  There is a growing interest in the stochastic processes of nonequilibrium
systems subject to non-conserved forces, such as the magnetic forces acting on
charged particles and the chiral self-propelled force acting on active
particles. In this paper, we consider the stationary transport of
non-interacting Brownian particles under a constant magnetic field in a
position-dependent temperature background. We demonstrate the existence of the
Nernst-like stationary density current perpendicular to both the temperature
gradient and magnetic field, induced by the intricate coupling between the
non-conserved force and the multiplicative noises due to the position-dependent
temperature.
