Symmetries of Stochastic Differential Equations: a geometric approach

TL;DR

This paper introduces a geometric framework for analyzing symmetries of stochastic differential equations, linking weak and strong symmetries through stochastic transformations, with applications to symmetric ODEs and Brownian motion.

Contribution

It proposes a new notion of stochastic transformation and establishes a correspondence between weak and strong symmetries of SDEs, advancing the theoretical understanding of stochastic symmetry analysis.

Findings

Established a correspondence between weak and strong symmetries under regularity conditions.

Applied the framework to a stochastic symmetric ODE and 2D Brownian motion.

Demonstrated the utility of the approach in specific stochastic models.

Abstract

A new notion of stochastic transformation is proposed and applied to the study of both weak and strong symmetries of stochastic differential equations (SDEs). The correspondence between an algebra of weak symmetries for a given SDE and an algebra of strong symmetries for a modified SDE is proved under suitable regularity assumptions. This general approach is applied to a stochastic version of a two dimensional symmetric ordinary differential equation and to the case of two dimensional Brownian motion.