Brownian motion, martingales and It\^o formula in Clifford analysis

TL;DR

This paper advances stochastic Clifford analysis by developing concepts like Brownian motion, martingales, and Itô calculus within Clifford analysis, aiming to bridge stochastic methods with Clifford-based mathematical objects.

Contribution

It introduces stochastic analysis tools such as Brownian motion and Itô formula in Clifford analysis, expanding the theoretical framework of the field.

Findings

Development of Brownian motion in Clifford analysis

Formulation of Itô formula in Clifford setting

Potential applications to PDEs and positive definite functions

Abstract

Clifford analysis has been the field of active research for several decades resulting in various methods to solve problems in pure and applied mathematics. However, the area of stochastic analysis has not been addressed in its full generality in the Clifford setting, since only a few contributions have been presented so far. Considering that the tools of stochastic analysis play an important role in the study of objects, such as positive definite functions, reproducing kernels and partial differential equations, it is important to develop tools for the study of these objects in the context of Clifford analysis. Therefore, in this work-in-progress paper, we present further steps towards stochastic Clifford analysis by studying random variables, martingales, Brownian motion, and It\^o formula in the Clifford setting, as well as their applications in Clifford analysis.