Random Walks on Virtual Persistence Diagrams

In the uniformly discrete case of virtual persistence diagram groups $K(X,A)$, we construct a translation-invariant heat semigroup. The kernels are supported on a countable subgroup $H$, and the restriction to $H$ has Fourier exponent ${\lambda }_H$ satisfying ${\lambda }_H(\theta )={\sum }_{\kappa \in H\setminus \{0\}}(1-\mathrm{\Re }\theta (\kappa ))\nu (\kappa ),$ for a symmetric $\nu \in {\mathrm{\ell }}^1(H\setminus \{0\})$. This gives a symmetric jump process on $H$. The exponent ${\lambda }_H$ determines heat kernels, which define reproducing kernel Hilbert spaces and their associated semimetrics. Convex orders on the mixing measures give monotonicity for the kernels, Hilbert spaces, and semimetrics.