[PPT] - Anomalous Diffusion Index for L evy Motions Chang C. Y. Dorea and PowerPoint Presentation

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References

Anomalous Diffusion Index for L´ evy Motions

Chang C. Y. Dorea

and Ary V. Medino

Departamento de Matem´ atica Universidade de Bras´ ılia

August 2005

Partially supported by CNPq, CAPES, FAPDF/PRONEX and FINATEC/UnB EVA 2005 Anomalous Diffusion Index for L´ evy Motions

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Infinitesimal Coefficients Scaled Random Walks

Diffusion Process : Infinitesimal Coefficients

lim

h↓0

1 hP (|X(t + h) − X(t)| > ǫ |X(t) = x ) = 0 lim

h↓0

1 hE {X(t + h) − X(t) |X(t) = x } = µX(x, t) lim

h↓0

1 hE

(X(t + h) − X(t))2 |X(t) = x
= σ2

X(x, t)

µ(·) = 0, σ2(·) = 1 : Brownian Motion

EVA 2005 Anomalous Diffusion Index for L´ evy Motions

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Infinitesimal Coefficients Scaled Random Walks

Diffusion Process : Infinitesimal Coefficients

lim

h↓0

1 hP (|X(t + h) − X(t)| > ǫ |X(t) = x ) = 0 lim

h↓0

1 hE {X(t + h) − X(t) |X(t) = x } = µX(x, t) lim

h↓0

1 hE

(X(t + h) − X(t))2 |X(t) = x
= σ2

X(x, t)

µ(·) = 0, σ2(·) = 1 : Brownian Motion

EVA 2005 Anomalous Diffusion Index for L´ evy Motions

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Infinitesimal Coefficients Scaled Random Walks

Diffusion Process : Infinitesimal Coefficients

lim

h↓0

1 hP (|X(t + h) − X(t)| > ǫ |X(t) = x ) = 0 lim

h↓0

1 hE {X(t + h) − X(t) |X(t) = x } = µX(x, t) lim

h↓0

1 hE

(X(t + h) − X(t))2 |X(t) = x
= σ2

X(x, t)

µ(·) = 0, σ2(·) = 1 : Brownian Motion

EVA 2005 Anomalous Diffusion Index for L´ evy Motions

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Infinitesimal Coefficients Scaled Random Walks

Diffusion Process : Infinitesimal Coefficients

lim

h↓0

1 hP (|X(t + h) − X(t)| > ǫ |X(t) = x ) = 0 lim

h↓0

1 hE {X(t + h) − X(t) |X(t) = x } = µX(x, t) lim

h↓0

1 hE

(X(t + h) − X(t))2 |X(t) = x
= σ2

X(x, t)

µ(·) = 0, σ2(·) = 1 : Brownian Motion

EVA 2005 Anomalous Diffusion Index for L´ evy Motions

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Infinitesimal Coefficients Scaled Random Walks

Scaled Random Walks

Sn = ξ1 + ξ2 + . . . + ξn E(ξj) = 0, E(ξ2

j ) = σ2 > 0

X (n)(t) = 1 σ√n{S[nt] + (nt − [nt])ξ[nt]+1} → B(t) X (n)(t) = an(S[knt] − bn) → X(t)

EVA 2005 Anomalous Diffusion Index for L´ evy Motions

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Infinitesimal Coefficients Scaled Random Walks

Scaled Random Walks

Sn = ξ1 + ξ2 + . . . + ξn E(ξj) = 0, E(ξ2

j ) = σ2 > 0

X (n)(t) = 1 σ√n{S[nt] + (nt − [nt])ξ[nt]+1} → B(t) X (n)(t) = an(S[knt] − bn) → X(t)

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Infinitesimal Coefficients Scaled Random Walks

Scaled Random Walks

Sn = ξ1 + ξ2 + . . . + ξn E(ξj) = 0, E(ξ2

j ) = σ2 > 0

X (n)(t) = 1 σ√n{S[nt] + (nt − [nt])ξ[nt]+1} → B(t) X (n)(t) = an(S[knt] − bn) → X(t)

