Stochastic processes for symmetric space-time fractional diffusion Gianni PAGNINI IKERBASQUE Research Fellow BCAM - Basque Center for Applied Mathematics Bilbao, Basque Country - Spain gpagnini@bcamath.org Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University
Main Goal of the Research Classical/normal diffusion is described by a Gaussian process. - Diffusion/Brownian motion Is it possible to describe anomalous diffusion/fractional diffusion with a stochastic process that is still based on a Gaussian process? - Time-fractional diffusion/grey Brownian motion (Schneider 199, 1992) - Erdélyi Kober fractional diffusion/generalized grey Brownian motion (Mura 28, Pagnini 212) - Space-time fractional diffusion/present stochastic process Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University
The Space-Time Fractional Diffusion Equation xdθ α K α,β θ (x; t) = td β Kα,β θ (x; t), K α,β θ (x; ) = δ(x), (1) < α 2, θ min{α, 2 α}, (2a) < β 1 or 1 < β α 2. (2b) xdθ α : Riesz Feller space-fractional derivative F { x D α θ f (x); κ} = κ α e i(sign κ)θπ/2 f (κ). (3) td β : Caputo time-fractional derivative { } m 1 L td β f (t); s = s β f (s) s β 1 k f (j) ( + ), (4) with m 1 < β m and m N. j= Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University
The Green Function Kα,β θ (x; t) Self-similarly law Symmetry relation K θ α,β (x; t) = t β/α K θ α,β K θ α,β which allows the restriction to x. Mellin Barnes integral representation Kα,β θ 1 (x; t) = αx 1 c+i 2πi c i ( x t β/α ). (5) θ ( x; t) = Kα,β (x; t), (6) Γ( q α )Γ(1 q α )Γ(1 q) ( x ) q dq, (7) Γ(1 β α q)γ(ρ q)γ(1 ρ q) t β/α where ρ = (α θ)/(2 α) and c is a suitable real constant. Mainardi, Luchko, Pagnini Fract. Calc. Appl. Anal. 21 Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University
Special Cases: x > K2,1 1 ( (x; t) = e x 2 x ) /(4t) = G(x; t) = t 1/2 G 4πt t 1/2, (8) K θ α,1 (x; t) = Lθ α(x; t) = t 1/α L θ α ( x t 1/α ), (9) K 2,β (x; t) = 1 2 M β/2(x; t) = 1 2 t β/2 M β/2 ( x t β/2 ), (1) K θ α,α(x; t) = t 1 π (x/t) α 1 sin[ π 2 (α θ)] 1 + 2(x/t) α cos[ π, (11) 2 (α θ)] + (x/t)2α K2,2 (x; t) = 1 δ(x t). (12) 2 Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University
Integral Representation Formulae for Kα,β θ (x; t) If x > then K θ α,β (x; t) = L θ α(x; τ)l β β t (t; τ) dτ, < β 1, (13) τβ Kα,β θ (x; t) = L θ α(x; τ) M β (τ; t) dτ, < β 1, (14) Kα,β θ (x; t) = Kα,α(x; θ τ) M β/α (τ; t) dτ, < β/α 1. (15) Mainardi, Luchko, Pagnini Fract. Calc. Appl. Anal. 21 Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University
Supplementary Results From formulae (13) and (14) it follows that ( ) 1 t L β τ 1/β β τ 1/β = τ β ( τ ) t 1+β M β t β, < β 1, τ, t >. (16) From formulae (8) and (9) K2,1 (x; t) = G(x; t) = L 2 (x; t). (17) From formulae (8) and (1) K 2,1 (x; t) = G(x; t) = 1 2 M 1/2(x; t). (18) Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University
Supplementary Results L θ α(x; t) = L ω η (x; ξ)l ν ν (ξ; t) dξ, α = ην, θ = ων, (19) < α 2, θ min{α, 2 α}, < η 2, ω min{η, 2 η}, < ν 1. In particular it holds L α(x; t) = = L 2 (x; ξ)l α/2 α/2 (ξ; t) dξ (2) G(x; ξ)l α/2 α/2 (ξ; t) dξ. (21) Mainardi, Pagnini, Gorenflo Fract. Calc. Appl. Anal. 23 Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University
Supplementary Results M ν (x; t) = M η (x; ξ)m β (ξ; t) dξ, ν = ηβ, (22) < ν, η, β 1. In particular it holds M β/2 (x; t) = 2 = 2 M 1/2 (x; ξ)m β (ξ; t) dξ (23) G(x; ξ)m β (ξ; t) dξ. (24) Mainardi, Pagnini, Gorenflo Fract. Calc. Appl. Anal. 23 Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University
New Integral Representation Formula for Kα,β θ (x; t) Consider formula (14), i.e. Kα,β θ (x; t) = L θ α(x; τ)m β (τ; t) dτ, then and by using (19) it follows { } Kα,β θ (x; t) = L ω η (x; ξ)l ν ν (ξ; t) dξ M β (τ, t) dτ = = { L ω η (x; ξ) } L ν ν (ξ; t)m β (τ; t) dτ dξ L ω η (x; ξ) K ν ν,β (ξ; t) dξ, α = ην, θ = ων, < α 2, θ min{α, 2 α}, < β 1, < η 2, ω min{η, 2 η}, < ν 1. Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University
