Continuous Time Random Walk Limits in Bounded Domains

Transcription

1 Continuous Time Random Walk Limits in Bounded Domains Erkan Nane DEPARTMENT OF MATHEMATICS AND STATISTICS AUBURN UNIVERSITY May 16, 2012

2 Acknowledgement Boris Baeumer, University of Otago, New Zealand. Zhen-Qing Chen, University of Washington. Mark M. Meerschaert, Michigan State University. Palaniappan Vellaisamy, IIT Bombay, India. Yimin Xiao, Michigan State University.

3 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.

4 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.

5 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.

6 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

7 Power law waiting times Wait between solar flares 1 < β < 2 Wait between raindrops β = 0.68 Wait between money transactions β = 0.6 Wait between s β 1.0 Wait between doctor visits β 1.4 Wait between earthquakes β = 1.6 Wait between trades of German bond futures β 0.95 Wait between Irish stock trades β = 0.4 (truncated)

8 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).

9 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 (1) 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.

10 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 )

11 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

12 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) (2) Zaslavsky (1994) used this to model Hamiltonian chaos. (2) 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.

13 Equivalence to higher order PDE s For any m = 2, 3, 4,... both the Cauchy problem t u(t, x) = m 1 j=1 t j/m 1 Γ(j/m) Lj xf (x) + L m x u(t, x); and the fractional Cauchy problem: u(0, x) = f (x) (3) 1/m t u(t, x) = L x u(t, x); u(0, x) = f (x), (4) have the same unique solution given by u(t, x) = E x [f (X (E 1/m (t)))] = 0 p(s, x)g E 1/m (t)(s) ds Allouba and Zheng (2001), Baeumer, Meerschaert and Nane (2007), Keyantuo and Lizama (2009), Li et al. (2010).

14 Connections to iterated Brownian motions Orsingher and Beghin (2004, 2008) show that the solution to is given by running 1/2n t u(t, x) = x u(t, x); u(0, x) = f (x), (5) I n+1 (t) = B 1 ( B 2 ( B 3 ( (B n+1 (t)) ) ) ) Where B j s are independent Brownian motions, i.e., u(t, x) = E x (f (I n+1 (t))) solves (5), and solves (3) for m = 2 n.

15 1 0.5 Brownian Motion Iterated Brownian Motion Brownian Motion Iterated Brownian Motion Space Time Figure: Simulations of iterated Brownian motions

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 (6) 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)] Joint work with Meerschaert and Vellaisamy (2009). 0 T D (l)f (x)g E(t) (l)dl

18 Analytic solution in intervals (0, M) R was obtained by Agrawal (2002). In this case, eigenfunctions and eigenvalues are ( ) 2 1/2 ϕ n (x) = sin(nπx/m), µ n = π2 n 2 M M 2 The time fractional diffusion on (0, M) has the solution here u(t, x) = n=1 ( ) 2 1/2 f (n) sin(nπx/m)m β ( µ n t β ) M M β (z) = k=0 z n Γ(1 + βn) For β = 1, M 1 ( z) = e z, and u coincides with the solution of the heat equation on (0, M).

19 Sketch of Proof Use Green s second identity and Dirichlet b.c. to write ϕ n (x) x u(t, x)dx = u(t, x) ϕ n (x) D D = µ n u(t, x)ϕ n (x)dx = µ n ū(t, n) Apply to both sides of the fractional Cauchy problem to get D β t ū(t, n) = µ n ū(t, n). (7) taking Laplace transforms on both sides of (7), we get s β û(s, n) s β 1 ū(0, n) = µ n û(s, n) (8) Collecting the like terms leads to û(s, n) = f (n)s β 1 s β +µ n.

20 Sketch of Proof (page2) By inverting the above Laplace transform, we obtain ū(t, n) = f (n)m β ( µ n t β ) in terms of the Mittag-Leffler function defined by z k M β (z) = Γ(1 + βk). k=0 Compute the Laplace transform of the hitting time density E(e µe(t) ) = 0 e µl g E(t) (l)dl = M β ( µt β ). Inverting now the ϕ n -transform, we get an L 2 -convergent solution of Equation (6) as (for each t 0 ) u(t, x) = f (n)ϕ n (x)m β ( µ n t β ) (9) n=1

21 Sketch of Proof (page3) To get the probabilistic form of the solution we proceed as u(t, x) = f (n)ϕ n (x)m β ( µ n t β ) = = = = n=1 f (n)ϕ n (x) n= e µnl g E(t) (l)dl ( ) f (n)e µ nl ϕ n (x) g E(t) (l)dl n=1 T D (l)f (x)g E(t) (l)dl E x [f (B(l))I (τ D > l)]g E(t) (l)dl = E x [f (B(E(t)))I (τ D (B) > E(t))] (10)

22 IBM in bounded domains The (classical) solution of t u(t, x) = 2 n 1 j=1 t j/2n 1 Γ(j/m) j xf (x) + 2n x u(t, x), x D, t > 0; u(t, x) = l xu(t, x) = 0, t 0, x D, l = 1, 2 n 1; u(0, x) = f (x), x D is given by (running I n+1 (t) = B 1 ( B 2 ( B n+1 (t) ) ) = B 1 ( I n (t) )) u(t, x) = E x [f (I n+1 (t))i (τ D (B 1 ) > I n (t) )] = 2 0 T D (l)f (x)h(t, l)dl, (11) where h(t, l) is the transition density of {I n (t)}. Proof: equivalence with fractional Cauchy problem for β = 1/2 n.

23 Corollary 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

24 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 (12) 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) (12)

25 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. 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)

26 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

27 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. and

28 ψ 2 ( x )ϕ n (x) = λ n ϕ n (x); ϕ n (x) = 0, x D (or x D C ). 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).

29 Further research Work completed recently for the symmetric stable process as the outer process. The corresponding space operator is ( ) α/2 for 0 < α 2. 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... Fractal properties of B(E(t)) and other time-changed processes Applications-interdisciplinary research

30 Thank You!