Kramers type law for L L evy noise induced transitions Peter - PowerPoint PPT Presentation

evy flights ∗ Kramers’ type law for L´ L´ evy noise induced transitions Peter Imkeller Ilya Pavlyukevich Humboldt-University of Berlin Department of Mathematics 4th Conference on Extreme Value Analysis , Gothenburg, 2005 ∗ Supported by the DFG Research Center Matheon (FZT86) in Berlin and the DFG Reaserch Project Stochastic Dynamics of Climate States

K RAMERS ’ TYPE LAW FOR L´ 1 EVY FLIGHTS 1. Motivation Greenland ice-core data allows to reconstruct Earth’s climate up to 200.000 years before present. International projects: GRIP (3028 m), GISP2 (3053.44 m), NGRIP (3084.99 m)

K RAMERS ’ TYPE LAW FOR L´ 2 EVY FLIGHTS 2. Paleo Proxy Data. Dansgaard-Oeschger Events Paleo Data Proxies: Oxygen isotopes, dust, volcanic markers etc. Global climate during the last glacial ( ∼ 120 000 -10 000 b. p.) has experienced at least 20 abrupt and large-amplitude shifts (Dansgaard-Oeschger events). − 34 − −34 0 � 0 17 20 1 19 14 12 16 8 11 3 15 4 7 18 10 6 13 − 38 5 2 ∆ T −38 9 �� � − 42 −42 − 20 0 100 100 80 60 40 20 0 80 60 20 40 Age (thousands of years before present) • rapid warming by 5-10 ◦ C within at most a few decades • plateau phase with slow cooling lasting several centuries Simulations: Ganopolsky/Rahmstorf, • rapid drop to cold stadial conditions Potsdam Institute for Climate Impact Research

K RAMERS ’ TYPE LAW FOR L´ 3 EVY FLIGHTS 3. Paleo Proxy Data. Detailed Look. The calcium (Ca) signal from the GRIP ice-core: about 80,000 data-points from 11 kyr to 91 kyr before present. Typical interjump time: 1000 – 2000 years, mean waiting time ∼ 1470 years What triggers the transitions? Langevin equation for climate dynamics dX ε t = − U ′ ( X ε t ) dt + ε dL t U – double-well potential, wells correspond to the climate states. P . Ditlevsen ( Geophys. Res. Lett. 1999 ): spectral analysis of the data. Noise L has α -stable component with α ≈ 1 . 75 .

K RAMERS ’ TYPE LAW FOR L´ 4 EVY FLIGHTS 4. Object of Study. Simple System with L´ evy Noise Small noise ( ε ↓ 0 ) asymptotics of transition times for systems with L´ evy noise � t X ε U ′ ( X ε t = x − s − ) ds + εL t , ε ↓ 0 . 0 • L — α -stable symmetric L´ evy motion ( 0 < α < 2 ) + Brownian Motion U(x) Potential U ∈ C (3) ( R ) : • U ′ ( x ) x ≥ 0 • U ′ ( x ) = 0 iff x = 0 • U ′′ (0) = M > 0 x −b a σ ( ε ) = inf { t ≥ 0 : X ε t / ∈ [ − b, a ] } , a, b < ∞ ( b = ∞ ) 0.2 5 10 15 20 25 30 -0.2 -0.4 -0.6 Ref.: P . Imkeller, I. Pavlyukevich, arXiv:math.PR/0409246 (2004)

K RAMERS ’ TYPE LAW FOR L´ 5 EVY FLIGHTS 5. The Driving Process L L – symmetric α -stable L´ evy process (plus Brownian motion). Marginal laws determined by L´ evy–Hinˇ cin’s formula � � − d � dy 2 λ 2 + ( e iλy − 1 − iλy I {| y | ≤ 1 } ) E e iλL 1 = exp , | y | 1+ α R \{ 0 } α ∈ (0 , 2) • Gaussian variance d ≥ 0 dy • L´ evy measure ν ( dy ) = | y | 1+ α , y � = 0 • ν ( R ) = ∞ ⇔ countably many (small) jumps on any finite time interval, jump times are dense � − d � 2 λ 2 − c ( α ) | λ | α E e iλL 1 = exp . c ( α ) → ∞ , α ↑ 2

