Journal article 1440 views
Equilibrium Diffusion on the Cone of Discrete Radon Measures
Diana Conache,
Yuri G. Kondratiev,
Eugene Lytvynov 
Potential Analysis, Volume: 44, Issue: 1, Pages: 71 - 90
Swansea University Author:
Eugene Lytvynov
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DOI (Published version): 10.1007/s11118-015-9499-9
Abstract
Let $K(R^d)$ denote the cone of discrete Radon measures on $R^d$.There is a natural differentiation on $K(R^d)$: for a differentiable function $F:K(R^d)\to R$, one defines its gradient $\nabla^K F $ as a vector field which assigns to each $\eta\in K(R^d)$ an element of a tangent space $T_\eta(K(R^d)...
| Published in: | Potential Analysis |
|---|---|
| ISSN: | 0926-2601 1572-929X |
| Published: |
2016
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| URI: | https://cronfa.swan.ac.uk/Record/cronfa23989 |
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<?xml version="1.0"?><rfc1807><datestamp>2019-08-12T13:02:22.9468799</datestamp><bib-version>v2</bib-version><id>23989</id><entry>2015-10-26</entry><title>Equilibrium Diffusion on the Cone of Discrete Radon Measures</title><swanseaauthors><author><sid>e5b4fef159d90a480b1961cef89a17b7</sid><ORCID>0000-0001-9685-7727</ORCID><firstname>Eugene</firstname><surname>Lytvynov</surname><name>Eugene Lytvynov</name><active>true</active><ethesisStudent>false</ethesisStudent></author></swanseaauthors><date>2015-10-26</date><deptcode>SMA</deptcode><abstract>Let $K(R^d)$ denote the cone of discrete Radon measures on $R^d$.There is a natural differentiation on $K(R^d)$: for a differentiable function $F:K(R^d)\to R$, one defines its gradient $\nabla^K F $ as a vector field which assigns to each $\eta\in K(R^d)$ an element of a tangent space $T_\eta(K(R^d))$ to $K(R^d)$ at point $\eta$. Let $\phi:R^d\times R^d\to\R$ be a potential of pair interaction, and let $\mu$ be a corresponding Gibbs perturbation of (the distribution of) a completely random measure on $R^d$. In particular, $\mu$ is a probability measure on $K(\R^d)$ such that the set of atoms of a discrete measure $\eta\in K(R^d)$ is $\mu$-a.s. dense in $R^d$. We consider the corresponding Dirichlet form$$\mathcal E^K(F,G)=\int_{K\R^d)}\langle\nabla^K F(\eta), \nabla^K G(\eta)\rangle_{T_\eta(K)}\,d\mu(\eta).$$Integrating by parts with respect to the measure $\mu$, we explicitly find the generator of this Dirichlet form. By using the theory of Dirichlet forms, we prove the main result of the paper: If $d\ge2$, there exists a conservative diffusion process on $K(R^d)$ which is properly associated with the Dirichlet form $\mathcal E^K$.</abstract><type>Journal Article</type><journal>Potential Analysis</journal><volume>44</volume><journalNumber>1</journalNumber><paginationStart>71</paginationStart><paginationEnd>90</paginationEnd><publisher/><issnPrint>0926-2601</issnPrint><issnElectronic>1572-929X</issnElectronic><keywords/><publishedDay>31</publishedDay><publishedMonth>12</publishedMonth><publishedYear>2016</publishedYear><publishedDate>2016-12-31</publishedDate><doi>10.1007/s11118-015-9499-9</doi><url/><notes></notes><college>COLLEGE NANME</college><department>Mathematics</department><CollegeCode>COLLEGE CODE</CollegeCode><DepartmentCode>SMA</DepartmentCode><institution>Swansea University</institution><apcterm/><lastEdited>2019-08-12T13:02:22.9468799</lastEdited><Created>2015-10-26T17:45:38.0789468</Created><path><level id="1">Faculty of Science and Engineering</level><level id="2">School of Mathematics and Computer Science - Mathematics</level></path><authors><author><firstname>Diana</firstname><surname>Conache</surname><order>1</order></author><author><firstname>Yuri G.</firstname><surname>Kondratiev</surname><order>2</order></author><author><firstname>Eugene</firstname><surname>Lytvynov</surname><orcid>0000-0001-9685-7727</orcid><order>3</order></author></authors><documents/><OutputDurs/></rfc1807> |
| spelling |
