Fractional Cauchy Problems for time-changed Processes

Transcription

1 Fractional Cauchy Problems for time-changed Processes Erkan Nane Department of Mathematics and Statistics Auburn University February 17, 211 Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

2 Scaling limits Continuous time random walks R d X 2 X 3 X 1 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. Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

3 Scaling limits 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 : T n t} is the number of jumps by time t >. Suppose P(J n > t) Ct β for < 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 > : D(u) > t}. The self-similar limit E(ct) d = c β E(t) is non-markovian. Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

4 Scaling limits 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 subordinated to (a renewal process) N(t) S(N(t)) = X 1 + X X N(t). S(N(t)) is particle location at time t >. CTRW scaling limit is a subordinated 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. Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

5 Scaling limits Scaling limit: Subordinated motion Limit retains long waiting times. Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

6 Scaling limits Power law waiting times Wait between solar flares 1 < β < 2 Wait between raindrops β =.68 Wait between money transactions β =.6 Wait between s β 1. Wait between doctor visits β 1.4 Wait between earthquakes β = 1.6 Wait between trades of German bond futures β.95 Wait between Irish stock trades β =.4 (truncated) Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

7 Scaling limits CTRW with serial dependence Particle jumps X n = j= c jz n j with Z n IID. Short range dependence: n=1 E(X nx ) < = the usual limit and PDE. Long range dependence: If Z n has light tails, subordinated fractional Brownian motion limit B H (E(t)). Hahn, Kobayashi and Umarov (21) established a governing equation. For heavy tails, subordinated linear fractional stable motion L α,h (E(t)). Open problems: Governing equations, dependent waiting times. Due to Meerschaert, Nane and Xiao (29). Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

8 Fractional Diffusion Fractional time derivative: Two approaches Riemann-Liouville fractional derivative of order < 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) = e st g(t)dt denotes the usual Laplace transform of g. Caputo fractional derivative of order < β < 1; D β t g(t) = 1 t Γ(1 β) 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() incorporates the initial value in the same way as the first derivative. Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

9 Fractional Diffusion 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 ) Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

10 Fractional Diffusion 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 >, x R d ; p(, 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 Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

11 Fractional Diffusion Time-fractional model for Anamolous sub-diffusion Let < β < 1. Nigmatullin (1986) gave a physical derivation of fractional diffusion β t u(t, x) = L x u(t, x); u(, 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)))] = p(s, x)g E(t) (s)ds where p(t, x) = E x [f (X (t))] and E(t) = inf{τ > : D τ > t}, D(t) is a stable subordinator with index β and E(e sd(t) ) = e tsβ (Baeumer and Meerschaert, 22). E x (B(E(t))) = xe(e(t) 1/2 ) xt β/2. Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

12 Fractional Diffusion Equivalence to higher order PDE s For any m = 2, 3, 4,... both the Cauchy problem t u(t, x) = m 1 j=1 and the fractional Cauchy problem: t j/m 1 Γ(j/m) Lj xf (x) + L m x u(t, x); u(, x) = f (x) (3) 1/m t u(t, x) = L x u(t, x); u(, x) = f (x), (4) have the same unique solution given by u(t, x) = E x [f (X (E 1/m (t)))] = p(s, x)g E 1/m (t)(s) ds Due to Allouba and Zheng (21), Baeumer, Meerschaert, and Nane (27), Keyantuo and Lizama (29), Li et al. (21). Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

13 Fractional Diffusion Connections to iterated Brownian motions Orsingher and Beghin (24, 28) show that the solution to is given by running 1/2n t u(t, x) = x u(t, x); u(, 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. Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

14 Initial-Boundary value problems 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) =, x D. τ D (X ) = inf{t : 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 µnt 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 >, u(t, x) =, x D; u(, x) = f (x), x D. Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

15 Initial-Boundary value problems Fractional diffusion in bounded domains β t u(t, x) = x u(t, x); x D, t > (6) u(t, x) =, x D, t > ; u(, 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)] T D (l)f (x)g E(t) (l)dl Joint work with Meerschaert and Vellaisamy (29). Analytic solution in intervals (, M) R was obtained by Agrawal (22). Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

16 Initial-Boundary value problems 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(, 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. Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

17 Initial-Boundary value problems 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= Compute the Laplace transform of the hitting time density E(e µe(t) ) = 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 ) u(t, x) = f (n)ϕ n (x)m β ( µ n t β ) (9) n=1 Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

18 Initial-Boundary value problems 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=1 e µ nl g E(t) (l)dl ( ) f (n)e µnl 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))] (1) Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

19 Initial-Boundary value problems 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 > ; u(t, x) = l xu(t, x) =, t, x D, l = 1, 2 n 1; u(, 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 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. Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

20 Initial-Boundary value problems Corollary u(t, ) L 2 CE β ( µ 1 t β ) C µ 1 t β In the Heat-equation case, since β = 1 we have E β ( µ 1 t β ) = e µ 1t so u(t, ) L 2 CE 1 ( µ 1 t) = Ce µ 1t Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

