CTRW Limits: Governing Equations and Fractal Dimensions

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1 CTRW Limits: Governing Equations and Fractal Dimensions Erkan Nane DEPARTMENT OF MATHEMATICS AND STATISTICS AUBURN UNIVERSITY August 19-23, 2013 Joint work with Z-Q. Chen, M. D Ovidio, M.M. Meerschaert, P. Vellaisamy, Y. Xiao

2 Continuous time random walks R d X 2 X 3 X 1 0 J 1 J 2 J 3 J 4 J 5 X 4 X 5 t The CTRW is a random walk with jumps X n separated by random waiting times J n. The random vectors (X n, J n) are i.i.d.

3 Waiting time process J n s are nonnegative iid. T n = J 1 + J J n is the time of the nth jump. N(t) = max{n 0 : T n t} is the number of jumps by time t > 0. Suppose P(J n > t) Ct β for 0 < β < 1. Scaling limit c 1/β T [ct] = D(t) is a β-stable subordinator. Since {T n t} = {N(t) n} c β N(ct) = E(t) = inf{u > 0 : D(u) > t}. The self-similar limit E(ct) d = c β E(t) is non-markovian.

4 Continuous time random walks (CTRW) S(n) = X X n particle location after n jumps has scaling limit c 1/2 S([ct]) = B(t) a Brownian motion. Number of jumps has scaling limit c β N(ct) = E(t). CTRW is a random walk time-changed by (a renewal process) N(t) S(N(t)) = X 1 + X X N(t). S(N(t)) is particle location at time t > 0. CTRW scaling limit is a time-changed process: c β/2 S(N(ct)) = (c β ) 1/2 S(c β c β N(ct)) (c β ) 1/2 S(c β E(t)) = B(E(t)). The self-similar limit B(E(ct)) d = c β/2 B(E(t)) is non-markovian.

5 X Figure : Typical sample path of the time-changed process B(E(t)). Here B(t) is a Brownian motion and E(t) is the inverse of a β = 0.8-stable subordinator. Graph has dimension 1 + β/2 = The limit process retains long resting times. t

6 CTRW with serial dependence Particle jumps X n = j=0 c jz n j with Z n IID. Short range dependence: n=1 E(X nx 0 ) < = the usual limit and PDE. Long range dependence: If Z n has light tails: time-changed fractional Brownian motion limit B H (E(t)). Hahn, Kobayashi and Umarov (2010) established a governing equation. For heavy tails: time-changed linear fractional stable motion L α,h (E(t)). Open problems: Governing equations, dependent waiting times. Meerschaert, Nane and Xiao (2009).

7 Hausdorff dimension For any α > 0, the α-dimensional Hausdorff measure of F R d is defined by { s α -m(f ) = lim inf (2r i ) α : F ϵ 0 i i=1 } B(x i, r i ), r i < ϵ, (1) where B(x, r) denotes the open ball of radius r centered at x. It is well-known that s α -m is a metric outer measure and every Borel set in R d is s α -m measurable. The Hausdorff dimension of F is defined by dim H F = inf { α > 0 : s α -m(f ) = 0 } = sup { α > 0 : s α -m(f ) = }.

8 Hausdorff dimension of image Let Z = {Z(t) = Y (E(t)), t 0} be the time-changed process with values in R d where the processes Y and E independent and E(t) is a nondecreasing continuous process. If E(1) > 0 a.s. and there exist a constant c 1 such that for all constants 0 < a < then almost surely dim H Y ([0, a]) = c 1, a.s. (2) dim H Z([0, 1]) = c 1. (3) Applying this result to the space-time process x (x, Y (x)): If there exist constants c 2 such that for all constants 0 < a <, dim H { GrY ([0, a]) = c 2 a.s., then } dim H (E(t), Y (E(t))) : t [0, 1] = dimh GrY ([0, 1]), a.s.

