Subgeometric rates of convergence of f-ergodic strong Markov processes

By Randal Douc, Gersende Fort, Arnaud Guillin. 2006

199 citations recorded by the source corpus.

We provide a condition in terms of a supermartingale property for a functional of the Markov process, which implies (a) f-ergodicity of strong Markov processes at a subgeometric rate, and (b) a moderate deviation principle for an integral (bounded) functional. An equivalent condition in terms of a drift inequality on the extended generator is also given. Results related to (f,r)-regularity of the process, of some skeleton chains and of the resolvent chain are also derived. Applications to specific processes are considered, including elliptic stochastic differential equations, Langevin diffusions, hypoelliptic stochastic damping Hamiltonian systems and storage models.

Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/b0867f1b02169f342e49cada3bd3567ff053751a

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Citations in the corpus

Poincaré inequalities and hitting times (2010, methodology): “This is the key result in [20] (also see [19]), unfortunately with an essentially non-explicit control of the constants.”
Subgeometric rates of convergence of Markov processes in the Wasserstein metric (2012, background): “In the continuous time setting we obtain the same convergence rate provided that there exists a “good” d-small set and the Douc–Fort–Guillin supermartingale condition holds.”
Functional Inequalities for Heavy Tailed Distributions and Application to Isoperimetry (2010, background): “decay to equilibrium of dynamical systems, see e.g. [ 33 , 38, 55, 4, 28].”
Quantitative Harris-type theorems for diffusions and McKean–Vlasov processes (2016, methodology): “The following example shows how V and η can be chosen explicitly; see also [16].”
Bernstein type's concentration inequalities for symmetric Markov processes (2010, unknown): “(1.1)”
Rate of convergence for ergodic continuous Markov processes : Lyapunov versus Poincaré (2007, methodology): “Remark however that as in [9], Röckner-Wang [25] also considers the case of Markov processes with jumps.”
Lyapunov conditions for Super Poincaré inequalities (2009, unknown): “[16, 18, 15]).”
Asymptotic coupling and a general form of Harris’ theorem with applications to stochastic delay equations (2009, methodology): “Subexponential convergence rates via Lyapunov functions has been explored for Harris chains in [MT93, Ver99, BCG08, DFG09 , Hai09].”
Transportation-information inequalities for Markov processes (2007, methodology): “Lyapunov conditions are widely used to study the speed of convergence of Markov chains [33] or Markov processes [18], large or moderate deviations and essential spectral radii [45, 47].”