21 Ultraslow diffusion Stochastic model for ultraslow diffusion Let suppν (, 1) be a finite measure. W (t), increasing Lévy process with laplace transform [ E[e sw (t) ] = e t ] (e sx 1)ϕ W (dx) The Lévy measure is ϕ W (t, ) = 1 = e t [ 1 sβ Γ(1 β)ν(dβ) ]. t β ν(dβ), (12) Let E(t) = inf{τ : W (τ) t} be the inverse process. B(E(t)) is a stochastic solution of D (ν) u(t, x) := 1 β t u(t, x)γ(1 β)ν(dβ) = x u(t, x) For ν(dβ) = β α 1 dβ, α >, E(B(E W (t)) 2 ) = E(E W (t)) C(α)(log t) α as t. Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

22 Ultraslow diffusion Ultraslow diffusion in bounded domains D (ν) u(t, x) = x u(t, x); x D, t > u(t, x) =, x D, t > ; u(, x) = f (x), x D. has the unique (classical) solution u(t, x) = f (n)ϕ n (x)h(t, µ n ) = 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)] where h(t, µ n ) = E(e µ ne(t) ). Joint work with Meerschaert and Vellaisamy (29). T D (l)f (x)g E(t) (l)dl Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

23 Ultraslow diffusion Eigenvalue problem for distributed order time derivative h(t, µ) = E(e µe(t) ) is the solution of D (ν) h(t, µ) = µh(t, µ); h(, µ) = 1. (13) In the case ν(dβ) = p(β)dβ; by inverse Laplace transform it has the representation h(t, µ) = µ π where for U(r) = 1 r β sin(βπ)γ(1 β)p(β)dβ r 1 e tr Φ(r, 1)dr (14) U(r) Φ(r, 1) = [ 1 r β cos(βπ)γ(1 β)p(β)dβ + µ] 2 + [U(r)]. 2 Due to Kochubei (28). Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

24 Tempered Fractional Model Tempered fractional model D λ (t), increasing Lévy process with laplace transform [ E[e sdλ(t) ] = e tψλ(s) = e t ] ] (e sx 1)ϕ Dλ (dx) [(s+λ) = e t β λ β. The Lévy measure is ϕ Dλ (t, ) = 1 Γ(1 β) t e λr βr β 1 dr, (15) Let E λ (t) = inf{τ : D λ (τ) t} be the inverse process. Define the caputo tempered fractional time derivative by ( ) β,λ g(t) = ψ λ ( t )g(t) g()ϕ Dλ (t, ) t = e λt 1 d Γ(1 β) dt g()ϕ Dλ (t, ) t e λs g(s) ds (t s) β λ β g(t) (16) Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

25 Tempered Fractional Model Tempered fractional cauchy problem Then E[f (B(E λ (t)))] is a stochastic solution of ( ) β,λ u(t, x) = x u(t, x); u(t, ) = f (x). t Due to Meerschaert and Sheffler (28). { 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., 28) Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

26 Tempered Fractional Model Tempered fractional cauchy problem in bounded domains ( ) β,λ u(t, x) = x u(t, x), x D, t > ; t u(t, x) =, x D, t > ; u(, x) = f (x), x D. has the unique (classical) solution u(t, x) = E x (f (B(E λ (t)))i (τ Dλ (B) > E λ (t)) = = T D (l)f (x)g λ (t, l)dl f (n)ϕ n (x)h λ (t, µ n ), n=1 where h λ (t, µ) = E(e µe λ(t) ) = e µx g λ (t, x) dx is the Laplace transform of E λ (t) (Meerschaert, Nane and Vellaisamy 21). Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

27 Tempered Fractional Model Eigenvalue problem for tempered fractional time derivative h λ (t, µ) = E(e µe λ(t) ) is the unique solution of ( ) β,λ h λ (t, µ) = µh λ (t, µ); h λ (, µ) = 1. (17) t For any µ, λ >, µ λ β, the function h λ (t, µ) has the representation where h λ (t, µ) = µ π (r + λ) 1 e t(r+λ) Φ(r, 1)dr, (18) r β sin(βπ) Φ(r, 1) = r 2β sin 2 (βπ) + (µ λ β + r β cos(βπ)) 2 Due to Meerschaert, Nane and Vellaisamy (21). Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

28 Stable subordinators Other Subordinators; Cauchy process X (t) a continuous Markov process with generator A, Y (t) be a Cauchy process independent of X (t). Then u(t, x) = E x [f (X ( Y (t) ))] is a solution of t 2 2Af (x) u(t, x) = A 2 u(t, x), t >, x R d ; πt u(, x) = f (x) x R d. Due to Nane (28). Proof uses the fact that the density p(t, s) of Y (t) solves ( 2 s + 2 t )p(t, s) =. Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

29 Stable subordinators Nonhomogeneous wave equation This reduces to nonhomogeneous wave equation in the case X is another Cauchy process independent of Y, The generator of X is A = ( ) 1/2, fractional Laplacian. u(t, x) = E x [f (X ( Y (t) ))] is a solution of t 2 u(t, x) = 2( )1/2 f (x) + u(t, x), t >, x R d ; πt u(, x) = f (x), x R d This is one of the most interesting PDE connections of these iterated processes. Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

