Underdamped scaled Brownian motion

Brownian motion with time-dependent diffusion coefficient is ubiquitous in nature. It has been observed for the mobility of proteins in cell membranes, motion of molecules in porous environment, water diffusion in brain measured in terms of magnetic resonance imaging and also in media with time-dependent temperature such as free cooling granular materials or melting snow.

We investigate a new type of anomalous diffusion processes governed by an underdamped Langevin equation with time-dependent diffusion and friction coefficients and discuss possible applications to real physical systems such as free cooling granular materials. We show that for certain range of parameter values the overdamped limit for the Langevin equation does not exist.