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Quasi-invariance of completely random measures
Habeebat O. Ibraheem,
Eugene Lytvynov 
Methods of Functional Analysis and Topology, Volume: 24, Issue: 3, Pages: 207 - 239
Swansea University Author:
Eugene Lytvynov
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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...
| Published in: | Methods of Functional Analysis and Topology |
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| ISSN: | 1029-3531 |
| Published: |
2018
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| Online Access: |
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| URI: | https://cronfa.swan.ac.uk/Record/cronfa38956 |
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2018-03-06T14:14:22Z |
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| last_indexed |
2018-10-19T13:09:05Z |
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cronfa38956 |
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2018-10-19T11:44:45.0282633 v2 38956 2018-03-06 Quasi-invariance of completely random measures e5b4fef159d90a480b1961cef89a17b7 0000-0001-9685-7727 Eugene Lytvynov Eugene Lytvynov true false 2018-03-06 MACS 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 $\mathfrak G$. Journal Article Methods of Functional Analysis and Topology 24 3 207 239 1029-3531 31 10 2018 2018-10-31 http://mfat.imath.kiev.ua/article/?id=1083 COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University 2018-10-19T11:44:45.0282633 2018-03-06T11:25:36.3513215 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Habeebat O. Ibraheem 1 Eugene Lytvynov 0000-0001-9685-7727 2 0038956-19102018114258.pdf 38956v2.pdf 2018-10-19T11:42:58.5930000 Output 349651 application/pdf Version of Record true 2018-10-18T00:00:00.0000000 Released under the terms of a Creative Commons Attribution-ShareAlike 4.0 International License (CC BY-SA). true eng |
| title |
Quasi-invariance of completely random measures |
| spellingShingle |
Quasi-invariance of completely random measures Eugene Lytvynov |
| title_short |
Quasi-invariance of completely random measures |
| title_full |
Quasi-invariance of completely random measures |
| title_fullStr |
Quasi-invariance of completely random measures |
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Quasi-invariance of completely random measures |
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Quasi-invariance of completely random measures |
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e5b4fef159d90a480b1961cef89a17b7 |
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e5b4fef159d90a480b1961cef89a17b7_***_Eugene Lytvynov |
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Eugene Lytvynov |
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Habeebat O. Ibraheem Eugene Lytvynov |
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Methods of Functional Analysis and Topology |
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2018 |
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Swansea University |
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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 $\mathfrak G$. |
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2018-10-31T07:19:51Z |
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11.047306 |