30 Stable subordinators Bounded domains Let D be a bounded domain. Then is a solution of u(t, x) = E x [f (B( Y (t) )I (τ D (B) > Y (t) )] ( ) = 2 e µ t ns f (n)ϕn (x) π(t 2 + s 2 ) ds n=1 t 2 2 f (x) u(t, x) = 2 u(t, x), t >, x D (19) πt u(, x) = f (x), x D, Due to Nane (21). u(t, x) = u(t, x) =, x D, t >. Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

31 Stable subordinators Stable subordinators: α 1 α (, 2) be rational α = l/m, where l and m are relatively prime. Y (t); a symmetric α-stable process Then u(t, x) = E x [f (B( Y (t) )] is a solution of l+1 2m l ( 2l 2i ( 1) t 2m u(t, x) = 2 i=1 s 2l 2i pα (t, s) s= 2l u(t, x), t >, x R d u(, x) = f (x), x R d. ) 2i 1 f (x) Where p α (t, s) is the transition density of symmetric α-stable process Y (t). The proof uses the fact that 2m t p α (t, x) = ( 2 x ) l p α (t, x) Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

32 Stable subordinators Bounded domains α (, 2) a rational α = l/m, where l and m are relatively prime. D a bounded domain. Then u(t, x) = E x [f (B( Y (t) )I (τ D (B) > Y (t) )] is a classical solution of l+1 2m l ( 2l 2i ( 1) t 2m u(t, x) = 2 i=1 s 2l 2i pα (t, s) s= 2l u(t, x), t >, x D u(, x) = f (x), x D. ) 2i 1 f (x) (2) u(t, x) = j u(t, x) =, x D, t >, j = 1,, 2l 1 Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

33 Stable subordinators Further research Work in progress for the subordinated Brownian motions, e.g. symmetric stable process as the outer process. The corresponding space operators are ( ) α/2 for < α 2 Extension to Neumann boundary conditions... Fractal properties of B(E(t)) and other subordinate processes Applications-interdisciplinary research Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

34 Stable subordinators Further research:a question? D ovidio and Orsingher (21) established the fact that the density of B H ( B(t) ) 1 d = B H (E(t)) for β = 1/2 q(t, x) = 2 is a solution of the first order PDE e x2 2s 2H 2πs 2H e s2 2t 2πt ds q(t, x) t = H (xq(t, x)), t >, x R. (21) t 2 x How about B H ( B 1 ( B 2 ( (B n (t) ) ) ) ) 1 d = B H (E(t)) for β = 1/2 n? Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

35 Stable subordinators By induction, the density of B H ( B 1 ( B 2 ( (B n (t) ) ) ) ) 1 d = B H (E(t)) is a solution to the first-order pde: The first order PDE q(t, x) t = H t 2 n (xq(t, x)), t >, x R. (22) x u(t, x) t = H 1 H 2 (xu(t, x)), t >, x R. (23) t x has a general solution of the form u(t, x) = 1 ( ) x x f t H, x R \ {}, t > 1H 2 with H 1, H 2 (, 1) and f C 1 (R). Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

36 Stable subordinators An approach for higher order PDES To get the Higher order PDEs for B H ( B 1 ( B 2 ( (B n (t) ) ) ) ) 1 d = B H (E(t)). Let β = 1 m then f E(t) (s) t k f E(t) () s k = ( 1) m m f E(t) (s) s m, s, t > ; = ( 1)k t (k+1)/m Γ(1 (k+1) m ), t >, k =, 1, 2,, m 1; k f E(t) (s) lim s s k =, t >, k =, 1, 2,, (m 1). By Keyantuo and Lizama (21). (24) Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

37 Stable subordinators In the case < H < 1 and α = 1 the density function q(t, x) = 2 of W (Y (t)) solves the PDE f H (s, x)p t (s)ds = 2 e x 2 2s 2H (2πs 2H ) d/2 t π(t 2 + s 2 ) ds 2 q(t, x) t 2 = 2HI (,1/2](H) δ(x) H(2H 1) G πt (2H 2),t q(t, x) H 2 2 G (4H 2),t q(t, x), x R d, t >, (25) where G γ,t q(t, x) = 2 s γ p t (s)f H (s, x)ds, γ, and G,t is the identity operator. Due to Nane, Xiao and Wu (21). Properties of G γ,t? Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

38 Stable subordinators Beghin, Orsingher and Sakhno (21) has established that the density W (Y (t)) in the case d = 1, < H < 1, α = 1 q(t, x) = 2 e x2 2s 2H 2πs 2H t π(t 2 + s 2 ) ds solves 2 [ 1 q(t, x) = t2 t 2 H(H 1) ] 2 x H2 x x 2 x 2 q(t, x) 2HI (,1/2](H) 2 δ(x) πt x 2. (26) Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39

39 Stable subordinators Thank You! Erkan Nane (Auburn University) Fractional Cauchy Problems February 17, / 39