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 $\eta \in \mathbb {K}(\mathbb {R}^{d})$ an element of a tangent space $T_{\eta }(\mathbb {K}(\mathbb {R}^{d}))$ to $\mathbb {K}(\mathbb {R}^{d})$ at point η. Let $\phi :\mathbb {R}^{d}\times \mathbb {R}^{d}\to \mathbb {R}$ 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 $\eta \in \mathbb {K}(\mathbb {R}^{d})$ is μ-a.s. dense in $\mathbb {R}^{d}$. We consider the corresponding Dirichlet form

$$\mathcal{E}^{\mathbb{K}}(F,G)={\int}_{\mathbb K(\mathbb{R}^{d})}\langle\nabla^{\mathbb{K}} F(\eta), \nabla^{\mathbb{K}} G(\eta)\rangle_{T_{\eta}(\mathbb{K})}\,d\mu(\eta). $$

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 ≥ 2, there exists a conservative diffusion process on $\mathbb {K}(\mathbb {R}^{d})$ which is properly associated with the Dirichlet form $\mathcal {E}^{\mathbb {K}}$.