On the canonical distributions of a thermal particle in the weakly confining potential of special type

We consider a thermal particle which is diffusing in velocity-space and in a weakly confining potential characterized by the inverse hyperbolic sine function of the particle velocity $v$ and the control parameter $v_c$. The stationary state of the Fokker-Planck equation is shown to be a canonical probability distribution. Furthermore an appropriate re-parametrization relates this stationary state with the $\kappa$-deformed Gaussian.