  We employ weak hypocoercivity methods to study the long-term behavior of
operator semigroups generated by degenerate Kolmogorov operators with variable
second-order coefficients, which solve the associated abstract Cauchy problem.
We prove essential m-dissipativity of the operator, which extends previous
results and is key to the rigorous analysis required. We give estimates for the
$L^2$-convergence rate by using weak Poincar\'e inequalities. As an
application, we obtain estimates for the (sub-)exponential convergence rate of
solutions to the corresponding degenerate Fokker-Planck equations and of weak
solutions to the corresponding degenerate stochastic differential equation with
multiplicative noise.