K RAMERS ’ TYPE LAW FOR L´ 6 EVY FLIGHTS 6. Symmetric α –stable L´ evy process L 6 1 4 0.5 5 10 15 20 2 -0.5 -1 5 10 15 20 -1.5 -2 α = 0 . 75 α = 1 . 75 1 1 α = 1 Cauchy process 1 + x 2 π 1 2 πe − x 2 √ α = 2 Brownian motion 2 In the physical literature: L´ evy Flights In a broader sense: Anomalous Diffusion

K RAMERS ’ TYPE LAW FOR L´ 7 EVY FLIGHTS 7. Exit Time. Results Theorem 1. There exist positive constants ε 0 , γ , δ , and C > 0 such that for 0 < ε ≤ ε 0 the following asymptotics holds � 1 � ε α a α + 1 � � e − u (1+ Cε δ ) (1 − Cε δ ) ≤ P ≤ e − u (1 − Cε δ ) (1 + Cε δ ) σ ( ε ) > u x b α α uniformly for all x ∈ [ − b + ε γ , a − ε γ ] and u ≥ 0 . Theorem 2. There exist positive constants ε 0 , γ and δ such that for 0 < ε ≤ ε 0 the following asymptotics holds � 1 � − 1 E x σ ( ε ) = α a α + 1 (1 + O ( ε δ )) ε α b α uniformly for all x ∈ [ − b + ε γ , a − ε γ ] .

K RAMERS ’ TYPE LAW FOR L´ 8 EVY FLIGHTS 8. Probabilistic Approach L t = ξ ε t + η ε t dy dy ν ε ν ε ξ ( dy ) = I {| y |≤ 1 √ ε } ( y ) η ( dy ) = I {| y | > 1 √ ε } ( y ) | y | 1+ α | y | 1+ α � ∞ | y | 1+ α = 2 dy αε α/ 2 = β ε ν ε ν ε ξ ( R ) = ∞ η ( R ) = 2 1 / √ ε t ) | ≤ √ ε . εξ ε is a sum of BM of intensity ε and a small-jumps process, | ∆( εξ ε t ) | > √ ε εη ε is a compound Poisson process, | ∆( εη ε εξ ε and εη ε are independent.

K RAMERS ’ TYPE LAW FOR L´ 9 EVY FLIGHTS 9. The Small- and Large-Jump Parts 0 = τ 0 , τ 1 , τ 2 , . . . arrival times of η ε T k = τ k − τ k − 1 independent i.d. inter-arrival times W k = η ε τ k − η ε τ k − independent i.d. jumps W k ∼ 1 ν ε T k ∼ exp( β ε ) η ( · ) β ε � x E T k = 1 = α 1 P ( W k ≤ x ) = 1 dy √ ε } ( y ) I {| y | > 1 ε α/ 2 | y | 1+ α β ε 2 β ε −∞ Between the big jumps X ε is driven by εξ ε : � t X ε U ′ ( X ε s − ) ds + εξ ε t = x − t , t ∈ [0 , τ 1 ) , 0 � t U ′ ( Y ε Y t = x − s ) ds. 0 On inter-jump intervals X ε is Y perturbed by εξ ε .

K RAMERS ’ TYPE LAW FOR L´ 10 EVY FLIGHTS 10. Behaviour on the Intervals between Big Jumps Y t ˙ Y t = − U ′ ( Y t ) R( ) ε T 1 γ T 1 ∼ exp( β ε ) ε −ε γ � � � � t − Y t | ≥ ε γ t | ≥ ε γ | X ε | εξ ε sup ≤ P sup P 2 C [0 ,T 1 ) [0 ,T 1 ) � ∞ � � t | ≥ ε γ β ε e − β ε u du ≤ e − 1 /ε δ | εξ ε ≤ sup P C [0 ,u ) 0 � x � x � δ dy dy dy T ( x, ε ) = | U ′ ( y ) | ≈ | U ′ ( y ) | + My ε γ / 2 ε γ / 2 δ ≈ Const + γ M | ln ε | ≤ R ( ε ) = O ( | ln ε | )

K RAMERS ’ TYPE LAW FOR L´ 11 EVY FLIGHTS 11. Predominant Behaviour X ε t a 0.6 ε W 3 0.4 ε W 2 0.2 0 1 2 3 4 5 ε W 1 -0.2 −b τ τ τ 1 2 3 E T k = 1 β ε = α 2 ε − α/ 2 Expected inter-jump time polynomial in ε Relaxation time R ( ε ) = O ( | ln ε | ) logarithmic in ε X ε is driven by “small jumps” εξ ε Between big jumps Deviation probability small Typically, X ε jumps from a neighbourhood of 0 by εW k at τ k . ⇒ Typically, X ε exits I = [ − b, a ] by jumping at times τ k .

