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Equilibrium Diffusion on the Cone of Discrete Radon Measures

Diana Conache, Yuri G. Kondratiev, Eugene Lytvynov Orcid Logo

Potential Analysis, Volume: 44, Issue: 1, Pages: 71 - 90

Swansea University Author: Eugene Lytvynov Orcid Logo

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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)...

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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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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
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title 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
document_store_str 0
active_str 0
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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score 10.99342