GEOMETRIC ERGODICITY IN A WEIGHTED SOBOLEV SPACE
成果类型：
Article
署名作者：
Devraj, Adithya; Kontoyiannis, Ioannis; Meyn, Sean
署名单位：
State University System of Florida; University of Florida; University of Cambridge
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
DOI：
10.1214/19-AOP1364
发表日期：
2020
页码：
380-403
关键词：
markov process expectations
large deviations
asymptotic evaluation
lyapunov exponents
SPECTRAL THEORY
LIMIT-THEOREMS
CONVERGENCE
Operators
inequalities
diffusion
摘要：
For a discrete-time Markov chain X = {X (t)} evolving on R-l 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-infinity(v,1) of functions with norm, parallel to f parallel to(v,1) = sup(x is an element of Rl)1/v(x)max{vertical bar f (x)vertical bar,vertical bar partial derivative(1) f (x)vertical bar, ..., a vertical bar partial derivative(l)f(x)vertical bar}, where v : R-l -> [1, infinity) is a Lyapunov function and partial derivative(i) :=partial derivative/partial derivative x(i) . The Markov chain is geometrically ergodic in L-infinity(v,1): There is a unique invariant probability measure pi and constants B < infinity and delta > 0 such that, for each f is an element of L-infinity(v,1), any initial condition X(0) = x, and all t >= 0: vertical bar E-x[f (X(t))] -pi(f)vertical bar <= B parallel to f parallel to(v),(-delta t)(1e) v(x), parallel to del E-x[f(X(t))]parallel to(2)<= B parallel to f parallel to(v),(-delta t)(1e) v(x), where pi(f) = f f d pi. (iii) For any function f is an element of L-infinity(v,1) there is a function h is an element of L-infinity(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. Relationships with topological coupling, in terms of the Wasserstein metric, are also explored.