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The multi-dimensional non-linear Langevin equation with multiplicativeGaussian white noises in Ito's sense is made covariant with respect tonon-linear transform of variables. The formalism involves no metric or affineconnection, works for systems with or without detailed balance, and issubstantially simpler than previous theories. Its relation with deterministictheory is clarified. The unitary limit and Hermitian limit of the theory areexamined. Some implications on the choices of stochastic calculus are alsodiscussed.

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