  We provide a complete elaboration of the $L^2$-Hilbert space hypocoercivity
theorem for the degenerate Langevin dynamics with multiplicative noise,
studying the longtime behaviour of the strongly continuous contraction
semigroup solving the abstract Cauchy problem for the associated backward
Kolmogorov operator. Hypocoercivity for the Langevin dynamics with constant
diffusion matrix was proven previously by Dolbeault, Mouhot and Schmeiser in
the corresponding Fokker-Planck framework, and made rigorous in the Kolmogorov
backwards setting by Grothaus and Stilgenbauer. We extend these results to
weakly differentiable diffusion coefficient matrices, introducing
multiplicative noise for the corresponding stochastic differential equation.
The rate of convergence is explicitly computed depending on the choice of these
coefficients and the potential giving the outer force. In order to obtain a
solution to the abstract Cauchy problem, we first prove essential
self-adjointness of non-degenerate elliptic Dirichlet operators on Hilbert
spaces, using prior elliptic regularity results and techniques from Bogachev,
Krylov and R\"ockner. We apply operator perturbation theory to obtain essential
m-dissipativity of the Kolmogorov operator, extending the m-dissipativity
results from Conrad and Grothaus. We emphasize that the chosen Kolmogorov
approach is natural, as the theory of generalized Dirichlet forms implies a
stochastic representation of the Langevin semigroup as the transition kernel of
a diffusion process which provides a martingale solution to the Langevin
equation with multiplicative noise. Moreover, we show that even a weak solution
is obtained this way.
