  We study the long-time dynamics of two-dimensional linear Fokker-Planck
equations driven by a drift that can be decomposed in the sum of a large shear
component and the gradient of a regular potential depending on one spatial
variable. The problem can be interpreted as that of a passive scalar advected
by a slightly compressible shear flow, and undergoing small diffusion. For the
corresponding stochastic differential equation, we give explicit homogenization
rates in terms of a family of time-scales depending on the parameter measuring
the strength of the incompressible perturbation. This is achieved by exploiting
an auxiliary Poisson problem, and by computing the related effective diffusion
coefficients. Regarding the long-time behaviour of the solution of the
Fokker-Planck equation, we provide explicit decay rates to the unique invariant
measure by employing a quantitative version of the classical hypocoercivity
scheme. From a fluid mechanics perspective, this turns out to be equivalent to
quantifying the phenomenon of enhanced diffusion for slightly compressible
shear flows.
