# Geometric ergodicity in a weighted sobolev space

Devraj, A and Kontoyiannis, I and Meyn, S (2020) Geometric ergodicity in a weighted sobolev space. Annals of Probability, 48. pp. 380-403. ISSN 0091-1798

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## Abstract

© Institute of Mathematical Statistics, 2020. For a discrete-time Markov chain X = [X(t)] evolving on Rl 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 v,1∞ of functions with norm, ∥f ∥ v,1 = sup x∈Rl max [[pipe]f (x) [pipe], [pipe] ∂1f (x) [pipe],..., [pipe] ∂lf (x) [pipe] ], where v: Rl →[1, ∞) is a Lyapunov function and ∂i:= ∂/∂xi. (ii) The Markov chain is geometrically ergodic in L v,1∞: There is a unique invariant probability measure π and constants B <∞ and δ > 0 such that, for each f ∈ L v,1∞, any initial condition X(0) = x, and all t ≥ 0: [pipe] Ex [f (X(t))]-π(f) [pipe] ≤ B∥f ∥ v,1e -δt v(x), ∥∇Ex [f (X(t))]∥ 2 ≤ B∥f ∥ v,1e -δt v(x), where π(f) = ∫ f dπ. (iii) For any function f ∈ L v,1∞ there is a function h ∈ L v,1∞ solving Poisson's equation: h-Ph = f-π(f). 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.

Item Type: Article math.PR math.PR 60J05, 60J35, 37A30, 47H20 UNSPECIFIED Div F > Signal Processing and Communications Cron Job 08 Jan 2018 20:12 18 Aug 2020 12:42 10.1214/19-AOP1364