Stochastic processes with Hodge Laplacian generators on k-forms

Develop a stochastic process on differential k-forms over compact Riemannian manifolds whose infinitesimal generator coincides with the Hodge Laplacian Δk, and ascertain its foundational properties and limiting behavior to bridge higher-form analysis with probabilistic dynamics.

Background

In contrast to graphs, where the normalized Laplacian serves as the generator of a random walk, and to manifolds, where the Laplace–Beltrami operator generates Brownian motion, no analogous stochastic process is currently established whose generator is the Hodge Laplacian acting on k-forms.

The paper highlights structural differences between random walks on graphs and Laplacian-based walks on higher-dimensional simplicial complexes, and notes the absence of a fully developed stochastic process tied to the Hodge Laplacian on forms. Establishing such a process would connect higher-order geometric analysis with stochastic dynamics and could illuminate spectral and topological features of differential forms.

References

Moreover, there is no fully developed stochastic process whose generator coincides with the Hodge Laplacian on general k-forms on manifolds. We omit the details here, but there remain many open problems in this direction for interested researchers.

— From the discrete to the continuous, from simplicial complexes to Riemannian manifolds. Approximating flows and cuts on manifolds by discrete versions (2512.05319 - Eidi et al., 4 Dec 2025) in Remark, Section 2.7 (Analogies and convergence)