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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

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Kramers’ type law for L´ evy flights∗

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

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KRAMERS’ TYPE LAW FOR L´

EVY FLIGHTS

1

  • 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)

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  • 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).

100 80 60 40 20 −42 −38 −34

Age (thousands of years before present)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 19 20 18

− 20

∆T

− 34 − 38 − 42 100 80 60 40 20

−

  • 1
  • rapid warming by 5-10 ◦C within at most

a few decades

  • plateau phase with slow cooling lasting

several centuries

  • rapid drop to cold stadial conditions

Simulations: Ganopolsky/Rahmstorf, Potsdam Institute for Climate Impact Research

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EVY FLIGHTS

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  • 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 dXε

t = −U ′(Xε t ) dt + ε dLt

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.

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KRAMERS’ TYPE LAW FOR L´

EVY FLIGHTS

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  • 4. Object of Study. Simple System with L´

evy Noise

Small noise (ε ↓ 0) asymptotics of transition times for systems with L´ evy noise Xε

t = x −

t U ′(Xε

s−) ds + εLt,

ε ↓ 0.

  • L — α-stable symmetric L´

evy motion (0 < α < 2) + Brownian Motion Potential U ∈ C(3)(R):

  • U ′(x)x ≥ 0
  • U ′(x) = 0 iff x = 0
  • U ′′(0) = M > 0

−b a x U(x)

σ(ε) = inf{t ≥ 0 : Xε

t /

∈ [−b, a]}, a, b < ∞ (b = ∞)

5 10 15 20 25 30

  • 0.6
  • 0.4
  • 0.2

0.2

Ref.: P . Imkeller, I. Pavlyukevich, arXiv:math.PR/0409246 (2004)

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KRAMERS’ TYPE LAW FOR L´

EVY FLIGHTS

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  • 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 EeiλL1 = exp

  • −d

2λ2 +

  • R\{0}

(eiλy − 1 − iλy I{|y| ≤ 1}) dy |y|1+α

  • ,

α ∈ (0, 2)

  • Gaussian variance d ≥ 0
  • L´

evy measure ν(dy) =

dy |y|1+α, y = 0

  • ν(R) = ∞ ⇔ countably many (small) jumps on any finite time interval,

jump times are dense EeiλL1 = exp

  • −d

2λ2 − c(α)|λ|α

  • .

c(α) → ∞, α ↑ 2

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KRAMERS’ TYPE LAW FOR L´

EVY FLIGHTS

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  • 6. Symmetric α–stable L´

evy process L

5 10 15 20

  • 2

2 4 6 5 10 15 20

  • 1.5
  • 1
  • 0.5

0.5 1

α = 0.75 α = 1.75 α = 1 Cauchy process 1 π 1 1 + x2 α = 2 Brownian motion 1 √ 2πe−x2

2

In the physical literature: L´ evy Flights In a broader sense: Anomalous Diffusion

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KRAMERS’ TYPE LAW FOR L´

EVY FLIGHTS

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  • 7. Exit Time. Results

Theorem 1. There exist positive constants ε0, γ, δ, and C > 0 such that for 0 < ε ≤ ε0 the following asymptotics holds e−u(1+Cεδ)(1 − Cεδ) ≤ P

x

εα α 1 aα + 1 bα

  • σ(ε) > u
  • ≤ e−u(1−Cεδ)(1 + Cεδ)

uniformly for all x ∈ [−b + εγ, a − εγ] and u ≥ 0. Theorem 2. There exist positive constants ε0, γ and δ such that for 0 < ε ≤ ε0 the following asymptotics holds Exσ(ε) = α εα 1 aα + 1 bα −1 (1 + O(εδ)) uniformly for all x ∈ [−b + εγ, a − εγ].

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KRAMERS’ TYPE LAW FOR L´

EVY FLIGHTS

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  • 8. Probabilistic Approach

Lt = ξε

t + ηε t

νε

ξ(dy) = I{|y|≤ 1

√ε}(y)

dy |y|1+α νε

η(dy) = I{|y|> 1

√ε}(y)

dy |y|1+α νε

ξ(R) = ∞

νε

η(R) = 2

∞

1/√ε

dy |y|1+α = 2 αεα/2 = βε εξε is a sum of BM of intensity ε and a small-jumps process, |∆(εξε

t)| ≤ √ε.

εηε is a compound Poisson process, |∆(εηε

t)| > √ε

εξε and εηε are independent.

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EVY FLIGHTS

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  • 9. The Small- and Large-Jump Parts

0 = τ0, τ1, τ2, . . . arrival times of ηε Tk = τk − τk−1 independent i.d. inter-arrival times Wk = ηε

τk − ηε τk− independent i.d. jumps

Tk ∼ exp(βε) Wk ∼ 1 βε νε

η(·)

ETk = 1 βε = α 2 1 εα/2 P(Wk ≤ x) = 1 βε x

−∞

I{|y|> 1

√ε}(y)

dy |y|1+α Between the big jumps Xε is driven by εξε: Xε

t = x −

t U ′(Xε

s−) ds + εξε t,

t ∈ [0, τ1), Yt = x − t U ′(Y ε

s ) ds.