EVA 2005 Anomalous Diffusion Index for L´ evy Motions

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Infinitesimal Coefficients Scaled Random Walks

Scaled Random Walks

Sn = ξ1 + ξ2 + . . . + ξn E(ξj) = 0, E(ξ2

j ) = σ2 > 0

X (n)(t) = 1 σ√n{S[nt] + (nt − [nt])ξ[nt]+1} → B(t) X (n)(t) = an(S[knt] − bn) → X(t)

EVA 2005 Anomalous Diffusion Index for L´ evy Motions

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References L´ evy Flights Anomalous Diffusion

Critical Plenomena / L´ evy Flights

Anomalous diffusive regime : X(t) non-Gaussian n−1E(S2

n) → ∞

L´ evy Flights : Muralidhar et al. (1990), Metzler and Klafte (2000) Ferrari et al. (2001), Costa et al. (2003) and Morgado et al. (2004),.. DX = lim

t→∞

E(X 2(t)) 2t (Diffusion Constant)

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References L´ evy Flights Anomalous Diffusion

Critical Plenomena / L´ evy Flights

Anomalous diffusive regime : X(t) non-Gaussian n−1E(S2

n) → ∞

L´ evy Flights : Muralidhar et al. (1990), Metzler and Klafte (2000) Ferrari et al. (2001), Costa et al. (2003) and Morgado et al. (2004),.. DX = lim

t→∞

E(X 2(t)) 2t (Diffusion Constant)

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References L´ evy Flights Anomalous Diffusion

Critical Plenomena / L´ evy Flights

Anomalous diffusive regime : X(t) non-Gaussian n−1E(S2

n) → ∞

L´ evy Flights : Muralidhar et al. (1990), Metzler and Klafte (2000) Ferrari et al. (2001), Costa et al. (2003) and Morgado et al. (2004),.. DX = lim

t→∞

E(X 2(t)) 2t (Diffusion Constant)

EVA 2005 Anomalous Diffusion Index for L´ evy Motions

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References L´ evy Flights Anomalous Diffusion

Critical Plenomena / L´ evy Flights

Anomalous diffusive regime : X(t) non-Gaussian n−1E(S2

n) → ∞

L´ evy Flights : Muralidhar et al. (1990), Metzler and Klafte (2000) Ferrari et al. (2001), Costa et al. (2003) and Morgado et al. (2004),.. DX = lim

t→∞

E(X 2(t)) 2t (Diffusion Constant)

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References L´ evy Flights Anomalous Diffusion

Critical Plenomena / L´ evy Flights

Anomalous diffusive regime : X(t) non-Gaussian n−1E(S2

n) → ∞

L´ evy Flights : Muralidhar et al. (1990), Metzler and Klafte (2000) Ferrari et al. (2001), Costa et al. (2003) and Morgado et al. (2004),.. DX = lim

t→∞

E(X 2(t)) 2t (Diffusion Constant)

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References L´ evy Flights Anomalous Diffusion

Anomalous Diffusion

DX = 0 ⇒ subdiffusion 0 < DX < ∞ ⇒ normal diffusion DX = ∞ ⇒ superdiffusion DB = σ2/2, B(·) : Brownian Motion E(X 2(t)) = ∞ ⇒ DX = ∞

EVA 2005 Anomalous Diffusion Index for L´ evy Motions

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References L´ evy Flights Anomalous Diffusion

Anomalous Diffusion

DX = 0 ⇒ subdiffusion 0 < DX < ∞ ⇒ normal diffusion DX = ∞ ⇒ superdiffusion DB = σ2/2, B(·) : Brownian Motion E(X 2(t)) = ∞ ⇒ DX = ∞

EVA 2005 Anomalous Diffusion Index for L´ evy Motions

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References L´ evy Flights Anomalous Diffusion

Anomalous Diffusion

DX = 0 ⇒ subdiffusion 0 < DX < ∞ ⇒ normal diffusion DX = ∞ ⇒ superdiffusion DB = σ2/2, B(·) : Brownian Motion E(X 2(t)) = ∞ ⇒ DX = ∞

