nonlocal pdes & non-gaussian...

Motivation Impressions of Stochastic Dynamics Non-Gaussian Dynamics Conclusions

Nonlocal PDEs & Non-Gaussian Dynamics

Jinqiao DuanInstitute for Pure and Applied Mathematics

Los Angeles&

Dept. of Applied Math, Illinois Institute of TechnologyChicago

mypages.iit.edu/~duan


Outline

1 Motivation

2 Impressions of Stochastic Dynamics

3 Non-Gaussian DynamicsEscape probabilityMean exit timeBifurcation

4 Conclusions


Recall —-

Lin, Gao, Duan & Ervin, 2000:

Asymptotic Dynamical Difference between the Nonlocal andLocal Swift-Hohenberg Models

Macroscopic – Macroscopic

Macroscopic → Microscopic


Dynamical systems

• Deterministic dyn systems: Geometric views on solutionsOrdinary diff eqns (ODEs)Partial diff eqns (PDEs)

• Stochastic dyn systems : Geometric views on solutionsStochastic diff eqns (SDEs)Stochastic partial diff eqns (SPDEs)


Stochastic dynamical systems

1970s: Stochastic differential equations (SDEs)Ikeda-Watanabe, Arnold, Friedman

1980s: Stochastic flows, cocyclesElworthy, Baxendale, Bismut, Ikeda, Kunita, ...

1990s: Dynamical systems approaches for SDEsL. Arnold, ......

Related development:• Ergodic theory• Statistical mechanics


Stochastic Dynamical Systems:

Dynamical systems with noise!• Environmental or intrinsic fluctuations! (seen in observations)

• Gaussian noise or non-Gaussian noise• Brownian motion or α−stable Lévy motions


Scientific observations:Many independent measurements and then “averaging"

Where do Gaussian random variables come from?X1,X2, · · · ,Xn are independent, identically distributed (iid)random variables with finite mean µ and variance σ2

Central limit theorem: Gaussian random variables come from“averaging"

limn→∞

X1 + · · ·+ Xn − nµσ√

n= X ∼ N (0,1) in distribution

Namely,

limn→∞

P(X1 + · · · + Xn − nµ

σ√

n≤ x) =

1√2π

∫ x

−∞e−x2/2dx


Scientific observations:Many independent measurements and then “averaging"

Where do stable random variables come from?

X1,X2, · · · ,Xn are independent, identically distributed (iid)random variables (whose mean and variance may be infinite)

Definition:X is a stable random variable if it is a limit (in distribution) of anaveraging sequence of X ′

i s:

limn→∞

X1 + · · · + Xn − bn

an= X in distribution

for some constants an,bn (an 6= 0)

Notation: X ∼ Sα(σ, β, µ)


Gaussian vs. Non-Gaussian random variables

Gaussian random variable X (ω):Probability density function (PDF) 1√

2πe−u2/2

P(X ≤ x) =∫ x−∞

1√2π

e−x2/2 dx

Non-Gaussian random variable X (ω):Probability density function (PDF) f (x) ≥ 0,

∫

Rf (x)dx = 1.

P(X ≤ x) =∫ x−∞ f (x) dx

Gaussian process & non-Gaussian process


Prob density function for a Gaussian random variable

Also called a normal random variable: X ∼ N (µ, σ2)

−20 −15 −10 −5 0 5 10 15 200

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

f(x)

Figure: “Bell shape"


Prob density function for a non-Gaussian random variable

0x


• Browniam motion is defined in terms of Gaussian r. v.

• α−stable Lévy motion is defined in terms of stable r. v.


Brownian motion Bt : A Gaussian process

Independent increments: Bt2 − Bt1 and Bt3 − Bt2independent

Stationary increments with Bt − Bs ∼ N(0, t − s)In particular, Bt ∼ N(0, t)

Continuous sample paths, but nowhere differentiable

Reference:I. Karatzas and S. E. Shreve,Brownian Motion and Stochastic Calculus


A sample path for Brownian motion Bt(ω)

0 5 10 15 20 25 30−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

t

Wt

Figure: Continuous path, but nowhere differentiable


α-stable Lévy Motion Lt : A non-Gaussian process

Definition: α-stable Lévy motion Lt(ω) with 0 < α < 2:(1) L0 = 0, a.s.;(2) Lt has independent increments;(3) Stationary increments Lt − Ls ∼ Sα(|t − s| 1

α , β,0);

Note 1. Paths are stoch. continuous (right continuous with leftlimit; countable jumps): Lt → Ls in prob. as t → sNote 2. α = 2: Brownian motion Bt andBt − Bs ∼ S2(|t − s| 1

