  We develop a new method for proving hypocoercivity for a large class of
linear kinetic equations with only one conservation law. Local mass
conservation is assumed at the level of the collision kernel, while transport
involves a confining potential, so that the solution relaxes towards a unique
equilibrium state. Our goal is to evaluate in an appropriately weighted $L^2$
norm the exponential rate of convergence to the equilibrium. The method covers
various models, ranging from diffusive kinetic equations like
Vlasov-Fokker-Planck equations, to scattering models like the linear Boltzmann
equation or models with time relaxation collision kernels corresponding to
polytropic Gibbs equilibria, including the case of the linear Boltzmann model.
In this last case and in the case of Vlasov-Fokker-Planck equations, any linear
or superlinear growth of the potential is allowed.
