Equilibrium Diffusion on the Cone of Discrete Radon Measures

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 $\mathcal E^K$