Zeng Lian , Courant Institute KeninG Lu, Brigham Young University

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Lyapunov Exponents for Infinite Dimensional Lyapunov Exponents for Infinite Dimensional

Random Dynamical Systems in a Banach SpaceRandom Dynamical Systems in a Banach Space Zeng Lian, Courant Institute Zeng Lian, Courant Institute KeninG Lu, Brigham Young UniversityKeninG Lu, Brigham Young University

International Conference on Random Dynamical SystemsInternational Conference on Random Dynamical Systems Chern Institute of Mathematics, Nankai, June 8-12 2009Chern Institute of Mathematics, Nankai, June 8-12 2009

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ContentContent

1. Random Dynamical Systems2. Basic Problems3. Linear Random Dynamical Systems4. Brief History5. Main Results6. Nonuniform Hyperbolicity

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1. Random Dynamical System1. Random Dynamical Systemss

Example 1: Quasi-period ODE

where is nonlinear

Let be the Haar measure on .

Then

(1) is a probability space and is a DS preserving

(2) Let be the solution of Then

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1. Random Dynamical System1. Random Dynamical Systemss

Example 2. Stochastic Differential Equations

where

•

• is the Brownian motion.

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The classical Wiener Space:

Open compact topology

is the Wiener measure

The dynamical system

is invariant and ergodic under

The solution operator generates RDS

1. Random Dynamical System1. Random Dynamical Systemss

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Example: Random Partial Differential Equation

where -Banach space and is a measurable

dynamical system over probability space

Random dynamical system – solution operator:

1. Random Dynamical System1. Random Dynamical Systemss

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1. Random Dynamical Systems1. 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

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1. Random dynamical systems1. Random dynamical systems

A map

is called a random dynamical system over if

(i) is measurable;

(ii) the mappings form a

cocycle over

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1. Random Dynamical Systems1. Random Dynamical Systems

Time-one map:

Random map generates

the random dynamical system

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2. Basic Problems 2. Basic Problems Mathematical QuestionsMathematical Questions

Two Fundamental Questions:

Mathematical Model

Question 1.

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2. Basic Problems 2. Basic Problems Mathematical QuestionsMathematical Questions

Mathematical Model

Computational Model

Question 2:

Can we trust what we see?

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2. Basic Problems 2. Basic Problems Mathematical QuestionsMathematical Questions

1. Stability

2. Sensitive dependence of initial data

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2. Basic Problems2. Basic Problems

Deterministic Dynamical Systems

Stationary solutions

Eigenvalues

Eigenvectors

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Random Dynamical SystemsRandom Dynamical Systems

Deterministic Dynamical Systems

Periodic Orbits

Floquet exponents

Floquet spaces

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Random Dynamical SystemsRandom Dynamical Systems

Random Dynamical Systems

Orbits

Linearized Systems

Lyapunov exponentsmeasure the average rate of separation of orbits starting from nearby initial points.

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3. 3. Dynamical Behavior of Linear RDSDynamical Behavior of Linear RDS

The Linear random dynamical system generated by S:

Basic Problem:

Find all Lyapunov exponents

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4. Brief History4. Brief History

Finite Dimensional Dynamical SystemsV. Oseledets, 1968 (31 pages)

Existence of Lyapunov exponents

Invarant subspaces,.

Multiplicative Ergodic Theorem

Different ProofsMillionshchikov; Palmer, Johnson, & Sell; Margulis; Kingman;Raghunathan; Ruelle; Mane; Crauel; Ledrappier; Cohen, Kesten, & Newman; Others.

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4.Brief History4.Brief History

Applications: Deterministic Dynamical Systems

Pesin Theory, 1974, 1976, 1977Nonuniform hyperbolicityEntropy formula, chaotic dynamics

Random Dynamical SystemsRuelle inequality, chaotic dynamicsEntropy Formula, Dimension Formula Ruelle, Ladrappia, L-S. Young, …Smooth conjugacyW. Li and K. Lu

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4. Brief History4. Brief History

Infinite Dimensional RDSRuelle, 1982 (Annals of Math)

Random Dynamical Systems in a Separable Hilbert Space.

Multiplicative Ergodic Theorem

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4. Brief History4. Brief History

Basic Problem:

Establish Multiplicative Ergodic Theorem for RDS

Banach space such as

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Brief HistoryBrief History

Infinite Dimensional RDS Mane, 1983

Multiplicative Ergodic Theorem

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4. Brief History4. Brief History

Infinite Dimensional RDS Thieullen, 1987

Multiplicative Ergodic Theorem

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4. Brief History4. Brief History

Infinite Dimensional RDS Flandoli and Schaumlffel, 1991

Multiplicative Ergodic Theorem

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4. Brief History4. Brief History

Infinite Dimensional RDS Schaumlffel, 1991

Multiplicative Ergodic Theorem

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Main ResultsMain 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

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5. Main Results5. Main Results

Settings and Assumptions: --- Separable Banach Space

Measurable metric dynamical system

is strongly measurable map

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5. Main Results5. Main Results

Measure of Noncompactness

Let Kuratowski measure of noncompactness

Index of noncompactness for a map

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5. Main Results5. Main Results

Measure of Noncompactness

If S is a bounded linear operator,

is the radius of essential specrtrum of S

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5. Main Results5. Main Results

Principal Lyapunov Exponent

Exponent of Noncompactness

When LRDS is compact

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5. Main Results5. Main Results

Theorem A (Lian & L, Memoirs of AMS 2009)

Assume that Then, ( -invariant subset of full measure)

(I) there are finitely many Lyapuniv exponents

and invariant splitting

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5. Main Results5. Main Results

such that(1) Invariance:

(2) Lyapunov exponents

for all

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5. Main Results5. Main Results

(3) Exponential decay rate in

and

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5. Main Results5. Main Results

(4) Measurability:

1. are measurable

2. All projections are strongly measurable

(5) All projections are tempered

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5. Main Results5. Main Results

(II) There are many Lyapuniv exponents

and many finite dimensional subpaces

and many infinite dimensional subpaces

such that

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5. Main Results5. Main Results

and(1) Invariance:

(2) Lyapunov exponents

for all

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5. Main Results5. Main Results

(3) Exponential decay rate in

and

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5. Main Results5. Main Results

Theorem B (Lian & L, Memoirs of AMS 2009)

Theorem A holds for continuous time random dynamical systems

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6. 6. Nonuniform HyperbolicityNonuniform Hyperbolicity

Theorem C: There are -invariant random variable >0

and tempered random variable K(¸ 1such that

where are the stable, unstable projections

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7. Random Stable and Unstable Manifolds7. Random Stable and Unstable Manifolds

Theorem D: (Lian & L, Memoirs of AMS 2009)

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7. Random Stable and Unstable Manifolds7. Random Stable and Unstable Manifolds

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88. . ApplicationApplication

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88. . ApplicationApplication

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88. . ApplicationApplication

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99. Outline of Proof. Outline of Proof

• Volume function• Kingman’s additive erogidc theorem• Kato’s space gap • Measurable selection theorem• Measurable Hahn-Banach theorem• Measure theory

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