Invariant measures for stochastic functional differential equations

We establish new general sufficient conditions for the existence of an invariant measure for stochastic functional differential equations and for exponential or subexponential convergence to the equilibrium. The obtained conditions extend Veretennikov--Khasminskii conditions for SDEs and are optimal in a certain sense.

[1]  M. Scheutzow,et al.  Lyapunov exponents and stationary solutions for affine stochastic delay equations , 1990 .

[2]  Michael Scheutzow.  EXPONENTIAL GROWTH RATES FOR STOCHASTIC DELAY DIFFERENTIAL EQUATIONS , 2005 .

[3]  Uwe Küchler,et al.  Langevins stochastic differential equation extended by a time-delayed term , 1992 .

[4]  Jonathan C. Mattingly,et al.  Asymptotic coupling and a general form of Harris’ theorem with applications to stochastic delay equations , 2009, 0902.4495.

[5]  B. Øksendal.  Stochastic differential equations : an introduction with applications , 1987 .

[6]  M. Riedle,et al.  Delay differential equations driven by Lévy processes: Stationarity and Feller properties , 2005 .

[7]  Kiyosi Itô,et al.  ON STATIONARY SOLUTIONS OF A STOCHASTIC DIFFERENTIAL EQUATION. , 1964 .

[8]  A. Veretennikov,et al.  On Polynomial Mixing and Convergence Rate for Stochastic Difference and Differential Equations , 2008 .

[9]  R. Douc,et al.  Subgeometric rates of convergence of f-ergodic strong Markov processes , 2006, math/0605791.

[10]  Miss A.O. Penney.  (b) , 1974, The New Yale Book of Quotations.

[11]  D. Bakry,et al.  Rate of convergence for ergodic continuous Markov processes : Lyapunov versus Poincaré , 2007, math/0703355.

[12]  Uwe Küchler,et al.  On stationary solutions of delay differential equations driven by a Lévy process , 2000 .

[13]  A. Veretennikov,et al.  ON SUBEXPONENTIAL MIXING RATE FOR MARKOV PROCESSES , 2005 .

[14]  A. Veretennikov,et al.  Bounds for the Mixing Rate in the Theory of Stochastic Equations , 1988 .

[15]  Jonathan C. Mattingly,et al.  Noise-Induced Stabilization of Planar Flows I , 2014, 1404.0957.

[16]  M. Yor,et al.  Continuous martingales and Brownian motion , 1990 .

[17]  I. I. Gikhman.  Convergence to Markov processes , 1969 .

[18]  Alexei Michajlovič Kulik,et al.  Introduction to Ergodic rates for Markov chains and processes , 2015 .

[19]  V. Bogachev,et al.  The Monge-Kantorovich problem: achievements, connections, and perspectives , 2012 .

[20]  M. Scheutzow,et al.  Generalized couplings and convergence of transition probabilities , 2015, 1512.06359.

[21]  Jonathan C. Mattingly,et al.  Yet Another Look at Harris’ Ergodic Theorem for Markov Chains , 2008, 0810.2777.

[22]  Richard L. Tweedie,et al.  Markov Chains and Stochastic Stability , 1993, Communications and Control Engineering Series.

[23]  M. Scheutzow.  Qualitative behaviour of stochastic delay equations with a bounded memory , 1984 .

[24]  Chenggui Yuan,et al.  Harnack inequalities for functional SDEs with multiplicative noise and applications , 2010, 1012.5688.

[25]  M. Renesse,et al.  Harnack Inequality for Functional SDEs with Bounded Memory , 2009, 0910.4536.

[26]  O. Butkovsky.  Subgeometric rates of convergence of Markov processes in the Wasserstein metric , 2012, 1211.4273.

[27]  M. Scheutzow,et al.  Existence and uniqueness of solutions of stochastic functional differential equations , 2008, 0812.1726.