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Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

Scaling Limits for some P.D.E. Systems with Random Initial Data

Gi-Ren Liu and Narn-Rueih Shieh

Department of Mathematics National Taiwan University

Academia Sinica, 2010

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

Outline The papers achieved Background Preliminaries Heat-type system Time-fractional Diffusion-Wave system Spatial-fractional Kinetic system Open problem

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

The papers achieved Scaling Limits for some P.D.E. Systems with Random Initial Conditions, Stochastic Analysis and Applications 28, 505-522, 2010. Scaling Limits for Time-fractional P.D.E. Systems with Random Initial Data, Stochastics and Dynamics 10 1-35, 2010. Homogenization of Fractional Kinetic Systems with Random Initial Data, 38 pp, submitted and under a major revision, arXiv.1004.4267v1, 2010.

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

The papers achieved Scaling Limits for some P.D.E. Systems with Random Initial Conditions, Stochastic Analysis and Applications 28, 505-522, 2010. Scaling Limits for Time-fractional P.D.E. Systems with Random Initial Data, Stochastics and Dynamics 10 1-35, 2010. Homogenization of Fractional Kinetic Systems with Random Initial Data, 38 pp, submitted and under a major revision, arXiv.1004.4267v1, 2010. From single P.D.E. to P.D.E. system is usually something worthwhile. The solution-vector random field is of dependent components, even when the initial data are independent. We employ a stochastic decoupling method to handle the above. The limiting field is expressed as a multiple Itˆ

• -Wiener integral,

and has long-range-dependence inherited from that of the initial data.

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

Background For each partial differential equation and/or system, there must be a physical phenomenon to support it. Its boundary and initial conditions play important roles in the behaviour of the solution. A.N. Kolmogorov (1941, K41 Theory in Physics) initiated the stochastic theory of Turbulence and NSE, since the evolution of the error caused by the initial measurement so important to realize the bahaviour of the resulting fluid dynamics. Besides K41, there are researches on the PDEs with random initial conditions which can be traced back to 1950’s, Kampe, J. de Feriet, Random solutions of the partial differential equations, in Proc. 3rd Berkeley Symp. Math. Stat. Prob., Volume III , pp. 199-208. In this paper, he studied the heat equation for an infinite rod with the initial temperature being a random function f(x; ω).

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

Heat-type equations

• N. N. Leonenko and W. A. Woyczynski (JSP,1998)
• V. V. Anh and N. N. Leonenko (SPA, 1999)

They use the spectral representation to describe the initial condition for the heat equation when it is a nonlinear function of a stationary Gaussian process.

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

Heat-type equations

• N. N. Leonenko and W. A. Woyczynski (JSP,1998)
• V. V. Anh and N. N. Leonenko (SPA, 1999)

They use the spectral representation to describe the initial condition for the heat equation when it is a nonlinear function of a stationary Gaussian process. Fractional heat-type equations

• V. V. Anh and N. N. Leonenko (JSP, 2001)
• V. V. Anh and N. N. Leonenko (PTRF, 2002)

The time derivative ∂

∂t is extended to the fractional ∂β ∂tβ . The spatial

derivative Laplacian ∆ is extended to the fractional Riesz-Bessel

• perator −(I − ∆)

γ 2 (−∆) α 2 .

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

Nonlinear case: Burgers’ equation

• M. Rosenblatt (JMP, 1968), Remark on the Burgers equation.
• Ya. G. Sinai (CMP, 1991-2), Statistics of shocks in solutions of

inviscid Burgers equation. She, Zhen-Su; Aurell, Erik; Frisch, Uriel (CMP, 1991-2), The inviscid Burgers equation with initial data of Brownian type. The authors consider the 1D Burger equation with the Brownian initial data.

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

Nonlinear case: Burgers’ equation

• M. Rosenblatt (JMP, 1968), Remark on the Burgers equation.
• Ya. G. Sinai (CMP, 1991-2), Statistics of shocks in solutions of

inviscid Burgers equation. She, Zhen-Su; Aurell, Erik; Frisch, Uriel (CMP, 1991-2), The inviscid Burgers equation with initial data of Brownian type. The authors consider the 1D Burger equation with the Brownian initial data. Due to the Hopf-Cole transformation, which is the bridge between Burgers′ equation and heat equation, there are works to make use of this bridge, for example,

• O. E. Barndorff-Nielsen, N. N. Leonenko (JAP, 2005) Burgers′

turbulence problem with linear or quadratic external potential.

