Superposition rules and stochastic Lie-Scheffers systems

TL;DR

This paper extends the classical Lie-Scheffers Theorem to stochastic differential equations, providing a characterization of superposition rules through involution properties of vector field distributions, with applications to stochastic Lie group systems.

Contribution

It introduces a stochastic version of the Lie-Scheffers Theorem, linking superposition rules to involution properties in stochastic systems, and improves classical results for deterministic cases.

Findings

Stochastic Lie-Scheffers systems can be characterized by involution properties.

Reduction to Lie group valued stochastic systems simplifies analysis.

Multiple examples illustrate the theoretical developments.

Abstract

This paper proves a version for stochastic differential equations of the Lie-Scheffers Theorem. This result characterizes the existence of nonlinear superposition rules for the general solution of those equations in terms of the involution properties of the distribution generated by the vector fields that define it. When stated in the particular case of standard deterministic systems, our main theorem improves various aspects of the classical Lie-Scheffers result. We show that the stochastic analog of the classical Lie-Scheffers systems can be reduced to the study of Lie group valued stochastic Lie-Scheffers systems; those systems, as well as those taking values in homogeneous spaces are studied in detail. The developments of the paper are illustrated with several examples.