On inter-jump intervals Xε is Y perturbed by εξε.

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EVY FLIGHTS

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  • 10. Behaviour on the Intervals between Big Jumps

˙ Yt = −U ′(Yt) T1 ∼ exp(βε)

ε −εγ

γ

T

1

R( ) ε Y

t

P

  • sup

[0,T1)

|Xε

t − Yt| ≥ εγ 2

  • ≤ P
  • sup

[0,T1)

|εξε

t| ≥ εγ C

  • ≤

∞ P

  • sup

[0,u)

|εξε

t| ≥ εγ C

  • βεe−βεu du ≤ e−1/εδ

T(x, ε) = x

εγ/2

dy |U ′(y)| ≈ x

δ

dy |U ′(y)| + δ

εγ/2

dy My ≈ Const + γ M | ln ε| ≤ R(ε) = O(| ln ε|)

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EVY FLIGHTS

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  • 11. Predominant Behaviour

1 2 3 4 5

  • 0.2

0.2 0.4 0.6

τ τ τ

1 2 3

εW

2

εW

1

εW

3

−b a Xε

t

Expected inter-jump time ETk = 1

βε = α 2ε−α/2

polynomial in ε Relaxation time R(ε) = O(| ln ε|) logarithmic in ε Between big jumps Xεis driven by “small jumps”εξε Deviation probability small ⇒ Typically, Xε jumps from a neighbourhood of 0 by εWk at τk. Typically, Xε exits I = [−b, a] by jumping at times τk.

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EVY FLIGHTS

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  • 12. Exit Time Law. Heuristic Proof of Theorem 2

τk = T1 + T2 + · · · + Tk, ET1 = 1/βε P(εW1 / ∈ [−b, a]) = 1 βε ∞

a ε

+ ∞

b ε

  • dy

y1+α = εα αβε 1 aα + 1 bα

  • Exσ(ε) ≈

∞

  • k=1

Eτk · P(σ(ε) = τk) ≈

∞

  • k=1

k · ET1 · P(εW1 ∈ I, . . . , εWk−1 ∈ I, εWk / ∈ I) =

∞

  • k=1

k · ET1 · [1 − P(εW1 / ∈ I)]k−1 · P(εW1 / ∈ I) = P(εW1 / ∈ I) βε 1 P(εW1 / ∈ I)2 = α εα 1 aα + 1 bα −1

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KRAMERS’ TYPE LAW FOR L´

EVY FLIGHTS

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  • 13. Exponential Exit. Heuristic Proof of Theorem 1

τk = T1 + T2 + · · · + Tk ∼ Gamma(βε, k) P(τk ∈ [t, t + dt]) = βεe−βεt(βεt)k−1 (k − 1)! dt P(εW1 / ∈ [−b, a]) = 1 βε ∞

a ε

+ ∞

b ε

  • dy

y1+α = 1 2εα/2 1 aα + 1 bα

  • P

x

  • εα

α

1

aα + 1 bα

  • σ(ε) > u
  • ≈

∞

  • k=1

P

  • εα

α

1

aα + 1 bα

  • τk > u
  • · P

x(σ(ε) = τk)

≈

∞

  • k=1

P

  • εα

α

1

aα + 1 bα

  • τk > u
  • · P(εW1 ∈ I, . . . , εWk−1 ∈ I, εWk /

∈ I) = exp (−u)

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KRAMERS’ TYPE LAW FOR L´

EVY FLIGHTS

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  • 14. Comparison with Gaussian Case

−b a x U(x) h h = U(−b) < U(a)

ˆ Xε

t = x −

t

0 U ′( ˆ

Xε

s) ds + εWt

Freidlin–Wentzell (large deviations): P

x(e(2h−δ)/ε2 < ˆ

σ < e(2h+δ)/ε2) → 1 Kramers’ law (’40, Williams, Bovier): Exˆ σ ≈

ε√π |U′(−b)|√ U′′(0)e2h/ε2

Exponential exit (Day, Bovier) P

x( ˆ σ Exˆ σ > u) ∼ exp (−u)

Diffusion ‘climbs up and out’

−b a x U(x) h h = U(−b) < U(a)

Xε

t = x −

t

0 U ′(Xε s−) ds + εLt

P

x( 1 εα−δ < σ < 1 εα+δ) → 1

Exσ ≈ α

εα[ 1 aα + 1 bα]−1

P

x( σ Exσ > u) ∼ exp (−u)

L´ evy motion driven SDE ‘jumps out’

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EVY FLIGHTS

15

  • 15. Double-well Potential

Xε

t = x −

t U ′(Xε

s−) ds + εLt,

ε ↓ 0.