EVA 2005 Anomalous Diffusion Index for L´ evy Motions

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References L´ evy Flights Anomalous Diffusion

Anomalous Diffusion

DX = 0 ⇒ subdiffusion 0 < DX < ∞ ⇒ normal diffusion DX = ∞ ⇒ superdiffusion DB = σ2/2, B(·) : Brownian Motion E(X 2(t)) = ∞ ⇒ DX = ∞

EVA 2005 Anomalous Diffusion Index for L´ evy Motions

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References L´ evy Flights Anomalous Diffusion

Anomalous Diffusion

DX = 0 ⇒ subdiffusion 0 < DX < ∞ ⇒ normal diffusion DX = ∞ ⇒ superdiffusion DB = σ2/2, B(·) : Brownian Motion E(X 2(t)) = ∞ ⇒ DX = ∞

EVA 2005 Anomalous Diffusion Index for L´ evy Motions

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Examples Results

Diffusion Index : γX

Centered process : P(X(t) ≤ x) = P(X(t) ≥ −x) γ0

X = sup{γ : γ > 0, lim inf t→∞

E|X(t)|1/γ t > 0} γX = inf{γ : 0 < γ ≤ γ0

X, lim sup t→∞

E|X(t)|1/γ t < ∞} lim inf

t→∞

E|X(t)|1/γ t = 0, ∀γ > 0 ⇒ γ0

X = 0

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Examples Results

Diffusion Index : γX

Centered process : P(X(t) ≤ x) = P(X(t) ≥ −x) γ0

X = sup{γ : γ > 0, lim inf t→∞

E|X(t)|1/γ t > 0} γX = inf{γ : 0 < γ ≤ γ0

X, lim sup t→∞

E|X(t)|1/γ t < ∞} lim inf

t→∞

E|X(t)|1/γ t = 0, ∀γ > 0 ⇒ γ0

X = 0

EVA 2005 Anomalous Diffusion Index for L´ evy Motions

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Examples Results

Diffusion Index : γX

Centered process : P(X(t) ≤ x) = P(X(t) ≥ −x) γ0

X = sup{γ : γ > 0, lim inf t→∞

E|X(t)|1/γ t > 0} γX = inf{γ : 0 < γ ≤ γ0

X, lim sup t→∞

E|X(t)|1/γ t < ∞} lim inf

t→∞

E|X(t)|1/γ t = 0, ∀γ > 0 ⇒ γ0

X = 0

EVA 2005 Anomalous Diffusion Index for L´ evy Motions

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Examples Results

Diffusion Index : γX

Centered process : P(X(t) ≤ x) = P(X(t) ≥ −x) γ0

X = sup{γ : γ > 0, lim inf t→∞

E|X(t)|1/γ t > 0} γX = inf{γ : 0 < γ ≤ γ0

X, lim sup t→∞

E|X(t)|1/γ t < ∞} lim inf

t→∞

E|X(t)|1/γ t = 0, ∀γ > 0 ⇒ γ0

X = 0

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Examples Results

Diffusion Index : γX

0 < γX < ∞ : 0 < γ < γX ⇒ lim sup

t→∞

E|X(t)|1/γ t = ∞ γ > γX ⇒ lim inf

t→∞

E|X(t)|1/γ t = 0 lim inf

t→∞

E|X(t)|1/γ t = 0 , ∀γ > 0 ⇒ γX = 0 lim sup

t→∞

E|X(t)|1/γ t = ∞ , ∀γ > 0 ⇒ γX = ∞

EVA 2005 Anomalous Diffusion Index for L´ evy Motions

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Examples Results

Diffusion Index : γX

0 < γX < ∞ : 0 < γ < γX ⇒ lim sup

t→∞

E|X(t)|1/γ t = ∞ γ > γX ⇒ lim inf

t→∞

E|X(t)|1/γ t = 0 lim inf

t→∞

E|X(t)|1/γ t = 0 , ∀γ > 0 ⇒ γX = 0 lim sup

t→∞

E|X(t)|1/γ t = ∞ , ∀γ > 0 ⇒ γX = ∞

