We study a periodic kinetic Fokker—Planck equation in which free transport in position is coupled to Ornstein—Uhlenbeck relaxation in velocity. Our aim is to give a transparent weighted L^2 analysis of the relaxation mechanism and to test it by a modal approximation. The equation is written in Maxwellian variables, the generator is decomposed into a skew transport part and a dissipative velocity part and its contraction semigroup is considered on the natural weighted Hilbert space. Using Gaussian and torus Poincaré inequalities, we prove mass conservation, microscopic coercivity and exponential decay on the zero-mass subspace through a modified first-order energy containing a spatial—velocity cross term. For the homogeneous problem, the entropy identity gives decay from the Gaussian logarithmic Sobolev inequality. A Fourier—Hermite discretization is then derived, its semi-discrete L^2-stability is shown and truncation tests quantify convergence in the Hermite index and the spectral abscissa of the first nonzero Fourier block. The results give a compact account of how velocity relaxation enforces global return to Maxwellian equilibrium in this model.