New Integral Representation Formula for Kα,β θ (x; t) K θ α,β (x; t) = L ω η (x; ξ) K ν ν,β (ξ; t) dξ, (25) < x < +, α = ην, θ = ων, < α 2, θ min{α, 2 α}, < β 1, < η 2, ω min{η, 2 η}, < ν 1. In the spatial symmetric case, i.e. η = 2 and ω = such that L 2 G, hence ν = α/2 and θ = and formula (25) gives K α,β (x; t) = G(x; ξ) K α/2 α/2,β (ξ; t) dξ. (26) < x < +, < α 2, < β 1. Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University
Special Cases: α = 2 and < β 1 From (14) it results that K 1 1,β (ξ; t) = = L 1 1 (ξ; τ) M β(τ; t) dτ δ(ξ τ) M β (τ; t) dτ = M β (ξ; t), (27) finally by using (24) K 2,β (x; t) = = G(x; ξ) K 1 1,β (ξ; t) dξ (28) G(x; ξ) M β dξ = 1 2 M β/2(x; t). (29) Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University
Special Cases: < α 2 and β = 1 From (14) it results that K α/2 α/2,1 (ξ; t) = = L α/2 α/2 (ξ; τ) M 1(τ; t) dτ L α/2 α/2 (ξ; τ) δ(τ t) dτ = L α/2(ξ; t), (3) α/2 finally by using (21) K α,1 (x; t) = = G(x; ξ) K α/2 α/2,1 (ξ; t) dξ (31) G(x; ξ) L α/2 α/2 (ξ; t) dξ = L α(x; t). (32) Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University
Definitions in Stochastic Process Theory Stationary stochastic processes: A stochastic process X(t) whose one-time statistical characteristics do not change in the course of time t, i.e. they are invariant relative to the time shifing t t + τ for any fixed value of τ, and two-time statistics (e.g. autocorrelation) depend solely on the elapsed time τ, X(t) n = X n (t + τ) = C n, X(t)X(t + τ) = R(τ). (33) Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University
Definitions in Stochastic Process Theory Stochastic processes with stationary increments: A stochastic process X(t) such that the statistical characteristics of its increments X(t) = X(t) X(t + τ) do not vary in the course of time t, i.e. they are invariant relative to the time shifting t t + s for any fixed value of s, X(t) =, ( X(t)) 2 = 2 [C 2 R(τ)]. (34) Because R(τ = ) = C 2, when R(τ ) C 2 τ 2H it holds ( X(t)) 2 = 2 [C 2 C 2 + τ 2H ] = 2 τ 2H. (35) Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University
Definitions in Stochastic Process Theory Self-similar stochastic process: A stochastic process X(t) whose statistics are the same at different scales of time or space, i.e. X(a t) and a H X(t) have equal statistical moments X n (a t) (a t) nh = a nh t nh, (36) [a H X(t)] n a nh X n (t) = a nh t nh. (37) Hurst exponent H: The half exponent of the power law governing the rate of changes of a random function by [X(t) X()] 2 t 2H. (38) Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University
H-sssi Stochastic Processes H-sssi: Hurst self-similar with stationary increments processes Example: The fractional Brownian motion, which is a continuos-time Gaussian process G 2H (t) without independent increments and the following correlation function G 2H (t)g 2H (s) = 1 2 [ t 2H + s 2H t s 2H]. (39) Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University
Product of Random Variables Let Z 1 and Z 2 be two real independent random variables whose PDFs are p 1 (z 1 ) and p 2 (z 2 ), respectively, with z 1 R and z 2 R +. Then, the joint PDF is p(z 1, z 2 ) = p 1 (z 1 )p 2 (z 2 ). Let Z = Z 1 Z γ 2 so that z = z 1z γ 2, then, carrying out the variable transformations z 1 = z/λ γ and z 2 = λ, it follows that p(z, λ) dz dλ = p 1 (z/λ γ )p 2 (λ) J dz dλ, where J = 1/λ γ is the Jacobian of the transformation. Integrating in dλ, the PDF of Z emerges to be p(z) = p 1 ( z λ γ ) p 2 (λ) dλ λ γ. (4) Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University