EVA 2005 Anomalous Diffusion Index for L´ evy Motions

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Examples Results

Diffusion Index : γX

0 < γX < ∞ : 0 < γ < γX ⇒ lim sup

t→∞

E|X(t)|1/γ t = ∞ γ > γX ⇒ lim inf

t→∞

E|X(t)|1/γ t = 0 lim inf

t→∞

E|X(t)|1/γ t = 0 , ∀γ > 0 ⇒ γX = 0 lim sup

t→∞

E|X(t)|1/γ t = ∞ , ∀γ > 0 ⇒ γX = ∞

EVA 2005 Anomalous Diffusion Index for L´ evy Motions

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Examples Results

Diffusion Index : γX

0 < γX < ∞ : 0 < γ < γX ⇒ lim sup

t→∞

E|X(t)|1/γ t = ∞ γ > γX ⇒ lim inf

t→∞

E|X(t)|1/γ t = 0 lim inf

t→∞

E|X(t)|1/γ t = 0 , ∀γ > 0 ⇒ γX = 0 lim sup

t→∞

E|X(t)|1/γ t = ∞ , ∀γ > 0 ⇒ γX = ∞

EVA 2005 Anomalous Diffusion Index for L´ evy Motions

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Examples Results

Diffusion Index : γX

0 < γX < ∞ : 0 < γ < γX ⇒ lim sup

t→∞

E|X(t)|1/γ t = ∞ γ > γX ⇒ lim inf

t→∞

E|X(t)|1/γ t = 0 lim inf

t→∞

E|X(t)|1/γ t = 0 , ∀γ > 0 ⇒ γX = 0 lim sup

t→∞

E|X(t)|1/γ t = ∞ , ∀γ > 0 ⇒ γX = ∞

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Examples Results

Examples

X(t) zero-mean and finite moments : γX = DX = 0 (subdiffusion) Bσ(t) Brownian motion with variance σ2/2 : σ2

Bσ(x, t) = σ2 ; DBσ = σ2

2 ; γBσ = 1 2 Langevin Equation dV(t) = −µV(t)dt + σdB(t) , V(0) = 0

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Examples Results

Examples

X(t) zero-mean and finite moments : γX = DX = 0 (subdiffusion) Bσ(t) Brownian motion with variance σ2/2 : σ2

Bσ(x, t) = σ2 ; DBσ = σ2

2 ; γBσ = 1 2 Langevin Equation dV(t) = −µV(t)dt + σdB(t) , V(0) = 0

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Examples Results

Examples

X(t) zero-mean and finite moments : γX = DX = 0 (subdiffusion) Bσ(t) Brownian motion with variance σ2/2 : σ2

Bσ(x, t) = σ2 ; DBσ = σ2

2 ; γBσ = 1 2 Langevin Equation dV(t) = −µV(t)dt + σdB(t) , V(0) = 0

EVA 2005 Anomalous Diffusion Index for L´ evy Motions

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Examples Results

Examples

Ornstein-Uhlenbeck : V(t) = σ t e−µ(t−x)dB(s) µV(x, t) = −µx ; σ2

V(x, t) = σ2 (infinitesimal

coefficients/local diffusion) E|V(t)|1/γ = e−µt/γ[ σ2 2µ(e2µt − 1)]1/2γE|B(1)|1/γ lim

t→∞

E|V(t)|1/γ t = 0, ∀γ > 0 ⇒ γV = 0 (subdiffusion : long-range behavior)

EVA 2005 Anomalous Diffusion Index for L´ evy Motions

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Examples Results

Examples

Ornstein-Uhlenbeck : V(t) = σ t e−µ(t−x)dB(s) µV(x, t) = −µx ; σ2

V(x, t) = σ2 (infinitesimal

coefficients/local diffusion) E|V(t)|1/γ = e−µt/γ[ σ2 2µ(e2µt − 1)]1/2γE|B(1)|1/γ lim

t→∞

E|V(t)|1/γ t = 0, ∀γ > 0 ⇒ γV = 0 (subdiffusion : long-range behavior)

