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

Convergence Rate for Degenerate Partial and Stochastic Differential Equations via weak Poincar\'e Inequalities