A graph discretized approximation of semigroups for diffusion with drift and killing on a complete Riemannian manifold

TL;DR

This paper demonstrates that the continuous semigroup generated by a Schrödinger operator with drift on a complete Riemannian manifold can be approximated by discrete semigroups from random walks on proximity graphs, with error estimates in the compact case.

Contribution

It introduces a novel approximation method for semigroups on manifolds using discrete random walks and provides quantitative convergence estimates.

Findings

Approximation of continuous semigroups by discrete random walks.

Error estimates for convergence on compact manifolds.

Examples on Euclidean and model manifolds.

Abstract

In the present paper, we prove that the $C_{0}$-semigroup generated by a Schr\"odinger operator with drift on a complete Riemannian manifold is approximated by the discrete semigroups associated with a family of discrete time random walks with killing in a flow on a sequence of proximity graphs, which are constructed by partitions of the manifold. Furthermore, when the manifold is compact, we also obtain a quantitative error estimate of the convergence. Finally, we give examples of the partition of the manifold and the drift term on two typical manifolds: Euclidean spaces and model manifolds.