  • L — α-stable symmetric L´

evy motion (0 < α < 2) + Brownian Motion Potential U ∈ C(3)(R):

  • U ′(−p) = U ′(0) = U ′(q) = 0
  • U ′′(−p), U ′′(q) > 0, U ′′(0) < 0
  • |U ′(x)| > c1|x|1+c2, x → ±∞

x q −p U(x)

U(−p) < U(q)

10 20 30 40 50

  • 0.6
  • 0.4
  • 0.2

0.2 0.4 0.6

Main inconvenience: Saddle Point 0, characteristic boundary Gaussian Case: Exˆ σ ≈

2π

√

U′′(0)|U′′(q)|e2|U(q)|/ε2

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KRAMERS’ TYPE LAW FOR L´

EVY FLIGHTS

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  • 16. Metastable Behaviour

Theorem 3. For any 0 < t1 < t2 < · · · < tn, (Xε

t1/εα, Xε t2/εα, · · · , Xε tn/εα) D

→ (Yt1, Yt2, · · · , Ytn), ε ↓ 0, where Y is a {−p, q}-valued Markov process with generator

  • − 1

αpα 1 αpα 1 αqα

− 1

αqα

  • , and Y0 =
  • −p, if x < 0,

q, if x > 0. Gaussian case (Kipnis and Newman ’85, Mathieu ’95). Assume the left well is deeper: U(−p) < U(q) < 0 There is a scaling λ(ε): ε2 ln λ(ε) → −2U(q) such that ( ˆ Xε

t1λ(ε), ˆ

Xε

t2λ(ε), · · · , ˆ

Xε

tnλ(ε)) D

→ ( ˆ Yt1, ˆ Yt2, · · · , ˆ Ytn), ε ↓ 0, where ˆ Y is a {−p, q}-valued Markov process with generator

  • 1

−1

  • , and ˆ

Y0 =

  • −p, if x < 0,

q, if x > 0.

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EVY FLIGHTS

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Sources and References

Page 1 Pictures: http://www.ngdc.noaa.gov/paleo/globalwarming/gallery/icecore_4.jpg http://ess.geology.ufl.edu/ess/Notes/Paleoclimatology/Paleoclimate Slides/greenland.gif Page 2 Pictures: Ganopolski, A. and Rahmstorf, S., 2002: Abrupt glacial climate changes due to stochastic resonance.

  • Phys. Rev. Let. 88(3), 038501.

Page 3 Picture: P . D. Ditlevsen, 1999: Observation of alpha-stable noise and a bistable climate potential in an ice-core record.

  • Geophys. Res. Lett. 26, 1441-1444.

[1] P. IMKELLER AND I. PAVLYUKEVICH, First exit times of solutions of non-linear stochastic differential equations driven by symmetric L´ evy process with α-stable component, arXiv:math.PR/0409246, 2004. [2] M. FREIDLIN AND A. WENTZELL, Random perturbations of dynamical systems, 1998. [3] H.A. KRAMERS, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica 7, 284–304, 1940. [4] M. WILLIAMS, Asymptotic exit time distributions, SIAM J. Appl. Math. 42, 149–154, 1982. [5] A. BOVIER, M. ECKHOFF, V. GAYRARD AND M. KLEIN, Metastability in reversible diffusion processes I. Sharp asymptotics for capacities and exit times, WIAS Berlin, Preprint No. 767, 2002, to appear in J. Eur. Math. Soc. [6] M. DAY, On the exponential exit law in the small parameter exit problem, Stochastics 8, 297–323, 1983. [7] I. PAVLYUKEVICH, Metastable Behaviour of L´ evy-Driven Diffusion (L´ evy Flights in a Double-Well Potential), in preparation, 2005 [8] C. KIPNIS AND C. NEWMAN, The metastable behavior of infrequently observed, weakly random, one-dimensional diffusion processes, SIAM J. Appl. Math. 45 (6), 972–982, 1985 [9] P. MATHIEU, Spectra, exit times and long time asymptotics in the zero white noise limit, Stoch. Stoch. Rep. 55, 1–20 1995 Peter Imkeller Institut f¨ ur Mathematik Humboldt Universit¨ at zu Berlin Rudower Chaussee 25 12489 Berlin Germany E.mail: imkeller@mathematik.hu-berlin.de http://www.mathematik.hu-berlin.de/˜imkeller Ilya Pavlyukevich Institut f¨ ur Mathematik Humboldt Universit¨ at zu Berlin Rudower Chaussee 25 12489 Berlin Germany E.mail: pavljuke@mathematik.hu-berlin.de http://www.mathematik.hu-berlin.de/˜pavljuke