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

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.