2019-08-12T13:02:22.9468799 v2 23989 2015-10-26 Equilibrium Diffusion on the Cone of Discrete Radon Measures e5b4fef159d90a480b1961cef89a17b7 0000-0001-9685-7727 Eugene Lytvynov Eugene Lytvynov true false 2015-10-26 SMA Let $K(R^d)$ denote the cone of discrete Radon measures on $R^d$.There is a natural differentiation on $K(R^d)$: for a differentiable function $F:K(R^d)\to R$, one defines its gradient $\nabla^K F $ as a vector field which assigns to each $\eta\in K(R^d)$ an element of a tangent space $T_\eta(K(R^d))$ to $K(R^d)$ at point $\eta$. Let $\phi:R^d\times R^d\to\R$ be a potential of pair interaction, and let $\mu$ be a corresponding Gibbs perturbation of (the distribution of) a completely random measure on $R^d$. In particular, $\mu$ is a probability measure on $K(\R^d)$ such that the set of atoms of a discrete measure $\eta\in K(R^d)$ is $\mu$-a.s. dense in $R^d$. We consider the corresponding Dirichlet form$$\mathcal E^K(F,G)=\int_{K\R^d)}\langle\nabla^K F(\eta), \nabla^K G(\eta)\rangle_{T_\eta(K)}\,d\mu(\eta).$$Integrating by parts with respect to the measure $\mu$, we explicitly find the generator of this Dirichlet form. By using the theory of Dirichlet forms, we prove the main result of the paper: If $d\ge2$, there exists a conservative diffusion process on $K(R^d)$ which is properly associated with the Dirichlet form $\mathcal E^K$. Journal Article Potential Analysis 44 1 71 90 0926-2601 1572-929X 31 12 2016 2016-12-31 10.1007/s11118-015-9499-9 COLLEGE NANME Mathematics COLLEGE CODE SMA Swansea University 2019-08-12T13:02:22.9468799 2015-10-26T17:45:38.0789468 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Diana Conache 1 Yuri G. Kondratiev 2 Eugene Lytvynov 0000-0001-9685-7727 3 |
| title |
Equilibrium Diffusion on the Cone of Discrete Radon Measures |
| spellingShingle |
Equilibrium Diffusion on the Cone of Discrete Radon Measures Eugene Lytvynov |
| title_short |
Equilibrium Diffusion on the Cone of Discrete Radon Measures |
| title_full |
Equilibrium Diffusion on the Cone of Discrete Radon Measures |
| title_fullStr |
Equilibrium Diffusion on the Cone of Discrete Radon Measures |
| title_full_unstemmed |
Equilibrium Diffusion on the Cone of Discrete Radon Measures |
| title_sort |
Equilibrium Diffusion on the Cone of Discrete Radon Measures |
| author_id_str_mv |
e5b4fef159d90a480b1961cef89a17b7 |
| author_id_fullname_str_mv |
e5b4fef159d90a480b1961cef89a17b7_***_Eugene Lytvynov |
| author |
Eugene Lytvynov |
| author2 |
Diana Conache Yuri G. Kondratiev Eugene Lytvynov |
| format |
Journal article |
| container_title |
Potential Analysis |
| container_volume |
44 |
| container_issue |
1 |
| container_start_page |
71 |
| publishDate |
2016 |
| institution |
Swansea University |
| issn |
0926-2601 1572-929X |
| doi_str_mv |
10.1007/s11118-015-9499-9 |
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Faculty of Science and Engineering |
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|
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facultyofscienceandengineering |
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Faculty of Science and Engineering |
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facultyofscienceandengineering |
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Faculty of Science and Engineering |
| department_str |
School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics |
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0 |
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| description |
Let $K(R^d)$ denote the cone of discrete Radon measures on $R^d$.There is a natural differentiation on $K(R^d)$: for a differentiable function $F:K(R^d)\to R$, one defines its gradient $\nabla^K F $ as a vector field which assigns to each $\eta\in K(R^d)$ an element of a tangent space $T_\eta(K(R^d))$ to $K(R^d)$ at point $\eta$. Let $\phi:R^d\times R^d\to\R$ be a potential of pair interaction, and let $\mu$ be a corresponding Gibbs perturbation of (the distribution of) a completely random measure on $R^d$. In particular, $\mu$ is a probability measure on $K(\R^d)$ such that the set of atoms of a discrete measure $\eta\in K(R^d)$ is $\mu$-a.s. dense in $R^d$. We consider the corresponding Dirichlet form$$\mathcal E^K(F,G)=\int_{K\R^d)}\langle\nabla^K F(\eta), \nabla^K G(\eta)\rangle_{T_\eta(K)}\,d\mu(\eta).$$Integrating by parts with respect to the measure $\mu$, we explicitly find the generator of this Dirichlet form. By using the theory of Dirichlet forms, we prove the main result of the paper: If $d\ge2$, there exists a conservative diffusion process on $K(R^d)$ which is properly associated with the Dirichlet form $\mathcal E^K$. |
| published_date |
2016-12-31T03:28:22Z |
| _version_ |
1763751077230936064 |
| score |
11.0302305 |