Covariant formulation of nonlinear Langevin theory with multiplicative Gaussian white noises

Abstract

The multidimensional nonlinear Langevin equation with multiplicative Gaussian white noises in Ito's sense is made covariant with respect to nonlinear 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.