Two-temperature Langevin dynamics in a parabolic potential

We study a planar two-temperature diffusion of a Brownian particle in a parabolic potential. The diffusion process is defined in terms of two Langevin equations with two different effective temperatures in the X and the Y directions. In the stationary regime the system is described by a non-trivial particle position distribution P(x,y), which we determine explicitly. We show that this distribution corresponds to a non-equilibrium stationary state, characterised by the presence of space-dependent particle currents which exhibit a non-zero rotor. Theoretical results are confirmed by the numerical simulations.