9 Hausdorff dimension of graph Let A(x) = (D(x), Y (x)), x 0, (4) where D = {D(x), x 0} is defined by D(x) = inf { t > 0 : E(t) > x }. (5) If E(1) > 0 a.s. and there exist a constant c 3 such that for all constants 0 < a < then dim H A([0, a]) = c 3 (6) dim H GrZ([0, 1]) = dim H {(t, Z(t)) : t [0, 1]} Meerschaert, Nane and Xiao (2013). = max { 1, dim H A([0, 1]) }, a.s. (7)

10 Examples Let Z = {Y (E(t)), t 0}, where Y = {Y (x) : x 0} is a stable Lévy motion of index α (0, 2] with values in R d and E(t) is the inverse of a stable subordinator of index 0 < β < 1, independent of Y. Then dim H Z([0, 1]) = dim H Y ([0, 1]) = min{d, α}, a.s. (8) { max{1, α} if α d, dim H GrZ([0, 1]) = 1 + β(1 1 α ) if α > d = 1, a.s. compare to { } dim H GrY ([0, 1]) = dim H (E(t), Y (E(t))) : t [0, 1] { max{1, α} if α d, = 2 1 α = a.s. α if α > d = 1. (9) (10)

11 Let Z = {Y (E(t)), t 0}, where Y is a fractional Brownian motion with values in R d of index H (0, 1) and E(t) is the the inverse of a β-stable subordinator D which is independent of Y. Then dim H Z([0, 1]) = dim H Y ([0, 1]) = min { d, 1 }, a.s. (11) H { 1 dim H GrZ([0, 1]) = H if 1 Hd, β + (1 Hβ)d = d + β(1 Hd) if 1 > Hd, (12) compare to { } dim H (E(t), Y (E(t))) : t [0, 1] = dimh GrY ([0, 1]) { 1 = H if 1 Hd, 1 + (1 H)d = d + (1 Hd) if 1 > Hd, a.s. (13)

12 Fractional time derivative: Two approaches Riemann-Liouville fractional derivative of order 0 < β < 1; [ D β 1 t ] ds t g(t) = g(s) Γ(1 β) t (t s) β with Laplace transform s β g(s), g(s) = 0 e st g(t)dt denotes the usual Laplace transform of g. Caputo fractional derivative of order 0 < β < 1; D β t g(t) = 1 Γ(1 β) t 0 0 dg(s) ds ds (t s) β (14) was invented to properly handle initial values (Caputo 1967). Laplace transform of D β t g(t) is s β g(s) s β 1 g(0) incorporates the initial value in the same way as the first derivative.

13 examples D β t (t p ) = D β t (e λt ) = λ β e λt Γ(1 + p) Γ(p + 1 β) tp β t β Γ(1 β)? D β t (sin t) = sin(t + πβ 2 )

14 Diffusion Let L x be the generator of some continuous Markov process X (t). Then p(t, x) = E x [f (X (t))] is the unique solution of the heat-type Cauchy problem t p(t, x) = L x p(t, x), t > 0, x R d ; p(0, x) = f (x), x R d Examples: X : Brownian motion then L x = x, BM is a stochastic solution of the heat equation X : Symmetric α-stable process then L x = ( ) α/2

15 Time-fractional model for Anamolous sub-diffusion Let 0 < β < 1. Nigmatullin (1986) gave a physical derivation of fractional diffusion β t u(t, x) = L x u(t, x); u(0, x) = f (x) (15) Zaslavsky (1994) used this to model Hamiltonian chaos. (??) has the unique solution u(t, x) = E x [f (X (E(t)))] = 0 p(s, x)g E(t) (s)ds where p(t, x) = E x [f (X (t))] and E(t) = inf{τ > 0 : D τ > t}, D(t) is a stable subordinator with index β and E(e sd(t) ) = e tsβ (Baeumer and Meerschaert, 2002). E x (B(E(t))) 2 = E(E(t)) t β.

16 Heat equation in bounded domains Denote the eigenvalues and the eigenfunctions of on a bounded domain D with Dirichlet boundary conditions by {µ n, ϕ n } n=1 ; ϕ n (x) = µ n ϕ n (x), x D; ϕ n (x) = 0, x D. τ D (X ) = inf{t 0 : X (t) / D} is the first exit time of a process X, and let f (n) = D f (x)ϕ n(x)dx. The semigroup given by T D (t)f (x) = E x [f (B(t))I (τ D (B) > t)] = e µ nt ϕ n (x) f (n) n=1 solves the heat equation in D with Dirichlet boundary conditions: t u(t, x) = x u(t, x), x D, t > 0, u(t, x) = 0, x D; u(0, x) = f (x), x D.

