Geometric Ergodicity in a Weighted Sobolev Space

For a discrete-time Markov chain $\{X(t)\}$ evolving on ${\mathrm{\Re }}^{\mathrm{\ell }}$ 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 $L_{\mathrm{\infty }}^{v,1}$ of functions with norm, $$\mathrm{\parallel }f{\mathrm{\parallel }}_{v,1}=\underset{x\in {\mathrm{\Re }}^{\mathrm{\ell }}}{\sup }\frac{1}{v(x)}\max \{\mathrm{\mid }f(x)\mathrm{\mid },\mathrm{\mid }{\mathrm{\partial }}_{1}f(x)\mathrm{\mid },\dots ,\mathrm{\mid }{\mathrm{\partial }}_{\mathrm{\ell }}f(x)\mathrm{\mid }\},$$ where $v\colon \Re^\ell \to [1,\infty)$isaLyapunovfunctionand$\partial_i:=\partial/\partial x_i$.
2. The Markov chain is geometrically ergodic in $L_{\mathrm{\infty }}^{v,1}$: There is a unique invariant probability measure $\pi$ and constants $B<\mathrm{\infty }$ and $\delta >0$ such that, for each $f\in L_{\mathrm{\infty }}^{v,1}$, any initial condition $X(0)=x$, and all $t\ge 0$: $$\mid {\text{E}}_x[f(X(t))]-\pi (f)\mid \le B{e}^{-\delta t}v(x),\mathrm{\parallel }\mathrm{\nabla }{\text{E}}_x[f(X(t))]{\mathrm{\parallel }}_2\le B{e}^{-\delta t}v(x),$$ where $\pi (f)=\int fd\pi$.
3. For any function $f\in L_{\mathrm{\infty }}^{v,1}$ there is a function $h\in L_\infty^{v,1}$ solving Poisson’s equation: $$h-Ph=f-\pi (f)\mathrm{.}$$ Part of the analysis is based on an operator-theoretic treatment of the sensitivity process that appears in the theory of Lyapunov exponents.