Geometric Ergodicity in a Weighted Sobolev Space

Abstract

For a discrete-time Markov chain $\{X(t)\}$ evolving on ${\Re }^{\ell }$ with transition kernel $P$, natural, general conditions are developed under which the following are established: 1. The transition kernel $P$ has a purely discrete spectrum, when viewed as a linear operator on a weighted Sobolev space $L_{\infty }^{v,1}$ of functions with norm, $$\Vert f{\Vert }_{v,1}=\underset{x\in {\Re }^{\ell }}{\sup }\frac{1}{v(x)}\max \{|f(x)|,|{\partial }_{1}f(x)|,\dots ,|{\partial }_{\ell }f(x)|\},$$ where $v:{\Re }^{\ell }\to [1,\infty )$ is a Lyapunov function and ${\partial }_i:=\partial /\partial x_i$. 2. The Markov chain is geometrically ergodic in $L_{\infty }^{v,1}$: There is a unique invariant probability measure $\pi$ and constants $B<\infty$ and $\delta >0$ such that, for each $f\in L_{\infty }^{v,1}$, any initial condition $X(0)=x$, and all $t\ge 0$: $$|{\text{E}}_x[f(X(t))]-\pi (f)|\le B{e}^{-\delta t}v(x),\Vert \nabla {\text{E}}_x[f(X(t))]{\Vert }_2\le B{e}^{-\delta t}v(x),$$ where $\pi (f)=\int fd\pi$. 3. For any function $f\in L_{\infty }^{v,1}$ there is a function $h\in L_{\infty }^{v,1}$ solving Poisson's equation: \[ h-Ph = f-π(f). \] Part of the analysis is based on an operator-theoretic treatment of the sensitivity process that appears in the theory of Lyapunov exponents.

33 pages; The paper has been accepted for publication in the Annals of Probability