17 Fractional diffusion in bounded domains t β u(t, x) = x u(t, x); x D, t > 0 (16) u(t, x) = 0, x D, t > 0; u(0, x) = f (x), x D. has the unique (classical) solution u(t, x) = f (n)ϕ n (x)m β ( µ n t β ) = n=1 = E x [f (B(E(t)))I (τ D (B) > E(t))] = E x [f (B(E(t)))I (τ D (B(E)) > t)] Meerschaert, Nane and Vellaisamy (2009). 0 T D (l)f (x)g E(t) (l)dl

18 Corollary Mittag-Leffler function is defined by M β (z) = k=0 z k Γ(1 + βk). u(t, ) L 2 CM β ( µ 1 t β ) In the Heat-equation case, since β = 1 we have M β ( µ 1 t β ) = e µ 1t so C µ 1 t β u(t, ) L 2 CM 1 ( µ 1 t) = Ce µ 1t

19 New time operators Laplace symbol: ψ(s) inverse subordinator time operator 0 (1 e sy )ν(dy) E ψ (t) ψ( t ) δ(0)ν(t, ) s β E(t) t β, Caputo sβ Γ(1 β)µ(dβ) E µ (t) 0 β t Γ(1 β)µ(dβ) (s + λ) β λ β E λ (t) t β,λ in (17) β,λ t g(t) = ψ λ ( t )g(t) g(0)ϕ λ (t, ) [ = e λt 1 t Γ(1 β) d t g(0) Γ(1 β) t 0 e λs g(s) ds (t s) β e λr βr β 1 dr. ] λ β g(t) (17)

20 Subdiffusion: 0 < β < 1, E x (B(E(t))) 2 = E(E(t)) t β. Ultraslow diffusion: For special µ RV 0 (θ 1) for some θ > 0: E x (B(E µ (t))) 2 = E(E µ (t)) (log t) θ for some θ > 0. Intermediate between subdiffusion and diffusion: Tempered fractional diffusion { E x (B(E λ (t))) 2 t β /Γ(1 + β), t << 1 t/β, t >> 1. B(E λ (t)) occupies an intermediate place between subdiffusion and diffusion (Stanislavsky et al., 2008)

21 New space operators Laplace exp.: ψ(s) subord. process Generator 0 (1 e sy )ν(dy) D ψ (t) B(D ψ (t)) ψ( ) s β D(t) B(D(t)) ( ) β (s + m 1/β ) β m T β (t, m) B(T β (t, m)) [( + m 1/β ) β m] log(1 + s β ) D log (t) B(D log (t)) log(1 + ( ) β ) B(t), Brownian motion B(D(t)), symmetric stable process B(T β (t, m)), relativistic stable process B(D log (t)), geometric stable process

22 Space-time fractional diffusion in bounded domains [ψ 1 ( t ) δ(0)ν(t, )]u(t, x) = ψ 2 ( x )u(t, x); x D, t > 0 u(t, x) = 0, x D (or x D C ), t > 0; u(0, x) = f (x), x D. has the unique (classical) solution u(t, x) = f (n)ϕ n (x)h ψ1 (t, λ n ) n=1 = E x [f (B(D ψ2 (E ψ1 (t))))i (τ D (B(D ψ2 (E ψ1 ))) > t)] h ψ1 (t, λ) = E(e λe ψ (t) 1 ) is the solution of [ψ 1 ( t ) δ(0)ν(t, )]h ψ1 (t, λ) = λh ψ1 (t, λ); h ψ1 (0, λ) = 1; ψ 2 ( x )ϕ n (x) = λ n ϕ n (x); ϕ n (x) = 0, x D (or x D C ).

23 ψ 2 (s) = s with ψ 1 (s) = s β : subdiffusion ψ 1 (s) = 1 0 sβ Γ(1 β)µ(dβ): Ultraslow diffusion ψ 1 (s) = (s + λ) β λ β : tempered fractional diffusion. ψ 2 (s) = s β 2 with the three ψ 1 s: space-time fractional diffusion, ( ) β 2 Joint work with Meerschaert, Vellaisamy and Chen (2009,2010, 2012).

24 Time fractional models in Manifolds Let (M, µ) be a smooth connected Riemannian manifold of dimension n 1 with Riemannian metric g. The associated Laplace-Beltrami operator = M in M is an elliptic, second order, differential operator defined in the space C0 (M). In local coordinates, this operator is written as = 1 g n i,j=1 x i ( g ij g ) x j where {g ij } is the matrix of the Riemannian metric, {g ij } and g are respectively the inverse and the determinant of {g ij }. (18)

25 Let M be a connected and compact manifold. The unique strong solution to the fractional Cauchy problem { β t u(m, t) = u(m, t), m M, t > 0 u(m, 0) = f (m), m M, f H s (19) (M) is given by u(m, t) = Ef (B m E t ) = j=1 M β ( t β λ j )ϕ j (m) ϕ j (y)f (y)µ(dy) M (20) where B m t is a Brownian motion in M and E t = E β t is inverse to a stable subordinator with index 0 < β < 1. D Ovidio and Nane (2013).

