernel a is continuous and belongs to \(L^1 ({\mathbbm {R}}^d) \cap L^\infty ({\mathbbm {R}}^d)\). If not explicitly stated otherwise, \(a(0)>0\). The immigration and mortality rates b and m are continuous and bounded.
The structure of the article is as follows. In Sect. 2, we introduce the necessary technicalities and then formulate the results: Theorems 2.4 and 2.5. Thereafter, we make a number of comments to them. In Sects. 3 and 4, we present the proofs of Theorems 2.4 and 2.5, respectively.
2 Preliminaries and the results
We begin by presenting some facts on the subject—a more detailed description of them can be found in [5, 7, 8] and in the literature quoted therein.
By \({\mathcal {B}}({\mathbbm {R}}^d)\) and \({\mathcal {B}}_\mathrm{b}({\mathbbm {R}}^d)\) we denote the sets of all Borel and bounded Borel subsets of \({\mathbbm {R}}^d\), respectively. The configuration space \(\Gamma \) is equipped with the vague topology, see [8], and thus with the corresponding Borel \(\sigma \)-field \({\mathcal {B}}(\Gamma )\), which makes it a standard Borel space. Note that \({\mathcal {B}}(\Gamma )\) is exactly the \(\sigma \)-field generated by the sets \(\Gamma ^{\Lambda ,n}\), mentioned in Introduction. By \({\mathcal {P}}(\Gamma )\) we denote the set of all probability measures on \((\Gamma , {\mathcal {B}}(\Gamma ))\).
2.1 Correlation functions
Definition 2.1
The set of sub-Poissonian states \({\mathcal {P}}_\mathrm{exp}(\Gamma )\) consists of all those states \(\mu \in {\mathcal {P}}(\Gamma )\) for which \(B_\mu \) can be continued, as a function of \(\theta \), to an exponential type entire function on \(L^1 ({\mathbb {R}}^d)\).
Definition 2.2
A measurable function \(G:\Gamma _0 \rightarrow {\mathbbm {R}}\) is said to have bounded support if: (a) there exists \(\Lambda \in {\mathcal {B}}_\mathrm{b} ({\mathbbm {R}}^d)\) such that \(G(\eta ) = 0\) whenever \(\eta \cap ({\mathbbm {R}}^d \setminus \Lambda )\ne \emptyset \); (b) there exists \(N\in {\mathbbm {N}}_0\) such that \(G(\eta )=0\) whenever \(|\eta | >N\). By \(\Lambda (G)\) and N(G) we denote the smallest \(\Lambda \) and N with the properties just mentioned. By \(B_\mathrm{bs}(\Gamma _0)\) we denote the set of all bounded functions with bounded support.
2.2 The Banach spaces
Proposition 2.3
2.3 Without competition
2.4 The statements
Theorem 2.4
Theorem 2.5
2.5 Comments and comparison
By (2.25) it follows that the global in time boundedness in the Surgailis model is possible only if \(m(x) \ge m_*>0\) for all \(x\in {\mathbbm {R}}^d\). As follows from our Theorem 2.5, adding competition to the Surgailis model with the zero intrinsic mortality rate yields the global in time boundedness. In this case, the competition rate a(0) appears to be an effective mortality, see (4.19) below and the comments following the proof of Theorem 2.5. Note also that the global boundedness as in Theorem 2.5 does not mean that the evolution \(k_{\mu _0} \mapsto k_t\) holds in one and the same \({\mathcal {K}}_\vartheta \) with sufficiently large \(\vartheta \). It does if \(m(x) \ge m_*>0\). Since Theorem 2.4 covers also the case \(a \equiv 0\), the solution in (2.21) is unique in the same sense. A partial result on the global boundedness in the model discussed here was obtained in [3, Theorem 1]. Therein, under quite a strong condition imposed on the competition kernel a (which, in particular, implies that it has infinite range), and under the assumption that the evolution of states \(\mu _0 \mapsto \mu _t\) exists, there was proved the fact which in the present notations can be formulated as \(\mu _t(N_\Lambda ) \le C_\Lambda \).
3 The existence of the evolution of states
- (i)
Defining the Cauchy problem (2.8) with \(k_{\mu _0 }\in {\mathcal {K}}_{\vartheta _0}\) in a given Banach space \({\mathcal {K}}_\vartheta \) with \(\vartheta >\vartheta _0\), see (2.12) and (2.14), and then showing that this problem has a unique solution \(k_t\in {\mathcal {K}} _\vartheta \) on a bounded time interval \([0,T(\vartheta , \vartheta _0))\) (Sect. 3.1).
- (ii)
Proving that the mentioned solution \(k_t\) has the properties (a) and (b) in (2.18) ((c) follows by the fact that \(k_t\in {\mathcal {K}}_\vartheta \)). Then \(k_t \in {\mathcal {K}}_\vartheta ^\star \) and hence also in \({\mathcal {K}}_\vartheta ^{+}\), see (2.20) and (2.19). By Proposition 2.3 it follows that \(k_t\) is the correlation function of a unique state \(\mu _t\) (Sect. 3.2).
- (iii)
Constructing a continuation of \(k_t\) from \([0,T(\vartheta , \vartheta _0))\) to all \(t>0\) by means of the fact that \(k_t \in {\mathcal {K}}_\vartheta ^{+}\) (Sect. 3.3).
