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### 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:

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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 $T_\eta(K(R^d)... Full description Published in: Potential Analysis 0926-2601 1572-929X 2016 https://cronfa.swan.ac.uk/Record/cronfa23989 No Tags, Be the first to tag this record! Abstract: Let$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\$. College of Science 1 71 90