Geometric ergodicity in a weighted Sobolev space

Abstract

For a discrete-time Markov chain $\boldsymbol{X}=\{X(t)\}$ evolving on $\mathbb{R}^{\ell}$ 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 $L_{\infty}^{v,1}$ of functions with norm, \begin{equation*}\Vert f\Vert_{v,1}=\mathop{\mathrm{sup}}_{x\in\mathbb{R}^{\ell}}\frac{1}{v(x)}\max\{\vert f(x)\vert ,\vert \partial_{1}f(x)\vert ,\ldots,\vert \partial_{\ell}f(x)\vert \},\end{equation*} where $v\colon\mathbb{R}^{\ell}\to[1,\infty)$ is a Lyapunov function and $\partial_{i}:=\partial/\partial x_{i}$.

(ii) 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 $t\geq0$: \begin{eqnarray*}\vert \mathsf{E}_{x}[f(X(t))]-\pi(f)\vert &\le&B\Vert f\Vert_{v,1}e^{-\delta t}v(x),\\\Vert \nabla\mathsf{E}_{x}[f(X(t))]\Vert_{2}&\le&B\Vert f\Vert_{v,1}e^{-\delta t}v(x),\end{eqnarray*} where $\pi(f)=\int f\,d\pi$.

(iii) For any function $f\in L_{\infty}^{v,1}$ there is a function $h\in L_{\infty}^{v,1}$ solving Poisson’s equation: \begin{equation*}h-Ph=f-\pi(f).\end{equation*}

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.