Equilibrium diffusion on the cone of discrete Radon measures

Abstract

Let $\mathbb K(\mathbb R^d)$ denote the cone of discrete Radon measures on $\mathbb R^d$. There is a natural differentiation on $\mathbb K(\mathbb R^d)$: for a differentiable function $F:\mathbb K(\mathbb R^d)\to\mathbb R$, one defines its gradient $\nabla^{\mathbb K} F $ as a vector field which assigns to each $η\in \mathbb K(\mathbb R^d)$ an element of a tangent space $T_η(\mathbb K(\mathbb R^d))$ to $\mathbb K(\mathbb R^d)$ at point $η$. Let $ϕ:\mathbb R^d\times\mathbb R^d\to\mathbb R$ be a potential of pair interaction, and let $μ$ be a corresponding Gibbs perturbation of (the distribution of) a completely random measure on $\mathbb R^d$. In particular, $μ$ is a probability measure on $\mathbb K(\mathbb R^d)$ such that the set of atoms of a discrete measure $η\in\mathbb K(\mathbb R^d)$ is $μ$-a.s.\ dense in $\mathbb R^d$. We consider the corresponding Dirichlet form $$ \mathscr E^{\mathbb K}(F,G)=\int_{\mathbb K(\mathbb R^d)}\langle\nabla^{\mathbb K} F(η), \nabla^{\mathbb K} G(η)\rangle_{T_η(\mathbb K)}\,dμ(η). $$ Integrating by parts with respect to the measure $μ$, 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 $\mathbb K(\mathbb R^d)$ which is properly associated with the Dirichlet form $\mathscr E^{\mathbb K}$.