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

Preliminaries We consider a coupled system w(t, x) = (u(t, x), v(t, x))T, t ∈ R+, x ∈ Rd, in the form Ltw(t, x) = Lxw(t, x) + Bw, w(t = 0, x) = (u0(x), v0(x)), (1) where Lt and Lx are operators on time and space, respectively, given by Lt = ∂ ∂tβ , β ∈ (0, 2], (2) Lx = −(I − ∆)

γ 2 (∆) α 2 ,

α > 0, γ ≥ 0, (3) and B is a diagonalizable constant 2 × 2 matrix, that is, B = b11 b12 b21 b22

• = PDP−1, D =

d1 d2

• , P =

p11 p12 p21 p22

• .

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

Random initial condition Here, we assume the initial data u0(x) = G1(ζ1(x)) and v0(x) = G2(ζ2(x)), where functions G1, G2 : R → R belong to the Gaussian Hilbert space L2(R,

1 √ 2πe− r2

2 dr).

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

Random initial condition Here, we assume the initial data u0(x) = G1(ζ1(x)) and v0(x) = G2(ζ2(x)), where functions G1, G2 : R → R belong to the Gaussian Hilbert space L2(R,

1 √ 2πe− r2

2 dr).

Gj(r) =

∞

• k=0

Cj,k Hk(r) √ k! , j ∈ {1, 2},

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

Random initial condition Here, we assume the initial data u0(x) = G1(ζ1(x)) and v0(x) = G2(ζ2(x)), where functions G1, G2 : R → R belong to the Gaussian Hilbert space L2(R,

1 √ 2πe− r2

2 dr).

Gj(r) =

∞

• k=0

Cj,k Hk(r) √ k! , j ∈ {1, 2}, Hermite rank of function Gj mj := inf{k > 0|Cj,k = 0} ≥ 1.

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

Random initial condition Here, we assume the initial data u0(x) = G1(ζ1(x)) and v0(x) = G2(ζ2(x)), where functions G1, G2 : R → R belong to the Gaussian Hilbert space L2(R,

1 √ 2πe− r2

2 dr).

Gj(r) =

∞

• k=0

Cj,k Hk(r) √ k! , j ∈ {1, 2}, Hermite rank of function Gj mj := inf{k > 0|Cj,k = 0} ≥ 1. and ζ1(x), ζ2(x) are distributed as two independent, homogeneous and isotropic standard Gaussian random fields on Rd.

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

Spectral representation for ζj(x) and Gj(ζj(x)) Based on the homogeneous of the Gaussian random fields ζj(x), j ∈ {1, 2}, Bochner-Khintchine theorem says Rηj(x) =

• Rd ei<λ,x>fj(λ)dλ,

ζj(x) =

• Rd ei<x,λ>

fj(λ)Wj(dλ), j ∈ {1, 2}, where Wj(A), A ∈ B(Rd), are centered complex Gaussian with EWi(dλ)Wj(dµ) = δj

iδ(λ − µ)dλdµ.

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

Spectral representation for ζj(x) and Gj(ζj(x)) Based on the homogeneous of the Gaussian random fields ζj(x), j ∈ {1, 2}, Bochner-Khintchine theorem says Rηj(x) =

• Rd ei<λ,x>fj(λ)dλ,

ζj(x) =

• Rd ei<x,λ>

fj(λ)Wj(dλ), j ∈ {1, 2}, where Wj(A), A ∈ B(Rd), are centered complex Gaussian with EWi(dλ)Wj(dµ) = δj

iδ(λ − µ)dλdµ.

Applying the Wiener-Itˆ

• expansion, the subordinated random field

Gj(ζj(x)) is a superposition of Hk(ζj(x)) =

• ′

Rd×k ei<x,λ1+...+λk> k

• l=1
• fj(λl)Wj(dλl), k ∈ N,

where ′ denotes the multiple Wiener integrals.

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

Initial data with long range dependence We assume the correlation functions of ζ1(x) and ζ2(x) satisfy Rζj(|x|) := E[ζj(x)ζj(0)] ∼ L(|x|) |x|κj , κj ∈ (0, d mj ), where L : R+ → R+ is slowly varying at the infinity lim

r→∞L(cr)/L(r) = 1, ∀c > 0.

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

Initial data with long range dependence We assume the correlation functions of ζ1(x) and ζ2(x) satisfy Rζj(|x|) := E[ζj(x)ζj(0)] ∼ L(|x|) |x|κj , κj ∈ (0, d mj ), where L : R+ → R+ is slowly varying at the infinity lim

r→∞L(cr)/L(r) = 1, ∀c > 0.

