Infinitesimal generators for a class of polynomial processes

TL;DR

This paper investigates the infinitesimal generators of polynomial process evolutions, specifically quadratic harnesses, by linking them to a commutation equation and deriving explicit formulas for free quadratic harnesses.

Contribution

It introduces a novel connection between infinitesimal generators and a commutation equation for polynomial processes, providing explicit formulas for free quadratic harnesses.

Findings

Derived explicit formulas for infinitesimal generators of free quadratic harnesses.

Linked the generators to solutions of a specific commutation equation.

Enhanced understanding of polynomial process dynamics through this mathematical framework.

Abstract

We study the infinitesimal generators of evolutions of linear mappings on the space of polynomials, which correspond to a special class of Markov processes with polynomial regressions called quadratic harnesses. We relate the infinitesimal generator to the unique solution of a certain commutation equation, and we use the commutation equation to find an explicit formula for the infinitesimal generator of free quadratic harnesses.