Covariant Formulation of Non-linear Langevin Theory with Multiplicative Gaussian White Noises

TL;DR

This paper develops a covariant formulation of multi-dimensional non-linear Langevin equations with multiplicative Gaussian white noise, simplifying previous theories and clarifying their relation to deterministic systems.

Contribution

It introduces a covariant formalism for non-linear Langevin equations with multiplicative noise that is simpler and more general than prior approaches.

Findings

The formalism is covariant under variable transformations.

It applies to systems with or without detailed balance.

Connections to deterministic theory and stochastic calculus are clarified.

Abstract

The multi-dimensional non-linear Langevin equation with multiplicative Gaussian white noises in Ito's sense is made covariant with respect to non-linear transform of variables. The formalism involves no metric or affine connection, works for systems with or without detailed balance, and is substantially simpler than previous theories. Its relation with deterministic theory is clarified. The unitary limit and Hermitian limit of the theory are examined. Some implications on the choices of stochastic calculus are also discussed.