2 ,0,0)Note 3. When β = 0, Lévy jump measure: να(du) = Cα

|u|1+α(du),

α-stable symmetric Lévy Motion. (Cα = αΓ((d+α)/2)21−α

√π Γ(1−α/2) )


α-stable symmetric Lévy Motion Lt

Lévy jump measure: να(du) = cα 1|u|1+α

(du)0 < α < 2: Lévy motion Lt

α = 2: Brownian motion Bt

Heavy tail for 0 < α < 2: Power law (non-Gaussian)

P(|Lt | > u) ∼ 1uα

Light tail for α = 2: Exponential law (Guassian)

P(|Bt | > u) ∼ e−u2/2√

2πu

Reference:D. Applebaum — Lévy Processes and Stochastic Calculus


A sample path for α−stable Lévy motion Lt(ω): Jumps

0 0.2 0.4 0.6 0.8 1−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

t

L tα

Figure: α−stable Lévy motion: α = 0.25



0 0.2 0.4 0.6 0.8 1−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

t

L tα




0 0.2 0.4 0.6 0.8 1−0.5

0

0.5

1

1.5

2

t

L tα



Summary on comparison of BM and α−stable LM

Brownian Motion α−stable Levy Motion

Independent increment Independent increment

Stationary increment Stationary increment

Bt − Bs ∼ S2(|t − s| 12 ,0,0) Lt − Ls ∼ Sα(|t − s| 1

α , β,0)

Continuous sample paths Stoch continuous paths (“jumps")

Triplet (a, d , 0) Triplet (a, 0 ,να)


Properties of paths of α−stable Lévy motion Lt

(i) Countable and dense jumps in time

(ii) Right continuous with left limit at each jump time


Why Lévy motion Lt(ω): Jumps or flights

Abrupt climate change such as Dansgaard-Oeschgerevents.

Ditlevsen 1999: Ice record for temperature

Diffusion of tracers in rotating annular flows: Pauses nearcoherent structures & jumps or “flights" in betweenPDF for flight times: Power lawsSwinney et al. 1995

Data in biology and other areasShlesinger et al.: Lévy Flights and Related Topics inPhysics, 1995

Woyczynski, Lévy processes in the physical science, 2001

Financial data


What is noise?

Stationary process X (t): mean EX (t) is a constant and theautocorrelation E(X (t1)X (t2)) depends only on the time lagt2 − t1.

Noise: ηt

A stationary stochastic process

Mean Eηt = 0

Covariance E(ηtηs) = K c(t − s) for all t and s, K is aconstant

White noise: c(t − s) = δ(t − s) Dirac Delta function

Colored noise: Non-white noise


Gaussian noise: ddt Bt

Brownian motion Bt is a Gaussian process with stationary (andalso independent) increments, together with mean zeroEBt = 0 and covariance E(BtBs) = t ∧ s = min{t , s}.

Increments : Bt+∆t − Bt ≈ ∆t Bt are stationaryMean: EBt ≈ E

Bt+∆t−Bt∆t = 0

∆t = 0Covariance :By the formal formula E(Xt Xs) = ∂2

E(XtXs)/∂t∂s, we see that

E(Bt Bs) = ∂2E(BtBs)/∂t∂s = ∂2(t ∧ s)/∂t∂s = δ(t − s).

Made rigorous: Theory of Generalized Functions


Why is it called white noise?

The spectral density function for Bt , i.e., the Fourier transformF for its covariance function E(Bt Bs), is constant

F(E(Bt Bs)) = F(δ(t − s)) =1

2π.

Thus ηt = Bt is taken as a mathematical model for white noise.

Analogy: White light in Optics


Non-Gaussian noise: ddt Lt


Lee & Shih 2008


Gaussian & Non-Gaussian Noise

Gaussian noise: ddt Bt



Dynamics under Noise

Dynamical Systems

Input Output

Gaussian/Brownian motion

Non−Gaussian/Levy motion


Impact of Gaussian noise on solutions

dXt = (−Xt + X 3t )dt + εdBt , X0 = 0.5.

A solution path:

−20 −15 −10 −5 0 5 10 15 20−0.2

0

0.2

0.4

0.6

0.8

1

1.2

t

Xt

Figure: A solution path with “wigglings" when ε = 0.05, on top of the


Impact of non-Gaussian noise on solutions

dXt = (−Xt + X 3t )dt + εdLt , X0 = 0.5.

