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, $$ \|f\|_{v,1} = \sup_{x \in \Re^\ell} \frac{1}{v(x)} \max \{|f(x)|, |\partial_1 f(x)|,\ldots,|\partial_\ell f(x)|\}, $$ where $v\colon \Re^\ell \to [1,\infty)$ 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 $t\geq 0$: $$\Big| \text{E}_x[f(X(t))] - \pi(f)\Big| \le Be^{-\delta t}v(x),\quad \|\nabla \text{E}_x[f(X(t))] \|_2 \le Be^{-\delta t} v(x), $$ 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: \[ h-Ph = f-\pi(f). \] Part of the analysis is based on an operator-theoretic treatment of the sensitivity process that appears in the theory of Lyapunov exponents.