In this case, by Tauberian theorem, the spectral density function fj(λ)

• f the correlation function Rζj(·) satisfies

fj(λ) = K(d, κj)|λ|κj−dL(| 1 λ|), as λ → 0, j ∈ {1, 2}. where K(d, κj) =

Γ(

d−κj 2

) 2κjπ

d 2 Γ( κj 2 ).

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

Heat-type system ∂ ∂t u v

• = µ∆

u v

• +

b11 b12 b21 b22 u v

• ,

u0(x) v0(x)

• =

G1(ζ1(x)) G2(ζ2(x))

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

Heat-type system ∂ ∂t u v

• = µ∆

u v

• +

b11 b12 b21 b22 u v

• ,

u0(x) v0(x)

• =

G1(ζ1(x)) G2(ζ2(x))

• Solution form :

u(x, t) v(x, t)

• = Q(t; d1, d2)

U(x, t) V(x, t)

• where

Q(t; d1, d2) := P ed1t ed2t

• P−1

and U(x, t) V(x, t)

• =

   

• Rd

1 (4πµt)

d 2 e− |x−y|2 4µt u0(y)dy

• Rd

1 (4πµt)

n 2 e− |x−y|2 4µt v0(y)dy

    .

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

Scaling limit with random initial data u0, v0 Case 1 m2κ2 > m1κ1 and d1 > d2 : [ε

m1α1 2 Lm1(ε− 1 2 )]− 1 2 e−d1 t ε

• u( x

√ε, t ε)

v( x

√ε, t ε)

• − Q( t

ε; d1, d2)

• C(1)

C(2)

• ⇒
• p11p22X(1)

m1 (x, t)

p21p22X(1)

m1 (x, t)

• ,

where X(1)

m1 (x, t) =

C(1)

m1

√m1!K(d, α1)

m1 2

• ′

Rd×m1

ei(x,z1+...+zm1)−µt|z1+...+zm1|2 (|z1| · · · |zm1|)

d−α1 2

m1

• l=1

W1(dzl).

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

Scaling limit with random initial data u0, v0 Case 2 m1α1 > m2α2 and d1 > d2 [ε

m2α2 2 Lm2(ε− 1 2 )]− 1 2 e−d1 t ε

• u( x

√ε, t ε)

v( x

√ε, t ε)

• − Q( t

ε; d1, d2)

• C(1)

C(2)

• ⇒
• −p11p12X(2)

m2 (x, t)

−p21p12X(2)

m2 (x, t)

• ,

where X(2)

m2 (x, t) =

C(2)

m2

√m2!K(d, α2)

m2 2

• ′

Rd×m2

ei(x,z1+...+zm2)−µt|z1+...+zm2|2 (|z1| · · · |zm2|)

d−α2 2

m2

• l=1

W2(dzl).

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

Scaling limit with random initial data u0, v0 Case 3 m1 = m2, α1 = α2 and d1 > d2 [ε

mα 2 Lm(ε− 1 2 )]− 1 2 e−d1 t ε

• u( x

√ε, t ε)

v( x

√ε, t ε)

• − Q( t

ε; d1, d2)

• C(1)

C(2)

• ⇒
• p11p22X(1)

m1 (x, t)

p21p22X(1)

m (x, t)

• +
• −p11p12X(2)

m2 (x, t)

−p21p12X(2)

m (x, t)

• ,

where X(1)

·

(·) and X(2)

·

(·) are defined in the previous cases with α1 = α2. Conclusion: 1. The “strength” of long range dependence of the initial data u0 and v0 determines the limiting vector field.

• 2. There are long range dependence between the components of

the limiting vector field.

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

Time-fractional diffusion-wave system This section features the super-diffusive, i.e. the time-fractional index 1 < β ≤ 2 (β = 2 is the classical wave system) and the space-dimension d = 3. ∂β ∂tβ u v

• = µ∆

u v

• +

b11 b12 b21 b22 u v

• ,

subject to initial conditions u(x, 0) v(x, 0)

• =

u0(x) v0(x)

• ,

ut(x, 0) vt(x, 0)

• =

u1(x) v1(x)

• .

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

Time-fractional diffusion-wave system This section features the super-diffusive, i.e. the time-fractional index 1 < β ≤ 2 (β = 2 is the classical wave system) and the space-dimension d = 3. ∂β ∂tβ u v

• = µ∆

u v

• +

b11 b12 b21 b22 u v

• ,

subject to initial conditions u(x, 0) v(x, 0)

• =

u0(x) v0(x)

• ,

ut(x, 0) vt(x, 0)

• =

u1(x) v1(x)

• .

