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Diffusion Process Critical Plenomena Diffusion Index : X 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 lia

Diffusion Process Critical Plenomena Diffusion Index : γ X 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
Diffusion Process Critical Plenomena Infinitesimal Coefficients Diffusion Index : γ X Scaled Random Walks Numerical Illustrations References Diffusion Process : Infinitesimal Coefficients 1 lim hP ( | X ( t + h ) − X ( t ) | > ǫ | X ( t ) = x ) = 0 h ↓ 0 1 lim hE { X ( t + h ) − X ( t ) | X ( t ) = x } = µ X ( x , t ) h ↓ 0 1 � ( X ( t + h ) − X ( t )) 2 | X ( t ) = x � = σ 2 lim hE X ( x , t ) h ↓ 0 µ ( · ) = 0 , σ 2 ( · ) = 1 : Brownian Motion EVA 2005 Anomalous Diffusion Index for L´ evy Motions
Diffusion Process Critical Plenomena Infinitesimal Coefficients Diffusion Index : γ X Scaled Random Walks Numerical Illustrations References Diffusion Process : Infinitesimal Coefficients 1 lim hP ( | X ( t + h ) − X ( t ) | > ǫ | X ( t ) = x ) = 0 h ↓ 0 1 lim hE { X ( t + h ) − X ( t ) | X ( t ) = x } = µ X ( x , t ) h ↓ 0 1 � ( X ( t + h ) − X ( t )) 2 | X ( t ) = x � = σ 2 lim hE X ( x , t ) h ↓ 0 µ ( · ) = 0 , σ 2 ( · ) = 1 : Brownian Motion EVA 2005 Anomalous Diffusion Index for L´ evy Motions
Diffusion Process Critical Plenomena Infinitesimal Coefficients Diffusion Index : γ X Scaled Random Walks Numerical Illustrations References Diffusion Process : Infinitesimal Coefficients 1 lim hP ( | X ( t + h ) − X ( t ) | > ǫ | X ( t ) = x ) = 0 h ↓ 0 1 lim hE { X ( t + h ) − X ( t ) | X ( t ) = x } = µ X ( x , t ) h ↓ 0 1 � ( X ( t + h ) − X ( t )) 2 | X ( t ) = x � = σ 2 lim hE X ( x , t ) h ↓ 0 µ ( · ) = 0 , σ 2 ( · ) = 1 : Brownian Motion EVA 2005 Anomalous Diffusion Index for L´ evy Motions
Diffusion Process Critical Plenomena Infinitesimal Coefficients Diffusion Index : γ X Scaled Random Walks Numerical Illustrations References Diffusion Process : Infinitesimal Coefficients 1 lim hP ( | X ( t + h ) − X ( t ) | > ǫ | X ( t ) = x ) = 0 h ↓ 0 1 lim hE { X ( t + h ) − X ( t ) | X ( t ) = x } = µ X ( x , t ) h ↓ 0 1 � ( X ( t + h ) − X ( t )) 2 | X ( t ) = x � = σ 2 lim hE X ( x , t ) h ↓ 0 µ ( · ) = 0 , σ 2 ( · ) = 1 : Brownian Motion EVA 2005 Anomalous Diffusion Index for L´ evy Motions
Diffusion Process Critical Plenomena Infinitesimal Coefficients Diffusion Index : γ X Scaled Random Walks Numerical Illustrations References Scaled Random Walks S n = ξ 1 + ξ 2 + . . . + ξ n j ) = σ 2 > 0 E ( ξ j ) = 0 , E ( ξ 2 1 X ( n ) ( t ) = σ √ n { S [ nt ] + ( nt − [ nt ]) ξ [ nt ]+ 1 } → B ( t ) X ( n ) ( t ) = a n ( S [ k n t ] − b n ) → X ( t ) EVA 2005 Anomalous Diffusion Index for L´ evy Motions
Diffusion Process Critical Plenomena Infinitesimal Coefficients Diffusion Index : γ X Scaled Random Walks Numerical Illustrations References Scaled Random Walks S n = ξ 1 + ξ 2 + . . . + ξ n j ) = σ 2 > 0 E ( ξ j ) = 0 , E ( ξ 2 1 X ( n ) ( t ) = σ √ n { S [ nt ] + ( nt − [ nt ]) ξ [ nt ]+ 1 } → B ( t ) X ( n ) ( t ) = a n ( S [ k n t ] − b n ) → X ( t ) EVA 2005 Anomalous Diffusion Index for L´ evy Motions
Diffusion Process Critical Plenomena Infinitesimal Coefficients Diffusion Index : γ X Scaled Random Walks Numerical Illustrations References Scaled Random Walks S n = ξ 1 + ξ 2 + . . . + ξ n j ) = σ 2 > 0 E ( ξ j ) = 0 , E ( ξ 2 1 X ( n ) ( t ) = σ √ n { S [ nt ] + ( nt − [ nt ]) ξ [ nt ]+ 1 } → B ( t ) X ( n ) ( t ) = a n ( S [ k n t ] − b n ) → X ( t ) EVA 2005 Anomalous Diffusion Index for L´ evy Motions
Diffusion Process Critical Plenomena Infinitesimal Coefficients Diffusion Index : γ X Scaled Random Walks Numerical Illustrations References Scaled Random Walks S n = ξ 1 + ξ 2 + . . . + ξ n j ) = σ 2 > 0 E ( ξ j ) = 0 , E ( ξ 2 1 X ( n ) ( t ) = σ √ n { S [ nt ] + ( nt − [ nt ]) ξ [ nt ]+ 1 } → B ( t ) X ( n ) ( t ) = a n ( S [ k n t ] − b n ) → X ( t ) EVA 2005 Anomalous Diffusion Index for L´ evy Motions
