Ergodicity for SDEs and approximations: Locally Lipschitz vector fields and degenerate noise

Stochastic Processes and their Applications

Volumn 101, Issue 2, 2002, Pages 185-232

  Mattingly, J C ^a   Stuart, A M ^b   Higham, D J ^c  

^a STANFORD UNIVERSITY   (United States)

^b UNIVERSITY OF WARWICK   (United Kingdom)

^c UNIVERSITY OF STRATHCLYDE   (United Kingdom)

Author keywords

Additive noise; Dissipative and gradient systems; Geometric erdogicity; Hypoelliptic and degenerate diffusions; Langevin equation; Monotone; Stochastic differential equations; Time discretization

Indexed keywords

EID: 0036789329     PISSN: 03044149     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0304-4149(02)00150-3     Document Type: Article

References (44)

1. Arnold, L., Kliemann, W., 1987. On unique ergodicity for degenerate diffusions. Stochastics 21, 41-61.
2. Bakry, D., 1997. On Sobolev and logarithmic Sobolev inequalities for Markov semigroups. In: Elworthy, K.D. (Ed.), New Trends in Stochastic Analysis (Charingworth, 1994). Longman Scientific & Technical, River Edge, NJ, pp. 43-75.
3. Bell, D.R., 1987. The Malliavin Calculus. Longman Scientific & Technical, Harlow.
4. Down, D., Meyn, S.P., Tweedie, R.L., 1995. Exponential and uniform ergodicity of Markov processes. Ann. Probab. 23, 1671-1691.
5. Durrett, R., 1996. Stochastic Calculus, A Practical Introduction. CRC Press, Boca Raton, FL.
6. E, W., Mattingly, J.C., 2001. Ergodicity for the Navier-Stokes equation with degenerate random forcing: finite dimensional approximation Comm. Pure Appl. Math. 54, 1386-1402.
7. Fayolle, G., Malyshev, V.A., Menshikov, M.V., 1995. Topics in the Constructive Theory of Countable Markov Chains. Cambridge Univ. Press, Cambridge.
8. Ford, G.W., Kac, M., 1987. On the quantum Langevin equation. J. Statist. Phys. 46, 803-810.
9. Grorud, A., Talay, D., 1996. Approximation of Lyapunov exponents of nonlinear stochastic differential equations. SIAM J. Appl. Math. 56, 627-650.
10. Hale, J.K., 1988. Asymptotic Behavior of Dissipative Systems. American Mathematical Society, Providence, RI.
11. Hansen, N.R., 2002. Geometric ergodicity of discrete time approximations to multivariate diffusions. Preprint.
12. Has'minskii, R.Z., 1980. Stochastic Stability of Differential Equations. Sijthoff and Noordhoff, Alphen a/d Rijin.
13. Kifer, Y., 1988. Random Perturbations of Dynamical Systems. Birkhäuser, Boston Inc., Boston, MA.
14. Kliemann, W., 1987. Recurrence and invariant measures for degenerate diffusions. Ann. Probab. 15, 690-707.
15. Kloeden, P.E., Platen, E., 1991. Numerical Solution of Stochastic Differential Equations. Springer, New York.
16. Kunita, H., 1978. Supports of diffusion processes and controllability problems. In: Itô, K. (Ed.), Proceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976). Wiley, New York, pp. 163-185.
17. Ledoux, M., Concentration of measure and logarithmic Sobolev inequalities. In: Séminaire de Probabilités, XXXIII, Lecture Notes in Math., vol. 1709, Springer, Berlin, 1999, pp. 120-216.
18. Mao, X., 1997. Stochastic Differential Equations and Applications. Horwood, Chichester.
19. The MathWorks, Inc., 1992. MATLAB User's Guide. Natick, MA.
20. Mattingly, J., Stuart, A.M., in Preparation. Ergodicity for adaptive approximations of SDEs.
21. Meyn, S., Tweedie, R.L., 1992. Stochastic Stability of Markov Chains. Springer, New York.
22. Meyn, S.P., Tweedie, R.L., 1993. Stability of Markovian processes, I, II and III Adv. Appl. Probab. 24 542-574, 25, 487-517 and 25, 518-548.
