Cauchy Problems in Bounded Domains and Iterated Processes

Transcription

1 Cauchy Problems in Bounded Domains and Iterated Processes Erkan Nane DEPARTMENT OF MATHEMATICS AND STATISTICS AUBURN UNIVERSITY August 18, 21

2 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.

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 : 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.

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 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.

5 Scaling limit: Subordinated motion Limit retains long waiting times.

6 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)

7 CTRW with serial dependence Particle jumps X n = j= Z n j with Z n IID. Short range dependence = the usual limit and PDE. Long range dependence: If Z n has light tails, subordinated fractional Brownian motion limit B H (E(t)). 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).

8 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 Γ(1 β) t 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.

9 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 )

10 Time fractional model for Anamolous sub-diffusion Nigmatullin (1986) gave a physical derivation of fractional diffusion β t u(t, x) = Lu(t, x); u(, x) = f (x) (2) where < β < 1 and L is the generator of some continuous Markov process X (t). Zaslavsky (1994) used this to model Hamiltonian chaos. When p(t, x) = T (t)f (x) is a solution of t p(t, x) = Lp(t, x)(with corresponding process X t, E(t) = inf{u : D u > t}, D(t) is a stable subordinator with index β), the formula ] u(t, x) = E x [f (X (E(t))) = p(s, x)g E(t) (s)ds yields a solution of (2). Due to Baeumer and Meerschaert (22). E x (B(E(t))) = xe(e(t) 1/2 ) xt β/2.

11 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(, 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) = p(s, x)g E 1/m (t)(s) ds Due to Baeumer, Meerschaert, and Nane (27).

12 Connections to iterated Brownian motions Orsingher and Benghin (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.

13 Heat equation in bounded domains Denote the eigenvalues and the eigenfunctions of on a bounded domain D with Diriclet boundary conditions by {λ n, ϕ n } n=1 ; ϕ n = λ n ϕ n. Define the first exit time of a process X to be τ D (X ) = inf{t : X (t) / D}. 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 > t)] = e λnt ϕ n (x) f (n) n=1 solves the heat equation in D with Dirichlet boundary conditions: t u(t, x) = u(t, x), x D, t >, u(t, x) =, x D, u(, x) = f (x), x D.

14 Fractional diffusion in bounded domains β t u(t, 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)e β ( λ 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).

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

16 Sketch of Proof (page2) By inverting the above Laplace transform, we obtain ū(t, n) = f (n)e β ( λ n t β ) in terms of the Mittag-Leffler function defined by z k E β (z) = Γ(1 + βk). k= Compute the Laplace transform of the hitting time density E(e λe(t) ) = e λu g E(t) (u)du = E β ( λt 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)e β ( λ n t β ) (9) Erkan Nane DEPARTMENT n=1 OF MATHEMATICS AND STATISTICS AUBURN UNIVERSITY

17 Sketch of Proof (page3) To get the probabilistic form of the solution we proceed as u(t, x) = f (n)ϕ n (x)e β ( λ n t β ) = = = = n=1 f (n)ϕ n (x) n=1 e λ nu g E(t) (u)du ( ) f (n)e λnu ϕ n (x) g E(t) (u)du n=1 T D (u)f (x)g E(t) (u)du E x [f (B(u))I (τ D > u)]g E(t) (u)du = E x [f (B(E(t)))I (τ D (B) > E(t))] (1)

18 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 f (x) + 2n u(t, x), x D, t > (11) ; u(t, x) = l u(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, (12) where h(t, l) is the transition density of {I n (t)}. Proof: equivalence with fractional Cauchy problem for β = 1/2 n.

19 Corollary u(t, x) L 2 C(x)E β ( λ 1 t β ) C(x) λ 1 t β In the Heat-equation case, since β = 1 we have E β ( λ 1 t β ) = e λ 1t so u(t, x) L 2 C(x)E 1 ( λ 1 t) = C(x)e λ 1t

20 Stochastic model for ultraslow diffusion Let suppµ (, 1) be a finite measure. B i iid with dist. µ. Ji c d = c 1/β J 1 nonnegative iid with P(J 1 > t) = 1 t β µ(dβ), t 1 and { P(c 1/β 1, u < c 1/β J 1 > u B 1 = β) = c 1 u β, u c 1/β Time of the nth jump at scale c has T (ct) (t) = [ct] i=1 Jc i = W (t), increasing Lévy process. The number of jumps by time t at scale c has scaling limit; Nt c = max{n : T c (n) t} = E(t) = inf{τ : W (τ) t}. Xi c iid jumps has scaling limit S c (ct) = X1 c + X 2 c + + X [ct] c = B(t) then S c (N c t ) = B(E(t))

21 Ultraslow diffusion E[e sw (t) ] = e tψ W (s) and Laplace exponent ψ W (s) = (e sx 1)ϕ W (dx) = The Lévy measure is ϕ W (t, ) = 1 1 s β Γ(1 β)µ(dβ). (13) t β µ(dβ), (14) Then stochastic model for ultraslow diffusion B(E(t)) is a stochastic solution of D (ν) u(t, x) := 1 β t u(t, x)ν(dβ) = u(t, x); For special ν: E x (B(E(t))) = xe(e(t) 1/2 ) x(log t) β/2. ν(dβ) = Γ(1 β)µ(dβ).

22 Ultraslow diffusion in bounded domains D (ν) u(t, 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, λ n ) = n=1 = E x [f (X (E(t)))I (τ D (X ) > E(t))] = E x [f (X (E(t)))I (τ D (X (E)) > t)] where h(t, λ n ) = E(e λ ne(t) ). Joint work with Meerschaert and Vellaisamy (29). T D (l)f (x)g E(t) (l)dl

23 Eigenvalue problem for distributed order time derivative h(t, λ) = E(e λe(t) ) is the solution of D (ν) h(t, λ) = λh(t, λ); h(, λ) = 1. (15) In the case µ(dβ) = p(β)dβ; by inverse laplace transform it has the representation h(t, λ) = λ π where for U(r) = 1 r β sin(βπ)γ(1 β)p(β)dβ r 1 e tr Φ(r, 1)dr (16) U(r) Φ(r, 1) = [ 1 r β cos(βπ)γ(1 β)p(β)dβ + λ] 2 + [U(r)]. 2 Due to Kochubei (28).

24 Extensions: Markov generators The Markov process X (t) with generator d d Lu = a ij (x) xi xj u + b i (x) xi u (17) i,j=1 i=1 solves dx (t) = σ(x (t))db(t) + b(x (t))dt with a = σσ T. Then p(t, x) = T D (t)f (x) = E[f (X (t))i (τ D (X ) > t)] solves the Cauchy problem t p(t, x) = Lp(t, x) with Dirichlet boundary conditions.

25 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... Extensions to other time operators; tempered fractional derivative... Fractal properties of B(E(t)) and other subordinate processes Applications-interdisciplinary research

26 Thank You!