Octonion random functions and integration of stochastic PDEs

TL;DR

This paper develops a framework for stochastic integration over octonion and Cayley-Dickson algebras, enabling the analysis and solution of higher-order stochastic PDEs of various types.

Contribution

It introduces a novel approach to stochastic integrals in non-associative algebras and applies it to solve complex stochastic PDEs beyond second order.

Findings

Established stochastic integrals over octonion and Cayley-Dickson algebras.

Applied the method to integrate higher-order stochastic PDEs.

Enabled analysis of parabolic, elliptic, and hyperbolic stochastic PDEs.

Abstract

In the article random functions in modules over the octonion algebra and Cayley-Dickson algebras are investigated. For their study transition measures with values in the octonion algebra and Cayley-Dickson algebras are used. Stochastic integrals over these algebras are studied. They are applied to integration of stochastic PDEs. This approach permits subsequently to analyze and integrate PDEs of orders higher than two of different types including parabolic, elliptic and hyperbolic.