  In this paper, hypocoercivity methods are applied to linear kinetic equations
with mass conservation and without confinement, in order to prove that the
solutions have an algebraic decay rate in the long-time range, which the same
as the rate of the heat equation. Two alternative approaches are developed: an
analysis based on decoupled Fourier modes and a direct approach where, instead
of the Poincar\'e inequality for the Dirichlet form, Nash's inequality is
employed. The first approach is also used to provide a simple proof of
exponential decay to equilibrium on the flat torus. The results are obtained on
a space with exponential weights and then extended to larger function spaces by
a factorization method. The optimality of the rates is discussed. Algebraic
rates of decay on the whole space are improved when the initial datum has
moment cancellations.