23. Norris, J., 1986. Simplified Malliavin calculus. In: Azéma, J., Yor, M. (Eds.), Séminaire de Probabilités, XX, 1984/85, Springer, Berlin, pp. 101-130.
24. Orey, S., 1971. Lecture Notes on Limit Theorems for Markov Chain Transition Probabilities. Van Nostrand Reinhold Co., London, Van Nostrand Reinhold Mathematical Studies, No. 34.
25. Pages, G., 2001. Sur quelques algorithmes recursifs pour les probabilities numeriques. ESAIM: Probab. Statist. 5, 141-170.
26. Roberts, G., Rosenthal, J.S., 1998b. Optimal scaling of discrete approximations to Langevin diffusions. J. Roy. Statist. Soc. 60, 255-268.
27. Roberts, G.O., Tweedie, R.L., 1996. Exponential convergence of Langevin diffusions and their discrete approximations. Bernoulli 2, 341-363.
28. Roberts, G.O., Tweedie, R.L., 1999. Bounds on regeneration times and convergence rates for Markov chains. Stochastic Proc. Applic. 80, 211-229.
29. Rogers, L.C.G., Williams, D., 2000. Diffusions, Markov processes and Martingales, Vol. 2, 2nd Edition. Cambridge University Press, Cambridge, reprinted.
30. Rosenthal, J.S., 1995. Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Statist. Assoc. 90, 558-566.
31. Sanz-Serna, J.M., Stuart, A.M., 1999. Ergodicity of dissipative differential equations subject to random impulses. J. Differential Equations 155, 262-284.
32. Shardlow, T., Stuart, A.M., 2000. A perturbation theory for ergodic Markov chains with application to numerical approximation. SIAM J. Numer. Anal. 37, 1120-1137.
33. Sparrow, C., 1982. The Lorenz Equations, Bifurcations, Chaos and Strange Attractors. Springer, Berlin.
34. Stramer, O., Tweedie, R.L., 1999a. Langevin-type models I: diffusions with given stationary distributions, and their discretizations Method. Comput. Appl. Probab. 1, 283-306.
35. Stramer, O., Tweedie, R.L., 1999b. Langevin-type models II: self-targeting candidates for MCMC algorithms Methodol. Comput. Appl. Probab. 1, 307-328.
36. Stroock, D.W., 1982. Lectures on Topics in Stochastic Differential Equations. Tata Institute of Fundamental Research, Bombay (With notes by Satyajit Karmakar).
37. Stroock, D.W., Varadhan, S.R.S., 1972. On the support of diffusion processes with applications to the strong maximum principle. Proceedings of the Sixth Berkeley Symposium on Math. Stat. and Prob., Vol. III, pp. 333-360.
38. Stuart, A.M., Humphries, A.R., 1996. Dynamical Systems and Numerical Analysis. Cambridge University Press, Cambridge.
39. Talay, D., 1990. Second-order discretization schemes for stochastic differential systems for the computation of the invariant law. Stochastics Stochastics Rep. 29, 13-36.
40. Talay, D., 1991. Approximation of upper Lyapunov exponents of bilinear stochastic differential systems. SIAM J. Numer. Anal. 28, 1141-1164.
41. Talay, D., 1999. Approximation of the invariant probability measure of stochastic Hamiltonian dissipative systems with non globally Lipschitz co-efficients. Appears In: Bouc, R., Soize, C. (Eds.), Progress in Stochastic Structural Dynamics, Vol. 152. Publication du L.M.A.-CNRS.
42. Talay, D., 2002. Stochastic Hamiltonian dissipative systems with non globally Lipschitz co-efficients: exponential convergence to the invariant measure and discretization by the implicit Euler scheme. Markov Proc. Rel. Fields, submitted.
43. Tropper, M.M., 1977. Ergodic properties and quasideterministic properties of finite-dimensional stochastic systems. J. Statist. Phys. 17, 491-509.
44. Veretennikov, A.Y., 1997. On polynomial mixing bounds for stochastic differential equations. Stochastic Process. Appl. 70, 115-127.