Product of Random Variables Hence by applying the changes of variable z = xt γω and λ = τt Ω, integral formula (4) becomes By setting ( x ) t γω p t γω = p 1 G, τ γ p 1 ( x τ γ ) t Ω p 2 ( τ t Ω ) dτ. (41) p 2 K α/2 α/2,β, (42) γ = 1/2, Ω = 2β/α, (43) formula (41) turns out to be identical to (26), i.e. p(x; t) = G(x; τ) K α/2 α/2,β (τ; t) dτ, hence p(x; t) Kα,β (x; t). (44) Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University
Product of Random Variables In terms of random variables it follows that Z = X t β/α and Z = Z 1 Z 1/2 2, (45) hence it holds X = Z t β/α = Z 1 t β/α Z 1/2 2 = G 2β/α (t) Λ α/2,β. (46) Since the random variable Z 1 is Gaussian, i.e., p 1 G, the stochastic process G 2β/α (t) = Z 1 t β/α is a standard fbm with Hurst exponent β/α < 1. The random variable Λ α/2,β = Z 2 emerges to be distributed according to p 2 K α/2 α/2,β. Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University
H-sssi Processes Following the same constructive approach adopted by Mura (PhD 28) to built up the generalized grey Brownian motion (Mura, Pagnini J. Phys. A 28), the following class of H-sssi processes is established. Let X α,β (t), t, be an H-sssi defined as X α,β (t) = d Λ α/2,β G 2β/α (t), < β 1, < β < α 2, (47) where = d denotes the equality of the finite-dimensional distribution, the stochastic process G 2β/α (t) is a standard fbm with Hurst exponent H = β/α < 1 and Λ α/2,β is an independent non-negative random variable with PDF K α/2 α/2,β (λ), λ, then the marginal PDF of X α,β (t) is Kα,β (x; t). Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University
H-sssi Processes Tthe finite-dimensional distribution of X α,β (t) is obtained from (4) according to f α,β (x 1, x 2,..., x n ; γ α,β ) = n 1 (2π) 2 det γα,β 1 ( λ n/2 G zn ) λ 1/2 K α/2 α/2,β (λ) dλ, (48) where z n is the n-dimensional particle position vector 1/2 n z n = x i γ 1 α,β (t i, t j ) x j, i,j=1 and γ α,β (t i, t j ) is the covariance matrix γ α,β (t i, t j ) = 1 2 (t2β/α i + t 2β/α j t i t j 2β/α ), i, j = 1,..., n. Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University
Stochastic Solution of Space-Time Fractional Diffusion For the one-point case, i.e., n = 1, formula (48) reduces to ( ) 1 f α,β (x; t) = λ 1/2 G x t β/α λ 1/2 K α/2 α/2,β (λ) dλ = K α,β (x t β/α ), (49) or, after the change of variable λ = τ t 2β/α, 1 ( x ) τ 1/2 G τ 1/2 ( K α/2 τ ) ( x ) α/2,β t β/α dτ = t β/α Kα,β t β/α. (5) This means that the marginal PDF of the H-sssi process X α,β (t) is indeed the solution of the symmetric space-time fractional diffusion equation (1). Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University
Stochastic Process Generation From (14) it follows that K α/2 α/2,β (ξ; t) = L α/2 α/2 (ξ; τ) M β(τ; t) dτ, < β 1, (51) and by using the self-similarity properties and the changes of variable ξ = t 2β/α λ and τ = t β y it holds K α/2 α/2,β (λ) = ( ) L α/2 λ α/2 y 2/α M β (y) dy, < β 1. (52) y 2/α Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University
Stochastic Process Generation Integral (52) suggests to obtain Λ α/2,β again by means of the product of two independent random variables, i.e. Λ α/2,β = Λ 1 Λ 2/α 2 = L ext α/2 M2/α β, (53) where Λ 1 = L ext α/2 and Λ 2 = M β are distributed according to the extremal stable density L α/2 α/2 (λ 1) and M β (λ 2 ), respectively, so that λ = λ 1 λ 2/α 2. Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University
Stochastic Process Generation Moreover, from (16) and setting t = 1, the random variable M β can be determined by an extremal stable random variable according to M β = [ L ext ] β β, (54) so that the random variable Λ α/2,β is computed by the product Λ α/2,β = L ext α/2 [L ] ext 2β/α β. (55) Finally, the desired H-sssi processes are established as follows X α,β (t) = L ext α/2 [L ] ext β/α β G2β/α (t). (56) Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University
