Ergodicity in Stochastic Process and Dynamical System
Khatia Sharia Supervised by : Ole Peters, Yuzuru Sato, Jeroen Lamb Ergodicity in Stochastic Process and
Stochastic Dynamical System Process
Dynamical System
Summary Khatia Sharia Supervised by : Ole Peters, Yuzuru Sato, Jeroen Lamb
LML, 2018 Ergodicity in Stochastic Process and Dynamical System
Khatia Sharia Supervised by : Ole Peters, Yuzuru Sato, Jeroen Lamb Definition: An observable is ergodic if its time average is the same as its expectation value. Stochastic Process T Dynamical 1 Z Z System lim f (x(t))dt = f (x)P(x)dx T →∞ T 0 Ω Summary | {z } | {z } Time average of f Expectation value of f Ergodicity in Stochastic Process and Dynamical System
Khatia Sharia Supervised by : Ole Peters, Yuzuru Sato, Jeroen Lamb
Stochastic • Process Ergodicity allows replacing time averages with expectation
Dynamical values. System • Studied in stochastic process and dynamical system. Summary Ergodicity in Stochastic Outline Process and Dynamical System
Khatia Sharia Supervised by : Ole Peters, Yuzuru Sato, Jeroen Lamb
Stochastic 1 Stochastic Process Process
Dynamical System Summary 2 Dynamical System
3 Summary Ergodicity in Stochastic Outline Process and Dynamical System
Khatia Sharia Supervised by : Ole Peters, Yuzuru Sato, Jeroen Lamb
Stochastic 1 Stochastic Process Process
Dynamical System Summary 2 Dynamical System
3 Summary Ergodicity in Stochastic Random Variable Process and Dynamical System
Khatia Sharia Supervised by : Ole Peters, Yuzuru Sato, Jeroen Lamb
Stochastic Process
Dynamical Random variable Z: System • set of possible values {z} Summary • probability distribution over P(z) Ergodicity in Stochastic Stochastic Process Process and Dynamical System
Khatia Sharia Supervised by : Ole Peters, Yuzuru Sato, Jeroen Lamb
Stochastic Process
Dynamical System A stochastic process Yz (t), is a family of random variables, one Summary for each time, t. Ergodicity in Stochastic Stochastic process visualisation Process and Dynamical System
Khatia Sharia Supervised by : Ole Peters, Yuzuru Sato, Jeroen Lamb
Stochastic Process
Dynamical System
Summary • Ergodic for µ > 0 • Non-ergodic for µ ≤ 0
Ergodicity in Stochastic Example: Ornstein-Uhlenbeck process Process and Dynamical System
Khatia Sharia Supervised by : Ole Peters, Yuzuru Sato, Jeroen Lamb
Stochastic Process dx = −µ(x − x )dt + σdω Dynamical 0 System
Summary • Non-ergodic for µ ≤ 0
Ergodicity in Stochastic Example: Ornstein-Uhlenbeck process Process and Dynamical System
Khatia Sharia Supervised by : Ole Peters, Yuzuru Sato, Jeroen Lamb
Stochastic Process dx = −µ(x − x )dt + σdω Dynamical 0 System • Ergodic for µ > 0 Summary Ergodicity in Stochastic Example: Ornstein-Uhlenbeck process Process and Dynamical System
Khatia Sharia Supervised by : Ole Peters, Yuzuru Sato, Jeroen Lamb
Stochastic Process dx = −µ(x − x )dt + σdω Dynamical 0 System • Ergodic for µ > 0 Summary • Non-ergodic for µ ≤ 0 Ergodicity in Stochastic Sample paths Process and Dynamical System
Khatia Sharia Supervised by : Ole Peters, Yuzuru Sato, Jeroen Lamb
Stochastic Process
Dynamical System
Summary Ergodicity in Stochastic Process and Dynamical System
Khatia Sharia Supervised by : Ole Peters, Yuzuru Sato, Jeroen Lamb Ergodicity can be broken, for example if: Stochastic • Process the time evarage does not exist because the quantity f
