  Statistical invariance of Wiener increments under SO(n) rotations provides a
notion of gauge transformation of state-dependent Brownian motion. We show that
the stochastic dynamics of non gauge-invariant systems is not unambiguously
defined. They typically do not relax to equilibrium steady states even in the
absence of extenal forces. Assuming both coordinate covariance and gauge
invariance, we derive a second-order Langevin equation with state-dependent
diffusion matrix and vanishing environmental forces. It differs from previous
proposals but nevertheless entails the Einstein relation, a Maxwellian
conditional steady state for the velocities, and the equipartition theorem. The
over-damping limit leads to a stochastic differential equation in state space
that cannot be interpreted as a pure differential (Ito, Stratonovich or else).
At odds with the latter interpretations, the corresponding Fokker-Planck
equation admits an equilibrium steady state; a detailed comparison with other
theories of state-dependent diffusion is carried out. We propose this as a
theory of diffusion in a heat bath with varying temperature. Besides
equilibrium, a crucial experimental signature is the non-uniform steady spatial
distribution.
