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Invariant transports of stationary random measures and mass-stationarity. (English) Zbl 1176.60036

The present paper treats invariant weighted transport-kernels \(T =\) \(T(\omega, s, \cdot)\) ( on \(\Omega \times G\)) balancing stationary random measures \(\{ \xi \}\) on a locally compact abelian group \(G\). Three theorems on interwoven aspects of invariant weighted transport-kernels are stated. The first main result is an associated fundamental invariance property of Palm measure, derived from a generalization of Neveu’s exchange fromula, [cf. J. Neveu, Probabilités XX, Proc. Sémin., Strasbourg, 1984/85, Lect. Notes Math. 1204, 503–514 (1986; Zbl 0609.60058)].
Theorem 1. Let \(\xi\) and \(\eta\) be invariant random measures on \(G\), and let \(T\) be an invariant weighted transport-kernel. Then \(T\) is \({\mathbb P}\)-a.e. \((\xi, \eta)\)-balancing if and only if
\[ {\mathbb E}_{ {\mathbb P}_{\xi} } \left\{ \int f( \theta_t ) T( 0, dt) \right\} = {\mathbb E}_{ {\mathbb P}_{\eta} } \{ T \} \tag{1} \]
holds for all measurable \(f :\) \(\Omega \to\) \([0, \infty)\), where \({\mathbb P}_{\xi}\) (resp. \({\mathbb P}_{\eta}\)) is the Palm measure of \(\xi\) (resp. \(\eta\)) respectively with respect to a \(\sigma\)-finite stationary measure \({\mathbb P}\) on \(( \Omega, {\mathcal F} )\).
The second main result is a sufficient and necessary criterion for the existence of balancing invariant transport-kernels.
Theorem 2. Let \(\xi\) and \(\eta\) be invariant random measures on \(G\) with positive and finite intensities. Then there exists a \({\mathbb P}\)-a.e. \(( \xi, \eta)\)-balancing transport-kernel if and only if
\[ {\mathbb E}_{ {\mathbb P} } \{ \xi(B) | {\mathcal I} \} = {\mathbb E}_{ {\mathbb P} }\{ \eta (B) | {\mathcal I} \}, \quad {\mathbb P}\text{-a.e.} \tag{2} \]
for some \(B \in {\mathcal G}\) satisfying \(0 < \lambda (B) < \infty\), where \({\mathcal I}\) \(\subset\) \({\mathcal F}\) is the invarinat \(\sigma\)-field of \({\mathcal F}\), and \(\lambda\) is a positive and finite Haar measure.
The third main result is about a necessary and sufficient condition for a \(\sigma\)-finite measure \({\mathbb Q}\) on \(( \Omega, {\mathcal F})\) to be the Palm measure of \(\xi\) with respect to some stationary \(\sigma\)-finite measure \({\mathbb P}\) on \(( \Omega, {\mathcal F})\). In order to describe the statement, the authors introduce the notion of mass-stationarity. In fact, the \(\sigma\)-finite measure \({\mathbb Q}\) on \(( \Omega, {\mathcal F})\) is called mass-stationary for \(\xi\) if \({\mathbb Q}( \xi(G) = 0 )\) \(=0\) and
\[ {\mathbb E}_{ {\mathbb Q} } \left\{ \iint 1_A( \theta_s, s+r) T_C( - r, ds) \lambda_C(dr) \right\} = {\mathbb Q} \otimes \lambda_C(A), \quad A \in {\mathcal F} \otimes {\mathcal G} \tag{3} \]
holds for all relatively compact sets \(C \in {\mathcal G}\) with \(\lambda(C) >0\) and \(\lambda( \partial C) = 0\), where \(T_C\) is an invariant transport-kernel for \(C \in {\mathcal G}\), defined by
\[ T_C(t, B) := \xi(C + t)^{-1} \xi( B \cap ( C + t) ), \quad t \in G, \quad B \in {\mathcal G} \tag{4} \]
if \(\xi(C + t) > 0\), and \(\lambda_C\) is the uniform distribution on \(G\), defined by \[ \lambda_C( B) := \lambda( B \cap C) / \lambda(C). \tag{5} \] Here is the last main theorem: Theorem 3. There exists a \(\sigma\)-finite stationary measure \({\mathbb P}\) on \(( \Omega, {\mathcal F})\) such that \({\mathbb Q} = {\mathbb P}_{\xi}\) if and only if \({\mathbb Q}\) is mass-stationary for \(\xi\).
The first result is motivated by Theorem 16 in the recent paper by A. E. Holroyd and Y. Peres [Ann. Probab. 33, No. 1, 31–52 (2005; Zbl 1097.60032)]. For other related work, [see e.g. M. Heveling and G. Last, Ann. Probab. 33, No. 5, 1698–1715 (2005; Zbl 1111.60029)], and G. Last [Adv. Appl. Probab. 38, No. 3, 602–620 (2006; Zbl 1121.60008)].

MSC:

60G57 Random measures
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60G60 Random fields
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