26 in the sphere In the sphere: x S 2 r can be represented as x = (r sin ϑ cos φ, r sin ϑ sin φ, r cos ϑ). The spherical Laplacian is defined by S 2 r = 1 [ 1 2 r 2 sin 2 ϑ φ sin ϑ ϑ ( sin ϑ ϑ )], ϑ [0, π], φ [0, 2π]. Sometimes S 2 r is called Laplace operator on the sphere. In this case the eigenfunctions of the spherical Laplacian are called Spherical harmonics obtained from Legendre polynomials.

27 We can use B m (E t ) as a process to introduce time dependence in a Gaussian random field T to model Cosmic Microwave Background (CMB) radiation. CMB radiation is thermal radiation filling the observable universe almost uniformly and is well explained as radiation associated with an early stage in the development of the universe. A further remarkable feature is that the new random field T (B m (E t )) has short-range dependence for β 1 and long-range dependence for β (0, 1).

28 Further research Work in progress for other Subordinated Brownian motions, e.g. relativistic stable process as the outer process... corresponding to the operator ( + m 1/β ) β m for 0 < α 2, m 0. Extension to Neumann boundary conditions... Extensions to anisotropic/nonsymmetric space operators.. of B(E(t)) and other time-changed processes Applications-interdisciplinary research solutions to the spde with space time white noise ξ: β t u(t, x) = 2 x u(t, x) + ξ; x R, t > 0; u(0, x) = f (x), x R.

29 References Baeumer, B., Meerschaert, M.M.: Stochastic solutions for fractional Cauchy problems. Fract. Calc. Appl. Anal. 4, (2001) Caputo, M.: Linear models of dissipation whose Q is almost frequency independent, Part II. Geophys. J. R. Astr. Soc. 13, (1967) Z.-Q. Chen, M.M. Meerschaert and E. Nane. Space-time fractional diffusion on bounded domains. Journal of Mathematical Analysis and Applications. Volume 393, Issue 2, 15 September 2012, Pages D Ovidio, M., Nane, E.: Time dependent random fields on spherical non-homogeneous surfaces, arxiv:

30 Hahn M., Kobayashi K. Umarov, S.: Fokker-Planck-Kolmogorov equations associated with time-changed fractional Brownian motion. Proc. Amer. Math. Soc. Volume 139, Number 2, February 2011, Pages Meerschaert, M.M., Scheffler, H.P.: Limit theorems for continuous time random walks with infinite mean waiting times. J. Appl. Probab. 41, (2004) Meerschaert, M.M., Scheffler, H.P.: Limit theorems for continuous time random walks with infinite mean waiting times. J. Appl. Probab. 41, (2004) Meerschaert, M.M., Nane, E., Vellaisamy, P.: Fractional Cauchy problems on bounded domains. Ann. Probab. 37, (2009) M.M. Meerschaert, E. Nane and P. Vellaisamy. Distributed-order fractional diffusions on bounded domains. Journal of Mathematical Analysis and Applications. 379 (2011)

31 M.M. Meerschaert, E. Nane and Y. Xiao. Correlated continuous time random walks. Stat.Probab.Lett. Volume 79, Issue 9, 1 May 2009, Pages Meerschaert, M.M., Nane, E., Xiao, Y.: Fractal dimensions for continuous time random walk limits. Statistics & Probability Letters, Vol. 83 (2013), No. 4, pp R.R. Nigmatullin. The realization of the generalized transfer in a medium with fractal geometry. Phys. Status Solidi B. 133, (1986) Penzias, A.A., Wilson, R. W.: A measurement of excess antenna temperature at 4080 mc/s. Astrophysical Journal. 142, (1965) Stanislavsky, A., Weron, K. and Weron, A.: Diffusion and relaxation controlled by tempered α-stable processes, Phys. Rev. E (3) 78 (2008), no. 5, Zaslavsky, G. (1994). Fractional kinetic equation for Hamiltonian chaos. Phys. D

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