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.
Citation
Adithya Devraj. Ioannis Kontoyiannis. Sean Meyn. "Geometric ergodicity in a weighted Sobolev space." Ann. Probab. 48 (1) 380 - 403, January 2020. https://doi.org/10.1214/19-AOP1364
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