Chernoff approximation of subordinate semigroups and applications

TL;DR

This paper develops a method using Chernoff's theorem to approximate subordinate semigroups, enabling explicit calculations and modeling of complex stochastic processes on various geometric structures.

Contribution

It introduces a novel approach to approximate subordinate semigroups via Chernoff's theorem, applicable to Feller processes and diffusions on multiple geometries.

Findings

Explicit operator-based approximations for subordinate semigroups

Representations via Feynman formulae for computational use

Applicability to processes on Euclidean spaces, graphs, and manifolds

Abstract

In this note the Chernoff Theorem is used to approximate evolution semigroups constructed by the procedure of subordination. The considered semigroups are subordinate to some original, unknown explicitly but already approximated by the same method, counterparts with respect to subordinators either with known transitional probabilities, or with known and bounded L\'evy measure. These results are applied to obtain approximations of semigroups corresponding to subordination of Feller processes, and (Feller type) diffusions in Euclidean spaces, star graphs and Riemannian manifolds. The obtained approximations are based on explicitly given operators and hence can be used for direct calculations and computer modelling. In several cases the obtained approximations are given as iterated integrals of elementary functions and lead to representations of the considered semigroups by Feynman formulae.