Random Walks on Virtual Persistence Diagrams

Abstract

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 $\hat{I}{»}_H$ satisfying $\hat{I}{»}_H(\hat{I}¸)={\sum }_{\hat{I}º\in H\setminus \{0\}}(1-\Re \hat{I}¸(\hat{I}º))\hat{I}½(\hat{I}º),$ for a symmetric $\hat{I}½\in {\ell }^1(H\setminus \{0\})$. This gives a symmetric jump process on $H$. The exponent $\hat{I}{»}_H$ 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.

40 pages