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Quasi-invariance of completely random measures


Abstract

Let $X$ be a locally compact Polish space. Let $\mathbb K(X)$ denote the space of discrete Radon measures on $X$. Let $\mu$ be a completely random discrete measure on $X$, i.e., $\mu$ is (the distribution of) a completely random measure on $X$ that is concentrated on $\mathbb K(X)$. We consider the multiplicative (current) group $C_0(X\to\mathbb R_+)$ consisting of functions on $X$ that take values in $\mathbb R_+=(0,\infty)$ and are equal to 1 outside a compact set. Each element $\theta\in C_0(X\to\mathbb R_+)$ maps $\mathbb K(X)$ onto itself; more precisely, $\theta$ sends a discrete Radon measure $\sum_i s_i\delta_{x_i}$ to $\sum_i \theta(s_i)s_i\delta_{x_i}$. Thus, elements of $C_0(X\to\mathbb R_+)$ transform the weights of discrete Radon measures. We study conditions under which the measure $\mu$ is quasi-invariant under the action of the current group $C_0(X\to\mathbb R_+)$ and consider several classes of examples. We further assume that $X=\mathbb R^d$ and consider the group of local diffeomorphisms $\operatorname{Diff}_0(X)$. Elements of this group also map $\mathbb K(X)$ onto itself. More precisely, a diffeomorphism $\varphi\in \operatorname{Diff}_0(X)$ sends a discrete Radon measure $\sum_i s_i\delta_{x_i}$ to $\sum_i s_i\delta_{\varphi(x_i)}$. Thus, diffeomorphisms from $\operatorname{Diff}_0(X)$ transform the atoms of discrete Radon measures. We study quasi-invariance of $\mu$ under the action of $\operatorname{Diff}_0(X)$. We finally consider the semidirect product $\mathfrak G:=\operatorname{Diff}_0(X)\times C_0(X\to \mathbb R_+)$ and study conditions of quasi-invariance and partial quasi-invariance of $\mu$ under the action of $\mathfrak G$.

Key words: Random measure, point process, Poisson point process, completely random measure, current group, diffeomorphism group.


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Article Information

TitleQuasi-invariance of completely random measures
SourceMethods Funct. Anal. Topology, Vol. 24 (2018), no. 3, 207-239
MilestonesReceived 15/08/2017; Revised 20/02/2018
CopyrightThe Author(s) 2018 (CC BY-SA)

Authors Information

Habeebat O. Ibraheem
Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, U.K.

Eugene Lytvynov
Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, U.K


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Citation Example

Habeebat O. Ibraheem and Eugene Lytvynov, Quasi-invariance of completely random measures, Methods Funct. Anal. Topology 24 (2018), no. 3, 207-239.


BibTex

@article {MFAT1083,
    AUTHOR = {Habeebat O. Ibraheem and Eugene Lytvynov},
     TITLE = {Quasi-invariance of completely random measures},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {24},
      YEAR = {2018},
    NUMBER = {3},
     PAGES = {207-239},
      ISSN = {1029-3531},
       URL = {http://mfat.imath.kiev.ua/article/?id=1083},
}


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