3.1 Solving the Cauchy problem
Proposition 3.1
- (i)for each \((t, t_1 , \dots , t_l) \in {\mathcal {T}}_l\), \(\Pi ^{(l)}_{\vartheta '\vartheta } (t, t_1 , \dots , t_l)\) is in \({\mathcal {L}}({\mathcal {K}}_{\vartheta }, {\mathcal {K}}_{\vartheta '})\) and the mapis continuous;$$\begin{aligned} (t, t_1 , \dots , t_l) \mapsto \Pi ^{(l)}_{\vartheta '\vartheta } (t, t_1 , \dots , t_l)\in {\mathcal {L}}({\mathcal {K}}_{\vartheta }, {\mathcal {K}}_{\vartheta '}) \end{aligned}$$
- (ii)for fixed \(t_1 , \dots , t_l\) and each \(\varepsilon >0\), the mapis continuously differentiable and such that, for each \(\vartheta ''\in (\vartheta , \vartheta ')\), the following holds$$\begin{aligned} (t_1 , t_1 + \varepsilon ) \ni t \mapsto \Pi ^{(l)}_{\vartheta '\vartheta } (t, t_1 , \dots , t_l)\in {\mathcal {L}}({\mathcal {K}}_{\vartheta }, {\mathcal {K}}_{\vartheta '}) \end{aligned}$$$$\begin{aligned} \frac{d}{dt} \Pi ^{(l)}_{\vartheta '\vartheta } (t, t_1 , \dots , t_l) = A_{\vartheta ' \vartheta ''} \Pi ^{(l)}_{\vartheta ''\vartheta } (t, t_1 , \dots , t_l). \end{aligned}$$(3.8)
Proposition 3.2
- (i)
the map \([0,T(\vartheta ', \vartheta )) \ni t \mapsto Q_{\vartheta ' \vartheta }(t) \in {\mathcal {L}}({\mathcal {K}}_{\vartheta }, {\mathcal {K}}_{\vartheta '})\) is continuous and \( Q_{\vartheta ' \vartheta }(0)\) is the embedding \({\mathcal {K}}_\vartheta \hookrightarrow {\mathcal {K}}_{\vartheta '}\);
- (ii)for each \(\vartheta '' \in (\vartheta , \vartheta ')\) and \(t < T(\vartheta '', \vartheta )\), the following holds$$\begin{aligned} \frac{d}{dt} Q_{\vartheta ' \vartheta }(t) = L^\Delta _{\vartheta '\vartheta ''} Q_{\vartheta '' \vartheta }(t). \end{aligned}$$(3.11)
Proof
Lemma 3.3
Proof
Remark 3.4
3.2 The identification
Our next step is based on the following statement.
Lemma 3.5
Let \(\{Q_{\vartheta '\vartheta }(t): (\vartheta , \vartheta ', t) \in \varTheta \}\) be the family as in Proposition 3.2. Then, for each \(\vartheta \) and \(\vartheta '\) and \(t\in [0, T(\vartheta ', \vartheta )/2)\), we have that \(Q_{\vartheta '\vartheta } (t): {\mathcal {K}}_{\vartheta }^\star \rightarrow {\mathcal {K}}_{\vartheta '}^\star \).
3.2.1 Auxiliary models
Proposition 3.6
Assume that \(Q^\sigma _{\vartheta _1\vartheta _0}:{\mathcal {K}}^\star _{\vartheta _0} \rightarrow {\mathcal {K}}^\star _{\vartheta _1}\) for all \(t< T(\vartheta _1, \vartheta _0)\). Then, for all \(t < T(\vartheta _1, \vartheta _0)/2\) and \(G\in B_\mathrm{bs}(\Gamma _0)\), the convergence in (3.22) holds.
Proof
3.2.2 Auxiliary evolutions
Proposition 3.7
The operator \((L^\dagger , {\mathcal {D}})\) defined in (3.34) and (3.36) is the generator of a substochastic semigroup \(S^\dagger = \{S^\dagger (t)\}_{t \ge 0}\) on \({\mathcal {G}}_0\), which leaves invariant each \({\mathcal {G}}_\vartheta \), \(\vartheta > 0\).
Proof
Proposition 3.8
For each \(\vartheta >0\), the operator \((\widehat{L}_\vartheta ,\widehat{{\mathcal {D}}}_\vartheta )\) is the generator of a \(C_0\)-semigroup \(\widehat{S}_\vartheta := \{\widehat{S}_\vartheta (t)\}_{t\ge 0}\) of bounded operators on \({\mathcal {G}}_\vartheta \).
Proof
Proposition 3.9
Proof
Proposition 3.10
Let \(k_t^{\Lambda ,N}\) and \(q_t\) be as in (3.47) and in (3.44), (3.45), respectively. Then, for all \(t \in [0,T(\vartheta _1, \vartheta _0))\), it follows that \(k_t^{\Lambda ,N}=q_t\).
Proof
A priori \(k_t^{\Lambda ,N}\) and \(q_t\) lie in different spaces: \({\mathcal {K}}_{\vartheta _1}\) and \({\mathcal {G}}_\vartheta \), respectively. Note that the latter \(\vartheta \) can be arbitrary positive. The idea is to construct one more evolution \(q^{\Lambda ,N}_0\mapsto u_t\) in some intersection of these two spaces, related to the evolutions in (3.47) and (3.44). Then the proof will follow by the uniqueness as in Proposition 3.9.
3.2.3 Proof of Lemma 3.5
3.3 Proof of Theorem 2.4
4 The global boundedness
Lemma 4.1
Proof
Proof of Theorem 2.5
Footnotes
- 1.
It is an independent thinning of \(\mu _0\).
Notes
Acknowledgements
The present research was supported by the European Commission under the project STREVCOMS PIRSES-2013-612669 and by the SFB 701 “Spektrale Strukturen and Topologische Methoden in der Mathematik”. The authors are also grateful to the referee whose suggestions helped to improve the presentation of the paper.
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