## 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

### Citations:

Zbl 0609.60058; Zbl 1097.60032; Zbl 1111.60029; Zbl 1121.60008
Full Text:

### References:

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