A solution path: α = 0.85

−20 −15 −10 −5 0 5 10 15 20−1

−0.5

0

0.5

1

1.5

2

2.5

3

t

X(t

)

Euler Scheme when noise α =0.85



Impact of non-Gaussian noise on solutions

dXt = (−Xt + X 3t )dt + εdLt , X0 = 0.5.

A solution path: α = 1.5

−20 −15 −10 −5 0 5 10 15 20−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

t

X(t

)

Euler Scheme when noise α =1.5



Impact of noise on pathwise uniqueness of solutions

x = f (x): a sufficient condition for uniqueness of solutions isthe local Lipschitz continuity condition for vector field f . Withoutthis condition, the uniqueness is often violated .

However, for x = f (x) + Lt with f being Borel measurable andbounded (no Lipschitz condition), the uniqueness holds .

Priola 2010 : Nonlocal PDE argument!


What is dynamics?

• Look at solutions collectively, as time goes on

• Examine solution mappings, as time goes on

• Discover structures as stepping stones for understanding


Deterministic dynamical system

Linear system:x ′ = Ax , x(0) = x0

Solution mapping (Matrix exponential ): ϕ(t , x0) , eAtx0

“Flow" property: ϕ(t + s, x0) = eA(t+s)x0 = ϕ(t , ϕ(s, x0))

Nonlinear system:x ′ = f (x) , x(0) = x0

Solution mapping: ϕ(t , x0)“Flow" property: ϕ(t + s, x0) = ϕ(t , ϕ(s, x0))


Stochastic dynamical systems

Langevin equation:

dx = −xdt + dBt , x(0) = x0

Random solution mapping:ϕ(t , x0, ω) = x0e−t +

∫ t0 e−(t−s)dBs(ω)

Linear stochastic systems: Random matrices


How to understand stochastic dynamics?

Topological methods: Invariants (e.g., Poincare index,Conley index)

Geometric methods: Invariant structures (e.g., invariantmanifolds)

Functional analytical methods: Nonlocal PDEs!

Duan

Rectangle


Topological methods: Invariants

Still in infancy!

Conley index: Liu 2008; Chen-Duan-Fu 2010

Difficulty due to the very nature of random invariant sets!


A caveat!

Topological : Brouwer fixed point theoremA continuous map from a convex compact subset K to K itselfhas a fixed point.

Does NOT hold for random mappings — Some continuousrandom mappings on compact intervals do not haverandom invariant points

Ochs and Oseledets 1999






Duan

Rectangle


Geometric methods: Invariant manifolds

Banach fixed point theoremA contraction mapping has a unique fixed point.

Invariant manifolds (Liapunov-Perron method)

Still hold for random mappings — contraction in meanSchmalfuss 1997


Basic definitions

Random invariant set M(ω):

ϕ(t , ω,M(ω)) = M(θtω) for t ∈ R

Stable manifold:If we can represent an invariant set as a graph of a Lipshitzmapping

hs(·, ω) : Hs → Hu

such that

W s(ω) = {ξ + hs(ξ, ω) | ξ ∈ Hs}

then W s(ω) is called a Lipschitz stable manifold.

Unstable manifold: Similar


Theorem: Impact of noise on stable manifold

Assumptions: dudt = Au + F (u) + ǫ u ◦ Lt

(i) Linear part A: exponential dichotomy (“saddle property")

(ii) Nonlinear part F (u): twice continuously Frechetdifferentiable with respect to u

(iii) Gap condition: K LF ( 1α − 1

β ) < 1


Theorem: Impact of noise on stable manifold

Random stable manifold: W s(ω) = {ξ + hs(ξ, ω)∣

∣

∣ξ ∈ Hs}

Approximating stable manifold:When ǫ is sufficiently small, W s can be expressed as

hs = h(d)(ξ) + ǫh(1)(ξ, ω) + Rs

where ‖Rs‖ ≤ C(ω)ǫ2 with C(ω) < ∞, h(d) is deterministicstable manifold, and

h(1)(ξ, ω) =

∫ 0

∞e−As{[Z (θr (ω))dr

+Z (θs(ω))]PuF (u0) + PuF u0(s)u [u1(s)− Z (θs(ω))u0(s)]}ds

Note: Z (ω) is the stationary solution ofdZ (t) + Z (t)dt = dB(t)


Example:What is impact of noise on invariant manifolds?