EVA 2005 Anomalous Diffusion Index for L´ evy Motions

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Examples Results

Examples

Ornstein-Uhlenbeck : V(t) = σ t e−µ(t−x)dB(s) µV(x, t) = −µx ; σ2

V(x, t) = σ2 (infinitesimal

coefficients/local diffusion) E|V(t)|1/γ = e−µt/γ[ σ2 2µ(e2µt − 1)]1/2γE|B(1)|1/γ lim

t→∞

E|V(t)|1/γ t = 0, ∀γ > 0 ⇒ γV = 0 (subdiffusion : long-range behavior)

EVA 2005 Anomalous Diffusion Index for L´ evy Motions

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Examples Results

Examples

Ornstein-Uhlenbeck : V(t) = σ t e−µ(t−x)dB(s) µV(x, t) = −µx ; σ2

V(x, t) = σ2 (infinitesimal

coefficients/local diffusion) E|V(t)|1/γ = e−µt/γ[ σ2 2µ(e2µt − 1)]1/2γE|B(1)|1/γ lim

t→∞

E|V(t)|1/γ t = 0, ∀γ > 0 ⇒ γV = 0 (subdiffusion : long-range behavior)

EVA 2005 Anomalous Diffusion Index for L´ evy Motions

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Examples Results

Examples

Ornstein-Uhlenbeck : V(t) = σ t e−µ(t−x)dB(s) µV(x, t) = −µx ; σ2

V(x, t) = σ2 (infinitesimal

coefficients/local diffusion) E|V(t)|1/γ = e−µt/γ[ σ2 2µ(e2µt − 1)]1/2γE|B(1)|1/γ lim

t→∞

E|V(t)|1/γ t = 0, ∀γ > 0 ⇒ γV = 0 (subdiffusion : long-range behavior)

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Examples Results

Examples

BH(t) : fractional Brownian motion, 0 < H < 1 cov(BH(t), BH(s)) = σ2 2 [t2H + s2H − |t − s|2H] , σ2 = var(BH(1))

EVA 2005 Anomalous Diffusion Index for L´ evy Motions

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Examples Results

Examples

DBH = lim

t→∞

t2Hσ2 2t =                      H < 1 2 σ2 2 H = 1 2 ∞ H > 1 2. Diffusion Index : γBH = H

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Examples Results

Examples

DBH = lim

t→∞

t2Hσ2 2t =                      H < 1 2 σ2 2 H = 1 2 ∞ H > 1 2. Diffusion Index : γBH = H

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Examples Results

Results :

Lemma 1 Let X(·) be a zero-mean stochastic process with stationary and independent increments. Then, if 0 < σ2

X(x, t) < ∞ we have

γX = 1/2. Lemma 2 If the diffusion constant 0 < DX < ∞ then γX = 1/2. Lemma 3 Let X(·) and Y(·) be centered stochastic processes. If a = 0 and b > 0 are constants, then γ|X| = γX , γaX = γX , γ|X|b = bγX and γX+Y ≤ max{γX, γY}.

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Examples Results

Results :

Lemma 1 Let X(·) be a zero-mean stochastic process with stationary and independent increments. Then, if 0 < σ2

X(x, t) < ∞ we have

γX = 1/2. Lemma 2 If the diffusion constant 0 < DX < ∞ then γX = 1/2. Lemma 3 Let X(·) and Y(·) be centered stochastic processes. If a = 0 and b > 0 are constants, then γ|X| = γX , γaX = γX , γ|X|b = bγX and γX+Y ≤ max{γX, γY}.

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Examples Results

Results :

Lemma 1 Let X(·) be a zero-mean stochastic process with stationary and independent increments. Then, if 0 < σ2

X(x, t) < ∞ we have

γX = 1/2. Lemma 2 If the diffusion constant 0 < DX < ∞ then γX = 1/2. Lemma 3 Let X(·) and Y(·) be centered stochastic processes. If a = 0 and b > 0 are constants, then γ|X| = γX , γaX = γX , γ|X|b = bγX and γX+Y ≤ max{γX, γY}.

