Random Walks on Virtual Persistence Diagrams

Abstract: Persistence diagrams, with the bottleneck and Wasserstein metrics, represent the interval decomposition of persistence modules constructed from filtered topological data. Finite persistence diagrams on a metric pair $(X,A)$ form the free translation-invariant commutative Lipschitz monoid $D(X,A)$. The Grothendieck group $K(X,A)$ is the free translation-invariant Abelian Lipschitz group of virtual persistence diagrams, and the canonical embedding $D(X,A) \hookrightarrow K(X,A)$ is isometric for the Wasserstein-1 distance with translation-invariant metric $ρ$. When the pointed metric space $(X/A,\overline d_1,[A])$ is uniformly discrete, $(K(X,A),ρ)$ is a discrete locally compact abelian group, but may be uncountable. We construct a symmetric, translation-invariant Markov semigroup $(P_t){t\ge0}$ on $\ell^2(K(X,A))$ as the projective limit of semigroups defined on the finitely generated subgroups $K(X_F,A)$, induced by the VPD metric $ρ$. Its convolution kernels $(p_t){t\ge0}$ define a random walk on $K(X,A)$, and for each $t\ge0$ the support of $p_t$ is contained in a countable subgroup $H\le K(X,A)$. On $H$, the semigroup $(p_t)_{t\ge0}$ has a Lévy-Khintchine representation $\widehat p_t(θ)=\exp(-tλ_H(θ))$ for $θ\in\widehat H$, and the kernels $k_t(x,y)=p_t(x-y)$ define reproducing kernel Hilbert spaces $\mathcal H_t$ with dense truncated subspaces. We show that a small collection of scalar random-walk invariants determined by the Lévy-Khintchine exponent -- including return probabilities, collision probabilities, and diagonal resolvent values -- controls global regularity properties of diagram functionals.