Inertial effects in the fractional diffusion of a Brownian particle in a double-well potential

The anomalous translational diffusion including inertial effects of nonlinear Brownian oscillators in a double well potential $V(x)=ax^2/2+bx^4/4$ is considered. An exact solution of the fractional Klein-Kramers (Fokker-Planck) equation is obtained which allows one to calculate via matrix continued fractions the positional autocorrelation function and dynamic susceptibility describing the position response to a small external field. The result is a generalization of the solution for the normal Brownian motion in a double well potential to fractional dynamics (giving rise to anomalous diffusion).