  Using the recently developed covariant Ito-Langevin dynamics, we develop a
non-equilibrium thermodynamic theory for small systems coupled to
multiplicative noises. The theory is based on Ito-calculus, and is fully
covariant under time-independent nonlinear transformation of variables.
Assuming instantaneous detailed balance, we derive expressions for various
thermodynamic functions, including work, heat, entropy production, and free
energy, both at ensemble level and at trajectory level, and prove the second
law of thermodynamics for arbitrary non-equilibrium processes. We relate
time-reversal asymmetry of path probability to entropy production, and derive
its consequences such as fluctuation theorem and non-equilibrium work relation.
For Langevin systems with additive noises, our theory is equivalent to the
common theories of stochastic energetics and stochastic thermodynamics. We also
discuss examples of multiplicative noises where the common theories are
inapplicable, but our theory yields correct results.
