Essential m-dissipativity and hypocoercivity of Langevin dynamics with multiplicative noise

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