Geometric ergodicity in a weighted Sobolev space

For a discrete-time Markov chain X={X(t)} evolving on R. with transition kernel P , natural, general conditions are developed under which the following are established: (i) The transition kernel P has a purely discrete spectrum, when viewed as a linear operator on a weighted Sobolev space Lv,1. of functions with norm, .f.v,1=supx.R.1v(x)max{|f(x)|,|.1f(x)|,.,|..f(x)|}, where v:R..[1,.) is a Lyapunov function and .i:=./.xi . (ii) The Markov chain is geometrically ergodic in Lv,1. : There is a unique invariant probability measure . and constants B<. and .>0 such that, for each f.Lv,1., any initial condition X(0)=x, and all t.0: |Ex[f(X(t))]..(f)|..Ex[f(X(t))].2..B.f.v,1e..tv(x),B.f.v,1e..tv(x) , where .(f)=.fd. . (iii) For any function f.Lv,1. there is a function h.Lv,1. solving Poisson.s equation: h.Ph=f..(f). Part of the analysis is based on an operator-theoretic treatment of the sensitivity process that appears in the theory of Lyapunov exponents. Relationships with topological coupling, in terms of the Wasserstein metric, are also explored.