K RAMERS ’ TYPE LAW FOR L´ 12 EVY FLIGHTS 12. Exit Time Law. Heuristic Proof of Theorem 2 τ k = T 1 + T 2 + · · · + T k , E T 1 = 1 /β ε � 1 �� ∞ � ∞ � y 1+ α = ε α ∈ [ − b, a ]) = 1 dy a α + 1 � P ( εW 1 / + b α β ε αβ ε a b ε ε ∞ � E x σ ( ε ) ≈ E τ k · P ( σ ( ε ) = τ k ) k =1 ∞ � ≈ k · E T 1 · P ( εW 1 ∈ I, . . . , εW k − 1 ∈ I, εW k / ∈ I ) k =1 ∞ ∈ I )] k − 1 · P ( εW 1 / � = k · E T 1 · [1 − P ( εW 1 / ∈ I ) k =1 � 1 � − 1 = P ( εW 1 / ∈ I ) 1 ∈ I ) 2 = α a α + 1 ε α b α β ε P ( εW 1 /

K RAMERS ’ TYPE LAW FOR L´ 13 EVY FLIGHTS 13. Exponential Exit. Heuristic Proof of Theorem 1 τ k = T 1 + T 2 + · · · + T k ∼ Gamma ( β ε , k ) P ( τ k ∈ [ t, t + dt ]) = β ε e − β ε t ( β ε t ) k − 1 ( k − 1)! dt � 1 �� ∞ � ∞ � ∈ [ − b, a ]) = 1 y 1+ α = 1 dy a α + 1 � 2 ε α/ 2 P ( εW 1 / + b α β ε a b ε ε � 1 � ε α � a α + 1 � σ ( ε ) > u P x b α α ∞ � 1 � � ε α � a α + 1 � ≈ τ k > u · P x ( σ ( ε ) = τ k ) P b α α k =1 ∞ � 1 � � ε α � a α + 1 � ≈ τ k > u · P ( εW 1 ∈ I, . . . , εW k − 1 ∈ I, εW k / ∈ I ) P b α α k =1 = exp ( − u )

K RAMERS ’ TYPE LAW FOR L´ 14 EVY FLIGHTS 14. Comparison with Gaussian Case U(x) U(x) h = U(−b) < U(a) h = U(−b) < U(a) h h x x −b a −b a � t � t ˆ 0 U ′ ( ˆ X ε X ε X ε 0 U ′ ( X ε t = x − s ) ds + εW t t = x − s − ) ds + εL t Freidlin–Wentzell (large deviations): x ( e (2 h − δ ) /ε 2 < ˆ σ < e (2 h + δ ) /ε 2 ) → 1 1 1 x ( ε α − δ < σ < ε α + δ ) → 1 P P Kramers’ law (’40, Williams, Bovier): ε √ π U ′′ (0) e 2 h/ε 2 | U ′ ( − b ) | √ E x σ ≈ α ε α [ 1 a α + 1 b α ] − 1 E x ˆ σ ≈ Exponential exit (Day, Bovier) x ( σ σ ˆ E x σ > u ) ∼ exp ( − u ) P x ( σ > u ) ∼ exp ( − u ) P E x ˆ L´ evy motion driven SDE ‘jumps out’ Diffusion ‘climbs up and out’

K RAMERS ’ TYPE LAW FOR L´ 15 EVY FLIGHTS 15. Double-well Potential � t X ε U ′ ( X ε t = x − s − ) ds + εL t , ε ↓ 0 . 0 • L — α -stable symmetric L´ evy motion ( 0 < α < 2 ) + Brownian Motion U(x) U(−p) < U(q) Potential U ∈ C (3) ( R ) : • U ′ ( − p ) = U ′ (0) = U ′ ( q ) = 0 −p q x • U ′′ ( − p ) , U ′′ ( q ) > 0 , U ′′ (0) < 0 • | U ′ ( x ) | > c 1 | x | 1+ c 2 , x → ±∞ 0.6 0.4 0.2 10 20 30 40 50 -0.2 -0.4 -0.6 Main inconvenience: Saddle Point 0 , characteristic boundary U ′′ (0) | U ′′ ( q ) | e 2 | U ( q ) | /ε 2 2 π √ Gaussian Case: E x ˆ σ ≈