  In this article we develop a new abstract strategy for proving ergodicity
with explicit computable rate of convergence for diffusions associated with a
degenerate Kolmogorov operator L. A crucial point is that the evolution
operator L may have singular and nonsmooth coefficients. This allows the
application of the method e.g. to degenerate and singular particle systems
arising in Mathematical Physics. As far as we know in such singular cases the
relaxation to equilibrium can't be discussed with the help of existing
approaches using hypoellipticity, hypocoercivity or stochastic Lyapunov type
techniques. The method is formulated in an L2-Hilbert space setting and is
based on an interplay between Functional Analysis and Stochastics. Moreover, it
implies an ergodicity rate which can be related to L2-exponential convergence
of the semigroup. Furthermore, the ergodicity method shows up an interesting
analogy with existing hypocoercivity approaches. In the first application we
discuss ergodicity of the N-particle degenerate Langevin dynamics with singular
potentials. The dual to this equation is also called the kinetic Fokker-Planck
equation with an external confining potential. In the second example we apply
the method to the so-called (degenerate) spherical velocity Langevin equation
which is also known as the fiber lay-down process arising in industrial
mathematics.