Numerical Generation (by P. Paradisi) Computer generation of extremal stable random variables of order < µ < 1 is obtained by using the method by Chambers, Mallows and Stuck L ext µ = sin[µ(r 1 + π/2)] (cos r 1 ) 1/µ { } cos[r1 µ(r 1 + π/2)] (1 µ)/µ, (57) ln r 2 where r 1 and r 2 are random variables uniformly distributed in ( π/2, π/2) and (, 1), respectively. Chambers, Mallows, Stuck J. Amer. Statist. Assoc. 1976 Weron Statist. Probab. Lett. 1996 Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University
Numerical Generation (by P. Paradisi) The Hosking direct method is applied for generating the fbm G 2H (t), < H < 1. In particular, first the so-called fractional Gaussian noise Y 2H is generated over the set of integer numbers with autocorrelation function Y 2H (k)y 2H (k + n) = 1 [ n 1 2H n 2H + n + 1 2H]. (58) 2 Finally, the fbm is then generated as a sum of stationary increments, i.e. Y 2H (n) = G 2H (n + 1) G 2H (n) G 2H (n + 1) = G 2H (n) + Y 2H (n). (59) Hosking Water Resour. Res. 1984 Dieker PhD Thesis Univ. of Twente, The Netherlands, 24 Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University
Numerical Set-up (by P. Paradisi) For a given set of parameter values (α,β), 1 4 trajectories are generated and the motion tracked for 1 3 time steps, which is stated equal to 1 following formula (59). Changing the time scale requires changing the time step, and the associated trajectories can be simply derived without any further numerical simulations by exploiting the self-similar property. Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University
Simulations (by P. Paradisi) =1 ; =.25 4 Sample Paths PDF, t=1 3 2 1 X -1-2 -3-4 2 4 t 6 8 1.5.1.15 =1 ; =1/3 4 Sample Paths PDF, t=1 3 2 1 X -1-2 -3-4 2 4 t 6 8 1.5.1 Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University
Simulations (by P. Paradisi) 1 Sample Paths =1 ; =.5 PDF, t=1 5 X -5-1 2 4 6 t 8 1.1.2.3.4.5 =1 ; =.9 3 Sample Paths PDF, t=1 2 1 X -1-2 -3 2 4 t 6 8 1.2.4.6 Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University
Simulations (by P. Paradisi) =.5 ; =.45 4 Sample Paths PDF, t=1 3 2 1 X -1-2 -3-4 2 4 6 t 8 1.1.3.5 X 3 2 1-1 -2 Sample Paths =1.5 ; =.5 PDF, t=1-3 2 4 t 6 8 1.2.6 Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University
Simulations (by P. Paradisi) =2 ; =.5 2 Sample Paths PDF, t=1 15 1 5 X -5-1 -15-2 2 4 t 6 8 1.2.6.1.12 X 6 4 2-2 -4 Sample Paths =2 ; =1 PDF, t=1-6 2 4 t 6 8 1.5.1.15.2 Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University
Simulations (by P. Paradisi).1 PDF, t=1, =.5, =.45 Numerical 3/ x 1.5.1 1e-5 1e-6 2 4 6 8 1 X Power-law decay of PDF tail (large x) according to 1/ x α+1. Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University
References Mainardi F., Luchko Yu., Pagnini G. 21 The fundamental solution of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 4, 153 192. Mainardi F., Pagnini G., Gorenflo R. 23 Mellin transform and subordination laws in fractional diffusion processes. Fract. Calc. Appl. Anal. 6, 441-459. A. Mura 28 Non-Markovian Stochastic Processes and Their Applications: From Anomalous Diffusion to Time Series Analysis. Ph.D. Thesis, University of Bologna. A. Mura, G. Pagnini 28 Characterizations and simulations of a class of stochastic processes to model anomalous diffusion. J. Phys. A: Math. Theor. 41, 2853. G. Pagnini, A. Mura, F. Mainardi 213 Two-particle anomalous diffusion: Probability density functions and self-similar stochastic processes. Phil. Trans. R. Soc. A 371, 212154. Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University
Acknowledgements Thank you for your attention! This research is supported by the Basque Government through the BERC 214 217 program and by the Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa accreditation SEV-213-323. Gianni PAGNINI - Workshop on particle transport; with emphasis on stochastics. 6 7 November, 214, Aarhus University