Consider a SDE system{

x = −x + ǫx ◦ Bt ,

y = y + x2 + ǫy ◦ Bt ,

0 < ǫ ≪ 1

W s: Stable manifold in a neighborhood around (0,0)

ǫ = 0: Deterministic stable manifold is

W s =

{(

xy

)

∣

∣

∣y = −x2

3

}


Deterministic stable manifold: ε = 0

x = −x

y = y + x2

Stable manifold W s : y = −13x2


Random stable manifold: ε = 0.01

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−1.5

−1

−0.5

0

x

y

Deterministic stable mfd

Random stable mfd (sample 1)

Random stable mfd (sample 2)


Random stable manifold: 0 < ε ≪ 1

Random stable manifold:

W s =

{(

xy

)

∣

∣

∣y = −x2

3− ǫ

x2

3

(∫ ∞

0e−3τ dBτ

)

+ O(ǫ2)

}

Remarks:With accuracy of O(ǫ2),

(i) Mean of random stable manifold is just the deterministicstable manifold

(ii) Variance of the random stable manifold is 154x4

Sun, Duan & Li: 2010Open problem: Non-Gaussian Lévy noise






Duan

Rectangle



Main Ideas:(i) Solutions of stochastic systems are Markov processes

(ii) Markov processes → Semigroups

(iii) Semigroups’ generators A: Nonlocal operators!

(iv) Non-Gaussian Lévy noise ∼ Nonlocal operators!


Generators of Markov process Xt : X0 = x

Semigroup : For observable ϕ

Ptϕ(x) , Eϕ(Xt)Pt+s = PtPs

Generator of semigroup : Derivative of semigroup at time 0

Aϕ(x) , ddt |t=0Ptϕ(Xt )

Duan

Text Box

Macroscopic description


Example: Generators of Bt & Lt

Brownian motion Bt : Generator is 12 ∆

α−stable Lévy motion: Generator is −Kα (−∆)α

2

Nonlocal operator:∫

Rd\{0}[u(x + y)− u(x)− I{|y |<1} yu′(x)] να(dy) = −Kα (−∆)α

2

Applebaum 2009



Escape probability

Mean exit time

Bifurcation


Escape probability from a domain D

dXt = f (Xt)dt + g(Xt)dLt , X0 = x ∈ D

p(x) : Likelihood that a “particle" first escapes D and lands in adomain U

D

x

U

Figure: Escape probability for SDEs driven by Lévy motions: anopen annular domain D, with its inner part U (which is in Dc) as atarget domain





D

x

U

Figure: Escape probability for SDEs driven by Lévy motions: ageneral open domain D, with a target domain U in Dc


Recall: Classical harmonic functions

∆h(x) = 0But ∆ is the generator of Brownian motion Bt

h(x): Harmonic function with respect to Brownian motion


What is a harmonic function with respect to α−stable Lévymotion?


Brownian motion Bt : Generator is 12∆

α−stable Lévy motion: Generator is −Kα (−∆)α

2


Harmonic function with respect to α−stable Lévy motion:−(−∆)

α

2 h(x) = 0


General harmonic functions

Harmonic function with respect to a Markov process withgenerator A:

Ah(x) = 0

Duan

Text Box

"Probability" feedbacks to "Analysis"!


Escape probability vs. harmonic functions


For

ϕ(x) ={

1, x ∈ U,0, x ∈ Dc \ U,

E[ϕ(XτDc (x))] =

∫

{ω:XτDc (x)∈U}ϕ(XτDc (x))dP(ω)

+

∫

{ω:XτDc (x)∈Dc\U}ϕ(XτDc (x))dP(ω)

= P{ω : XτDc (x) ∈ U}= p(x).

Left hand side is a harmonic function: Liao 1989

Duan

Callout

Macroscopic description



Liao 1989; Kan, Qiao & Duan, 2011



TheoremEscape probability is solution of Balayage-Dirichlet problem

Ap = 0,p|U = 1,p|Dc\U = 0,

where A is the generator for this system.


Example: Pure jump system

Blumenthal, Getoor & Ray 1961

dXt = 0dt + dLt , X0 = x

p(x) : Likelihood that a “particle" first escapes D = (−2,2) andlands in a domain U = [2,∞).

-2 -1 0 1 20.00.20.40.60.81.0

x

pHxL

Α=0.25

-2 -1 0 1 20.00.20.40.60.81.0

x

pHxL

Α=1

-2 -1 0 1 20.00.20.40.60.81.0

x

pHxL

Α=1.25

-2 -1 0 1 20.00.20.40.60.81.0

x

pHxL

Α=1.99


Another example: Double-well system

Gao, Duan, Li & Song 2011: A cross-over phenomenon

dXt = (Xt − X 3t )dt + dLt , X0 = x

p(x) : Likelihood that a “particle" first escapes D = (−1.1,0)and lands in a domain U = [0,∞); i.e., transition from stablestate {−1} to another stable state {1}.

