  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 $\eta$.
Let $\phi:\mathbb R^d\times\mathbb R^d\to\mathbb R$ 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 $$
\mathscr E^{\mathbb K}(F,G)=\int_{\mathbb K(\mathbb R^d)}\langle\nabla^{\mathbb
K} F(\eta), \nabla^{\mathbb K} G(\eta)\rangle_{T_\eta(\mathbb K)}\,d\mu(\eta).
$$ Integrating by parts with respect to the measure $\mu$, 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 $\mathbb K(\mathbb R^d)$ which is properly associated with
the Dirichlet form $\mathscr E^{\mathbb K}$.