The fractional derivative in time ∂β

∂tβ is taken in the Caputo-Djrbashian

sense: dβf dtβ (t) =

• f (m)(t)

if β = m ∈ N

1 Γ(m−β)

t

f (m)(τ) (t−τ)β+1−m dτ

if β ∈ (m − 1, m), where f (m)(t) denotes the ordinary derivative of order m of a causal function f(t) (i.e., f is vanishing for t < 0).

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

Solution form and Mittag-Leffler function Its solution vector can be represented as u v

• (x, t) =

1

• k=0
• R3

P Gβ,k+1(x − y, t; d1) Gβ,k+1(x − y, t; d2)

• P−1

uk(y) vk(y)

• dy,

where the Green function Gβ,k+1(·) is defined by the Fourier transformation tkEβ,k+1(−µ|λ|2tβ + djtβ) =

• R3 ei<x,λ>Gβ,k+1(x, t; dj)dx,

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

Solution form and Mittag-Leffler function Its solution vector can be represented as u v

• (x, t) =

1

• k=0
• R3

P Gβ,k+1(x − y, t; d1) Gβ,k+1(x − y, t; d2)

• P−1

uk(y) vk(y)

• dy,

where the Green function Gβ,k+1(·) is defined by the Fourier transformation tkEβ,k+1(−µ|λ|2tβ + djtβ) =

• R3 ei<x,λ>Gβ,k+1(x, t; dj)dx,

Mittag-Leffler functions: Eβ,γ(z) =

∞

• l=0

zl Γ(βl + γ), β, γ > 0, z ∈ C.

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

Randomness of u1(·) and v1(·) If we also randomize the initial velocity u1(·) and v1(·), then we get, in particular, for the case βm(1)

0 α(1)

< min{βm(2)

0 α(2) 0 , βm(1) 1 α(1) 1

− 4, βm(2)

1 α(2) 1

− 4} then the finite-dimensional distributions of the rescaled random field (ε

βm(1) α(1) 2

Lm(1)

0 (ε− β 2 ))− 1 2

• w(ε− β

2 x, ε−1t; εβB) − C0(ε−1t; εβd) − C1(ε−1t; εβd)

• ,

t > 0, x ∈ R3, converge weakly, as ε → 0, to the finite-dimensional distributions of the random field U0(x, t) :=       p11p22Xd1(x, t;

C(1)

0,m(1)

q m(1)

0 !

Hm(1)

0 , ζ(1)

0 ) − p12p21Xd2(x, t; C(1)

0,m(1)

q m(1)

0 !

Hm(1)

0 , ζ(1)

0 )

p21p22Xd1(x, t;

C(1)

0,m(1)

q m(1)

0 !

Hm(1)

0 , ζ(1)

0 ) − p21p22Xd2(x, t; C(1)

0,m(1)

q m(1)

0 !

Hm(1)

0 , ζ(1)

0 )

      .

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

In the limiting vector field, Xdi(·), i ∈ {1, 2}, are defined by Xdi(x, t; C(j)

0,m(j)

• m(j)

0 !

Hm(j)

0 , ζ(j)

0 ) :=

C(j)

0,m(j)

• m(j)

0 !

K(3, α(j)

0 )

m(j) 2 ×

• ′

R3×m(j)

e

i<x,λ1+...+λ

m(j)

> Eβ,1(−µ|λ1 + ... + λm(j)

0 |2tβ + ditβ)

(|λ1| · · · |λm(j)

0 |) 3−α(j) 2

m(j)

• l=1

W(j)

0 (dλl),

where mj, αj, etc, i ∈ {1, 2}, are parameters related to the initial data u0, v0, u1, v1.

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

Fractional kinetic system This section features two scaling procedures, the micro and the macro, in which the two space-fractional indices, the Riesz and the Bessel, play distinctive roles. ∂ ∂t u v

• = −µ(I − ∆)

γ 2 (−∆) α 2

u v

• + B

u v

• ,

where −(I − ∆)

γ 2 (−∆) α 2 is defined by the Fourier transform F as

F{−(I − ∆)

γ 2 (−∆) α 2 g}(λ) = −(1 + |λ|2) γ 2 |λ|α

for any function g ∈ S(Rd).