−1 −0.8 −0.6 −0.4 −0.2 00

0.2

0.4

0.6

0.8

1

x

PE(x

)

(a)

α=0.5

α=1

α=1.5

α=2

−2 −1.5 −1 −0.5 00

0.2

0.4

0.6

0.8

1

x

PE(x

)

(b)

α=0.5

α=1

α=1.5

α=2



Escape probability

Mean exit time

Bifurcation


Mean exit time from a domain

dXt = f (Xt)dt + g(Xt)dLt , X0 = x

x

D


How to quantify mean exit time?


f (·): Deterministic vector field

Exit time from a domain D: σx (ω) = inf{t : Xt ∈ Dc}

Mean exit time (for a ‘particle’ starting at x) from a domain D:u(x) = Eσx(ω)

Theorem: Mean exit time u(x) satisfiesAu = −1 for x ∈ D, u|Dc = 0,where A is the generator for this system.


One-dim system with α−stable Lévy motion: (0,d , να)

dXt = f (Xt)dt + dLt , X0 = x ∈ D

Generator:

Au = f (x)u′

(x) +d2

u′′(x)

+

∫

R\{0}[u(x + y)− u(x)− I{|y |<1} yu′(x)] να(dy)

να(dx) = Cα|x |−(1+α) dx with Cα =α

21−α√π

Γ(1+α2 )

Γ(1 − α2 )

u(x) : Mean exit time for a “particle" first escaping D

Au = −1 for x ∈ D, u|Dc = 0


Example: Mean exit time for α−stable Lévy motion

dXt = 0dt + dLt , X0 = x

Getoor 1961: Mean exit time from interval D = (−r , r)

u(x) =√π

2αΓ(1 + α2 )Γ(

12 + α

2 )(r2 − x2)

α

2

00.5

11.5

2

−1

−0.5

0

0.5

10

0.5

1

1.5

αx

u

Figure: Mean exit time from the interval (−0.75, 0.75)


Example: Mean exit time for Ornstein-Uhlenbeck system withα−stable Lévy motion

dXt = −Xtdt + dLt , X0 = x

Lt : Triplet (0,1, να)

−1 −0.5 0 0.5 10

0.2

0.4

0.6

0.8

x

u(x)

(a)

α=0.5

α=1

α=1.5

α=2

−1.5 −1 −0.5 0 0.5 1 1.50

0.5

1

1.5

xu(

x)

(b)

α=0.5

α=1

α=1.5

α=2

−2 −1 0 1 20

0.5

1

1.5

2

x

u(x)

(c)

α=0.5

α=1

α=1.5

α=2

−4 −2 0 2 40

2

4

6

8

10

x

u(x)

(d)

α=0.5

α=1

α=1.5

α=2



Escape probability

Mean exit time

Bifurcation


Bifurcation under non-Gaussian noise

dXt = f (b,Xt)dt + dLt

Bifurcation: How does this system evolve as parameters b, αvary?

Space of paths: A big mess!

Space of probability measures: Order emerges out oforderless!


Bifurcation under non-Gaussian noise

dXt = f (b,Xt)dt + ǫdLt .

Fokker-Planck equation for the stationary probability densityfunction p(x) :

−[f (b, x)p(x)]′

+ ǫ

∫

R\{0}[p(x + y)− p(x)− I{|y |<1}yp′(x)]

dy|y |1+α

= 0

under p(x) ≥ 0,∫

Rp(x)dx = 1.

Chen, Duan & Zhang 2011


Example: Bifurcation under non-Gaussian noise

dXt = (bXt − X 3t )dt + dLt for b ∈ R

1

Result: Stationary prob. density function b < 0 unimodal →b > 0 bimodal

−10 −5 0 5 100

0.05

0.1

0.15

0.2

0.25 (a)

−10 −5 0 5 100

0.1

0.2

0.3

0.4 (b)

−10 −5 0 5 100

0.05

0.1

0.15

0.2 (c)

−10 −5 0 5 100

0.05

0.1

0.15

0.2 (d)

Figure: α = 1.99. (a) b = −30; (b) b = −50; (c) b = 20; (d) b = 30


Conclusions

Nonlocal PDEs & Non-Gaussian Dynamics

Escape probability

Mean exit time

Bifurcation

Computation!

nonlocal pdes & non-gaussian...

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