dc.contributor.author Devraj, A dc.contributor.author Kontoyiannis, I dc.contributor.author Meyn, S dc.date.accessioned 2019-07-31T11:52:35Z dc.date.available 2019-07-31T11:52:35Z dc.date.issued 2020 dc.identifier.issn 0091-1798 dc.identifier.uri https://www.repository.cam.ac.uk/handle/1810/295116 dc.description.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. dc.language.iso en dc.publisher Institute of Mathematical Statistics dc.subject Markov chain dc.subject stochastic Lyapunov function dc.subject discrete spectrum dc.subject sensitivity process dc.subject weighted Sobolev space dc.subject Lyapunov exponent dc.title Geometric ergodicity in a weighted sobolev space dc.type Article prism.endingPage 403 prism.issueIdentifier 1 prism.publicationDate 2020 prism.publicationName Annals of Probability prism.startingPage 380 prism.volume 48 dc.identifier.doi 10.17863/CAM.42187 rioxxterms.versionofrecord 10.1214/19-AOP1364 rioxxterms.version AM rioxxterms.licenseref.uri http://www.rioxx.net/licenses/all-rights-reserved rioxxterms.licenseref.startdate 2020-01-01 dc.contributor.orcid Kontoyiannis, Ioannis [0000-0001-7242-6375] dc.identifier.eissn 2168-894X rioxxterms.type Journal Article/Review cam.issuedOnline 2020-01-01
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