Journal article 1360 views
Equilibrium Diffusion on the Cone of Discrete Radon Measures
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 |
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ISSN: | 0926-2601 1572-929X |
Published: |
2016
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URI: | https://cronfa.swan.ac.uk/Record/cronfa23989 |
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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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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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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 |
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1763751077230936064 |
score |
11.021648 |