An It\^o calculus for a class of limit processes arising from random walks on the complex plane

TL;DR

This paper extends stochastic calculus to certain limit processes from complex plane random walks, enabling new analytical tools for differential equations and complex analysis.

Contribution

It introduces a generalized Itô calculus for higher order diffusion processes related to complex plane random walks, expanding the mathematical framework.

Findings

Develops a stochastic calculus for higher order diffusions

Provides a Feynman-Kac formula for these processes

Offers a representation for higher derivatives of analytic functions

Abstract

Within the framework of the previous paper [8]. we develop a generalized stochastic calculus for processes associated to higher order diffusion operators. Applications to the study of a Cauchy problem, a Feynman-Kac formula and a representation formula for higher derivatives of analytic functions are also given.