Dynamical grows System • or the expectation value diverges Summary • or the distribution P(x(t)) has no well-defined time limit Ergodicity in Stochastic Outline Process and Dynamical System
Khatia Sharia Supervised by : Ole Peters, Yuzuru Sato, Jeroen Lamb
Stochastic 1 Stochastic Process Process
Dynamical System Summary 2 Dynamical System
3 Summary Ergodicity in Stochastic Dynamical system Process and Dynamical System
Khatia Sharia Supervised by : Ole Peters, Yuzuru Sato, Jeroen Lamb
Stochastic Process
Dynamical System
Summary Ergodicity in Stochastic “Probability” Process and Dynamical System
Khatia Sharia Supervised by : Ole Peters, Yuzuru Sato, Jeroen Lamb
Stochastic Process Probability, P(x), in dynamical systems: Dynamical System • Run the dynamical system for a long time with initial Summary condition x0 • The probability of an event A is the relative frequency with which that event is observed. • c irrational: dynamics is ergodic • c rational: dynamics is non-ergodic
Ergodicity in Stochastic Example: The Rotation Map Process and Dynamical System
Khatia Sharia Supervised by : Ole Peters, Yuzuru Sato, Jeroen Lamb
Stochastic Process
Dynamical xn+1 = (xn + c) mod 1 System
Summary • c rational: dynamics is non-ergodic
Ergodicity in Stochastic Example: The Rotation Map Process and Dynamical System
Khatia Sharia Supervised by : Ole Peters, Yuzuru Sato, Jeroen Lamb
Stochastic Process
Dynamical xn+1 = (xn + c) mod 1 System • c irrational: dynamics is ergodic Summary Ergodicity in Stochastic Example: The Rotation Map Process and Dynamical System
Khatia Sharia Supervised by : Ole Peters, Yuzuru Sato, Jeroen Lamb
Stochastic Process
Dynamical xn+1 = (xn + c) mod 1 System • c irrational: dynamics is ergodic Summary • c rational: dynamics is non-ergodic Ergodicity in Stochastic Process and Dynamical System
Khatia Sharia Supervised by : Ole Peters, Yuzuru Sato, Rational c Jeroen Lamb 1 xn+1 = xn + 5 mod 1 Stochastic Process
Dynamical System 1 x1 = x0 + mod 1 Summary 5 1 x2 = x1 + 5 mod 1 ··· Ergodicity in Stochastic Process and Dynamical System
Khatia Sharia Supervised by : Ole Peters, Yuzuru Sato, Jeroen Lamb
Stochastic Process
Dynamical System
Summary Ergodicity in Stochastic Process and Dynamical System
Khatia Sharia Supervised by : Ole Peters, Yuzuru Sato, Jeroen Lamb 1 Pi=4 Stochastic Time average: 5 i=0 f (xi ) Process 1 Expectation value : R 1 f (x) δ(x − x ) dx Dynamical 0 5 i System | {z } Summary P(x) Ergodicity in Stochastic How ergodicity can be broken? Process and Dynamical System
Khatia Sharia Supervised by : Ole Peters, Yuzuru Sato, Jeroen Lamb
Stochastic Process Dynamical • time average does not exist System
Summary • sample space split into different components depending on initial condition x0 Ergodicity in Stochastic Outline Process and Dynamical System
Khatia Sharia Supervised by : Ole Peters, Yuzuru Sato, Jeroen Lamb
Stochastic 1 Stochastic Process Process
Dynamical System Summary 2 Dynamical System
3 Summary Ergodicity in Stochastic Summary Process and Dynamical System
Khatia Sharia Supervised by : Ole Peters, Yuzuru Sato, Thoughts: Jeroen Lamb • Stochastic process: Birkhoff’s equation holds for any Stochastic Process trajectory. Dynamical System • Dynamical system: Birkhoff’s equation holds for any
Summary x(t = 0). • No time in random variable • Dynamical systems has a closed space in mind (some blob we draw), whereas stochastic processes take place, typically, on the infinite real line, so you run into different problems.