  We study stochastic motion under a nonlinear frictional force that levels off
with increasing velocity. Specifically, our frictional force is of the
so-called Coulomb-tanh type. At small speed, it increases approximately
linearly with velocity, while at large speed it approaches a constant
magnitude, similarly to solid (dry, Coulomb) friction. In one spatial
dimension, a formal analogy between the associated Fokker-Planck equation and
the Schr\"odinger equation for a quantum-mechanical oscillator in a nonharmonic
P\"oschl-Teller potential is revealed. Then, the stationary velocity statistics
can be treated analytically. From such analytical considerations, we determine
associated diffusion coefficients, which we confirm by agent-based simulations.
Moreover, from such simulations and from numerically solving the associated
Fokker-Planck equation, we find that the spatial distribution function,
starting from an initial Gaussian shape, develops tails that appear exponential
at intermediate time scales. At small magnitudes of stochastic driving, the
velocity distribution resembles the case of linear friction, while at large
magnitudes it rather approaches the case of solid (dry, Coulomb) friction. The
same is true for diffusion coefficients. In a certain sense, thus interpolating
between the two extreme cases of linear friction and solid (dry, Coulomb)
friction, our approach should be useful to describe several cases of practical
relevance. For instance, a reduced increase in friction with increasing
relative speed is typical of shear-thinning behavior. Therefore, driven motion
in shear-thinning environments is one specific example to which our description
may be applied.