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Examples Results

Lα,σ(t) : symmetric L´

evy stable process Lemma 4 For the L´ evy stable process L = Lα,σ, if α = 2, then σ2

L(x, t) = σ2, DL = σ2

2 and γL = 1 2. If 0 < α < 2 (superdiffusion), we have σ2

L(x, t) = DL = ∞

and γL = 1 α > 1 2. Moreover, for ϕt(x) = P(L(t) > x), we have ϕt(·) is slowly varying if α = 2 and ϕt(·) is (− 1 γL ) regularly varying if 0 < α < 2

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Examples Results

Lα,σ(t) : symmetric L´

evy stable process Lemma 4 For the L´ evy stable process L = Lα,σ, if α = 2, then σ2

L(x, t) = σ2, DL = σ2

2 and γL = 1 2. If 0 < α < 2 (superdiffusion), we have σ2

L(x, t) = DL = ∞

and γL = 1 α > 1 2. Moreover, for ϕt(x) = P(L(t) > x), we have ϕt(·) is slowly varying if α = 2 and ϕt(·) is (− 1 γL ) regularly varying if 0 < α < 2

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Examples Results

Lα,σ(t) : symmetric L´

evy stable process Lemma 4 For the L´ evy stable process L = Lα,σ, if α = 2, then σ2

L(x, t) = σ2, DL = σ2

2 and γL = 1 2. If 0 < α < 2 (superdiffusion), we have σ2

L(x, t) = DL = ∞

and γL = 1 α > 1 2. Moreover, for ϕt(x) = P(L(t) > x), we have ϕt(·) is slowly varying if α = 2 and ϕt(·) is (− 1 γL ) regularly varying if 0 < α < 2

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Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Brazil : real vs US dollar Mexico : peso vs US dollar Switzerland : swiss franc vs US dollar

Numerical Illustrations

Daily Foreign Exchange Rates ( http://www.federalreserve.gov ) January/2000 to January/2005 : Brazil, Switzerland, Mexico

γL = 1

α Table1

Country ˆ αn ˆ σn value(k) value(m) value(ǫ) Brazil(Real) 1.62424198 0.008866886626 180 15 0.005 Switzerland(Franc) 1.4737571 .49918254e-2 180 6 0.002 Mexico(Peso) 1.2723789 0.20784845e-1 200 5 0.01 EVA 2005 Anomalous Diffusion Index for L´ evy Motions

SLIDE 47

Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Brazil : real vs US dollar Mexico : peso vs US dollar Switzerland : swiss franc vs US dollar

Brazil : real vs US dollar

EVA 2005 Anomalous Diffusion Index for L´ evy Motions

SLIDE 48

Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Brazil : real vs US dollar Mexico : peso vs US dollar Switzerland : swiss franc vs US dollar

Brazil : real vs US dollar

EVA 2005 Anomalous Diffusion Index for L´ evy Motions

SLIDE 49

Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Brazil : real vs US dollar Mexico : peso vs US dollar Switzerland : swiss franc vs US dollar

Mexico : peso vs US dollar

EVA 2005 Anomalous Diffusion Index for L´ evy Motions

SLIDE 50

Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References Brazil : real vs US dollar Mexico : peso vs US dollar Switzerland : swiss franc vs US dollar

Switzerland : swiss franc vs US dollar

EVA 2005 Anomalous Diffusion Index for L´ evy Motions

SLIDE 51

Diffusion Process Critical Plenomena Diffusion Index : γX Numerical Illustrations References

References

Dorea, C.C.Y. and Medino, A.V. (2005) - Anomalous diffusion index for L´ evy motions, Journal of Statistical Physics, (to appear). Costa, I.V.L.; Morgado, R.; Lima, M.V.B.T. and Oliveira, F .A. (2003) - The Fluctuation - Dissipation Theorem fails for fast superdiffusion, Europhysics Letters, vol. 63, 173-179. Metzler, R. and Klafter, J. (2000) - The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Physics Reports, vol. 339, 1-77.

EVA 2005 Anomalous Diffusion Index for L´ evy Motions