Equilibrium diffusion on the cone of discrete Radon measures

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 ${\mathrm{\nabla }}^{\mathbb{K}}F$ 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 $\eta$. Let $\varphi :{\mathbb{R}}^d\times {\mathbb{R}}^d\to \mathbb{R}$ be a potential of pair interaction, and let $\mu$ be a corresponding Gibbs perturbation of (the distribution of) a completely random measure on ${\mathbb{R}}^d$. In particular, $\mu$ 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 $\mu$-a.s.
dense in ${\mathbb{R}}^d$. We consider the corresponding Dirichlet form $${\mathcal{E}}^{\mathbb{K}}(F,G)={\int }_{\mathbb{K}({\mathbb{R}}^d)}⟨{\mathrm{\nabla }}^{\mathbb{K}}F(\eta ),{\mathrm{\nabla }}^{\mathbb{K}}G(\eta ){⟩}_{T_{\eta }(\mathbb{K})}d\mu (\eta )\mathrm{.}$$ 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 $d\ge 2$, there exists a conservative diffusion process on $\mathbb{K}({\mathbb{R}}^d)$ which is properly associated with the Dirichlet form ${\mathcal{E}}^{\mathbb{K}}$.