Stochastic dynamics with state-dependent diffusion arise broadly in confined, anisotropic, and hydrodynamically coupled systems. The conventional Langevin equation contains a spurious drift associated with multiplicative noise. Since the restricted free energy is generally nonscalar, the covariance is not explicit. Here, we reformulate state-dependent Langevin dynamics using a scalar free energy and a diffusion metric defined by the inverse diffusion tensor. The conventional spurious drift is then expressed as a Christoffel contribution. Although our formulation is mathematically equivalent to the conventional one through the relation between the restricted and scalar free energies, it makes coordinate covariance explicit and enables the phenomenological construction of thermodynamic potentials directly from system symmetries. We demonstrate its consistency in representative examples of state-dependent diffusion arising from coordinate transformations, geometric confinement, and projection from curved to flat spaces.

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• Brownian motion
• Stochastic processes
• Stochastic thermodynamics
• Fokker–Planck equation
• Langevin equation