Diffusion Process Critical Plenomena L´ evy Flights Diffusion Index : γ X Anomalous Diffusion Numerical Illustrations References Critical Plenomena / L´ evy Flights Anomalous diffusive regime : X ( t ) non-Gaussian n − 1 E ( S 2 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),.. E ( X 2 ( t )) D X = lim (Diffusion Constant) 2 t t →∞ EVA 2005 Anomalous Diffusion Index for L´ evy Motions
Diffusion Process Critical Plenomena L´ evy Flights Diffusion Index : γ X Anomalous Diffusion Numerical Illustrations References Critical Plenomena / L´ evy Flights Anomalous diffusive regime : X ( t ) non-Gaussian n − 1 E ( S 2 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),.. E ( X 2 ( t )) D X = lim (Diffusion Constant) 2 t t →∞ EVA 2005 Anomalous Diffusion Index for L´ evy Motions
Diffusion Process Critical Plenomena L´ evy Flights Diffusion Index : γ X Anomalous Diffusion Numerical Illustrations References Critical Plenomena / L´ evy Flights Anomalous diffusive regime : X ( t ) non-Gaussian n − 1 E ( S 2 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),.. E ( X 2 ( t )) D X = lim (Diffusion Constant) 2 t t →∞ EVA 2005 Anomalous Diffusion Index for L´ evy Motions
Diffusion Process Critical Plenomena L´ evy Flights Diffusion Index : γ X Anomalous Diffusion Numerical Illustrations References Critical Plenomena / L´ evy Flights Anomalous diffusive regime : X ( t ) non-Gaussian n − 1 E ( S 2 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),.. E ( X 2 ( t )) D X = lim (Diffusion Constant) 2 t t →∞ EVA 2005 Anomalous Diffusion Index for L´ evy Motions
Diffusion Process Critical Plenomena L´ evy Flights Diffusion Index : γ X Anomalous Diffusion Numerical Illustrations References Critical Plenomena / L´ evy Flights Anomalous diffusive regime : X ( t ) non-Gaussian n − 1 E ( S 2 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),.. E ( X 2 ( t )) D X = lim (Diffusion Constant) 2 t t →∞ EVA 2005 Anomalous Diffusion Index for L´ evy Motions
Diffusion Process Critical Plenomena L´ evy Flights Diffusion Index : γ X Anomalous Diffusion Numerical Illustrations References Anomalous Diffusion D X = 0 ⇒ subdiffusion 0 < D X < ∞ ⇒ normal diffusion D X = ∞ ⇒ superdiffusion D B = σ 2 / 2 , B ( · ) : Brownian Motion E ( X 2 ( t )) = ∞ ⇒ D X = ∞ EVA 2005 Anomalous Diffusion Index for L´ evy Motions
Diffusion Process Critical Plenomena L´ evy Flights Diffusion Index : γ X Anomalous Diffusion Numerical Illustrations References Anomalous Diffusion D X = 0 ⇒ subdiffusion 0 < D X < ∞ ⇒ normal diffusion D X = ∞ ⇒ superdiffusion D B = σ 2 / 2 , B ( · ) : Brownian Motion E ( X 2 ( t )) = ∞ ⇒ D X = ∞ EVA 2005 Anomalous Diffusion Index for L´ evy Motions
Diffusion Process Critical Plenomena L´ evy Flights Diffusion Index : γ X Anomalous Diffusion Numerical Illustrations References Anomalous Diffusion D X = 0 ⇒ subdiffusion 0 < D X < ∞ ⇒ normal diffusion D X = ∞ ⇒ superdiffusion D B = σ 2 / 2 , B ( · ) : Brownian Motion E ( X 2 ( t )) = ∞ ⇒ D X = ∞ EVA 2005 Anomalous Diffusion Index for L´ evy Motions
Diffusion Process Critical Plenomena L´ evy Flights Diffusion Index : γ X Anomalous Diffusion Numerical Illustrations References Anomalous Diffusion D X = 0 ⇒ subdiffusion 0 < D X < ∞ ⇒ normal diffusion D X = ∞ ⇒ superdiffusion D B = σ 2 / 2 , B ( · ) : Brownian Motion E ( X 2 ( t )) = ∞ ⇒ D X = ∞ EVA 2005 Anomalous Diffusion Index for L´ evy Motions
Diffusion Process Critical Plenomena L´ evy Flights Diffusion Index : γ X Anomalous Diffusion Numerical Illustrations References Anomalous Diffusion D X = 0 ⇒ subdiffusion 0 < D X < ∞ ⇒ normal diffusion D X = ∞ ⇒ superdiffusion D B = σ 2 / 2 , B ( · ) : Brownian Motion E ( X 2 ( t )) = ∞ ⇒ D X = ∞ EVA 2005 Anomalous Diffusion Index for L´ evy Motions
Diffusion Process Critical Plenomena Examples Diffusion Index : γ X Results Numerical Illustrations References Diffusion Index : γ X Centered process : P ( X ( t ) ≤ x ) = P ( X ( t ) ≥ − x ) E | X ( t ) | 1 /γ γ 0 X = sup { γ : γ > 0 , lim inf > 0 } t t →∞ E | X ( t ) | 1 /γ γ X = inf { γ : 0 < γ ≤ γ 0 X , lim sup < ∞} t t →∞ E | X ( t ) | 1 /γ γ 0 lim inf = 0 , ∀ γ > 0 ⇒ X = 0 t t →∞ EVA 2005 Anomalous Diffusion Index for L´ evy Motions

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