Zeng Lian , Courant Institute KeninG Lu, Brigham Young University

1. ZengLian, Courant InstituteKeninG Lu, Brigham Young University Lyapunov Exponents for Infinite Dimensional Random Dynamical Systems in a Banach Space International Conference on Random Dynamical Systems Chern Institute of Mathematics, Nankai, June 8-12 2009

2. Content 1. Random Dynamical Systems 2. Basic Problems 3. Linear Random Dynamical Systems 4. Brief History 5. Main Results 6. Nonuniform Hyperbolicity

3. 1. Random Dynamical Systems Example 1:Quasi-period ODE where is nonlinear Let be the Haar measure on . Then • is a probability space and is a DS preserving • Let be the solution of Then

4. 1. Random Dynamical Systems Example 2.Stochastic Differential Equations where • is the Brownian motion.

5. 1. Random Dynamical Systems • The classical Wiener Space: Open compact topology is the Wiener measure • The dynamical system • is invariant and ergodic under The solution operator generates RDS

6. 1. Random Dynamical Systems Example: Random Partial Differential Equation where -Banach space and is a measurable dynamical system over probability space Random dynamical system – solution operator:

7. 1. Random Dynamical Systems Metric Dynamical System • Let be a probability space. • Let be a metric dynamical system: (i) (ii) (iii) preserves the probability measure Evolution of Noise

8. 1. Random dynamical systems A map is called a random dynamical system over if • is measurable; • the mappings form a cocycle over

10. 1. Random Dynamical Systems • Time-one map: • Random mapgenerates the random dynamical system

11. 2. Basic Problems Mathematical Questions Two Fundamental Questions: Mathematical Model Question 1.

12. 2. Basic Problems Mathematical Questions Mathematical Model Computational Model Question 2: Can we trust what we see?

13. 2. Basic Problems Mathematical Questions • Stability • Sensitive dependence of initial data

14. 2. Basic Problems • Deterministic Dynamical Systems • Stationary solutions Eigenvalues Eigenvectors

15. Random Dynamical Systems • Deterministic Dynamical Systems • Periodic Orbits Floquet exponents Floquet spaces

16. Random Dynamical Systems • Random Dynamical Systems • Orbits • Linearized Systems Lyapunov exponents measure the average rate ofseparation of orbits starting from nearby initial points.

17. 3. Dynamical Behavior of Linear RDS • The Linear random dynamical system generated by S: • Basic Problem: Find all Lyapunov exponents

18. 4. Brief History • Finite Dimensional Dynamical Systems V. Oseledets, 1968 (31 pages) Existence of Lyapunov exponents Invarant subspaces,. Multiplicative Ergodic Theorem Different Proofs Millionshchikov; Palmer, Johnson, & Sell; Margulis; Kingman;Raghunathan; Ruelle; Mane; Crauel; Ledrappier; Cohen, Kesten, & Newman; Others.

19. 4.Brief History • Applications: • Deterministic Dynamical Systems Pesin Theory, 1974, 1976, 1977 Nonuniform hyperbolicity Entropy formula, chaotic dynamics • Random Dynamical Systems Ruelle inequality, chaotic dynamics Entropy Formula, Dimension Formula Ruelle, Ladrappia, L-S. Young, … Smooth conjugacy W. Li and K. Lu

20. 4. Brief History • Infinite Dimensional RDS • Ruelle, 1982 (Annals of Math) Random Dynamical Systems in a Separable Hilbert Space. Multiplicative Ergodic Theorem

21. 4. Brief History • Basic Problem: Establish Multiplicative Ergodic Theorem for RDS Banach space such as

22. Brief History • Infinite Dimensional RDS • Mane, 1983 Multiplicative Ergodic Theorem

23. 4. Brief History • Infinite Dimensional RDS • Thieullen, 1987 Multiplicative Ergodic Theorem

24. 4. Brief History • Infinite Dimensional RDS • Flandoli and Schaumlffel, 1991 Multiplicative Ergodic Theorem

25. 4. Brief History • Infinite Dimensional RDS • Schaumlffel, 1991 Multiplicative Ergodic Theorem

26. Main Results • Infinite Dimensional RDS • Lian & L, Memoirs of AMS 2009 Multiplicative Ergodic Theorem Difficulties: Random Dynamical Systems No topological structure of the base space Banach Space No inner product structure

27. 5. Main Results Settings and Assumptions: • --- Separable Banach Space • Measurable metric dynamical system • is strongly measurable map

28. 5. Main Results • Measure of Noncompactness Let Kuratowski measure of noncompactness Index of noncompactness for a map

29. 5. Main Results • Measure of Noncompactness If S is a bounded linear operator, is the radius of essential specrtrum of S

30. 5. Main Results • Principal Lyapunov Exponent • Exponent of Noncompactness When LRDS is compact

31. 5. Main Results Theorem A(Lian & L, Memoirs of AMS 2009) Assume that Then,(  -invariant subset of full measure) • there are finitely many Lyapuniv exponents and invariant splitting

32. 5. Main Results such that (1) Invariance: (2) Lyapunov exponents for all

33. 5. Main Results (3)Exponential decay rate in and

34. 5. Main Results • Measurability: 1. are measurable 2. All projections are strongly measurable (5) All projections are tempered

35. 5. Main Results (II) There are manyLyapuniv exponents and many finite dimensional subpaces and many infinite dimensional subpaces such that

36. 5. Main Results and (1) Invariance: (2) Lyapunov exponents for all

37. 5. Main Results (3) Exponential decay rate in and

38. 5. Main Results Theorem B(Lian & L, Memoirs of AMS 2009) Theorem A holds for continuous time random dynamical systems

39. 6. Nonuniform Hyperbolicity Theorem C:There are  -invariant random variable() >0 and tempered random variable K() ¸ 1such that where are the stable, unstable projections

40. 7. Random Stable and Unstable Manifolds Theorem D: (Lian & L, Memoirs of AMS 2009)

41. 7. Random Stable and Unstable Manifolds

42. 8. Application

43. 8. Application

44. 8. Application

45. 9. Outline of Proof • Volume function • Kingman’s additive erogidc theorem • Kato’s space gap • Measurable selection theorem • Measurable Hahn-Banach theorem • Measure theory

46. 谢谢!