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

Solution form It has the unique solution given by u(t, x) v(t, x)

• = Q(t; d1, d2)

U(t, x) V(t, x)

• where

Q(t; d1, d2) := P ed1t ed2t

• P−1,

and U(t, x) V(t, x)

• =
• Rd G(t, y; α, γ)

u0(x − y) v0(x − y)

• dy,

where the Green function G(t, y; α, γ) is defined by the Fourier transform

• Rn ei<x,λ>G(t, x; α, γ)dx = exp[−µt|λ|α(1 + |λ|2)

γ 2 ].

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

I: Macro-scaling Let w := (u, v)(t, x) be the solution of the previous fraction kinetic system, then when the spatial and time variables tend to the infinity under a suitable order and m2κ2 > m1κ1 and d1 > d2 [ε

m1κ1 α Lm1(ε− 1 α )]− 1 2 e−d1 t ε

• w( t

ε, x ε

1 α

; w0(·)) − Q( t ε; d1, d2)

• C(1)

C(2) ⇒

• p11p22X(1)

m1 (t, x)

p21p22X(1)

m1 (t, x)

• ,

where X(1)

m1 (t, x) =

C(1)

m1

√m1!K(d, κ1)

m1 2

• Rd×m1

′ ei<x,z1+...+zm1>−µt|z1+...+zm1|α

(|z1| · · · |zm1|)

d−κ1 2

m1

• l=1

W1(dzl).

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

II(new even for the single eq case): Micro-scaling In the limiting field

• f Part I, only the Riesz parameter α plays the important role.

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

II(new even for the single eq case): Micro-scaling In the limiting field

• f Part I, only the Riesz parameter α plays the important role.

Question: When does the Bessel-parameter γ play a important role ?

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

II(new even for the single eq case): Micro-scaling In the limiting field

• f Part I, only the Riesz parameter α plays the important role.

Question: When does the Bessel-parameter γ play a important role ? For solving this problem, we take a rescaling on space and time variables t → εt, x → ε

1 α+γ x, t > 0.

and replace the initial condition as follows (u0, v0)(x) → (u0, v0)(ε−

1 α+γ −χ · x), x ∈ Rd,

where χ > 0 is an arbitrary constant and the form of u0(·), v0(·) is the same as the previous.

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

In this way, when ε → 0, if m2κ2 > m1κ1 then the finite-dimensional distributions of the rescaled random field [εm1κ1χLm1(ε−χ)]− 1

2

• w(εt, ε

1 α+γ x; w0(ε− 1 α+γ −χ·)) − Q(εt; d1, d2)

• C(1)

C(2) converge weakly to the finite-dimensional distributions of the random field

• X(1)

m1 (t, x)

• where

X(1)

m1 (t, x) =

C(1)

m1

√m1!K(d, κ1)

m1 2

• ′

Rd×m1

ei<x,z1+...+zm1>−µt|z1+...+zm1|α+γ (|z1| · · · |zm1|)

d−κ1 2

m1

• l=1

W1(dzl).

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

Open question Non-linear P.D.E. system with random initial data, such as, the gradient system of a Hamilton-Jacobi equation. Sample function behaviour, such as, the distribution of the hot points.

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

Open question Non-linear P.D.E. system with random initial data, such as, the gradient system of a Hamilton-Jacobi equation. Sample function behaviour, such as, the distribution of the hot points. Liu: stochastic Maxwell Eq’s (connect to A. Fannjiang, UC Davis)

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

Open question Non-linear P.D.E. system with random initial data, such as, the gradient system of a Hamilton-Jacobi equation. Sample function behaviour, such as, the distribution of the hot points. Liu: stochastic Maxwell Eq’s (connect to A. Fannjiang, UC Davis) Shieh: to return the calender....

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

The Exponential Ornstein-Uhlenbeck-type Stationary Processes Multifractality of products of geometric Ornstein-Uhlenbeck type processes. Adv. Appl. Probab. 40, 1129-1156. Joint work with V. Anh and N.N. Leonenko. On the exponentials of fractional Ornstein-Uhlenbeck Processes.

• Elec. J. Probab. 14, Paper no. 23, 594-611. Joint work with M.

Matsui. Simulation of Multifractal Products of Ornstein-Uhlenbeck type

• processes. Nonlinearity 23, 823-843. Joint work with V. Anh,

N.N. Leonenko, and E. Taufer. On the Exponential Process associated with a CARMA-type Process, working paper. Joint work with M. Matsui. The Ornstein-Uhlenbeck transform of fractional Brownian sheets, working project. Joint work with A. Ayache. Complete multifractal analysis for the IP of the exponential OU process (driven by BM), working project. Joint work with P. M¨

• rters.

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic

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