  We study stochastic thermodynamics of a Brownian particle which is subjected
to a temperature gradient and is confined by an external potential. We first
formulate an over-damped Ito-Langevin theory in terms of local temperature,
friction coefficient, and steady state distribution, all of which are
experimentally measurable. We then study the associated stochastic
thermodynamics theory. We analyze the excess entropy production (EP) both at
trajectory level and at ensemble level, and derive the Clausius inequality as
well as the transient fluctuation theorem (FT). We also use molecular dynamics
to simulate a Brownian particle inside a Lennard-Jones fluid and verify the FT.
Remarkably we find that the FT remains valid even in the under-damped regime.
We explain the possible mechanism underlying this surprising result.
