Skew Brownian Motion and Applications in Fluid Dispersion Ed - - PowerPoint PPT Presentation

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Skew Brownian Motion and Applications in Fluid Dispersion Ed - - PowerPoint PPT Presentation

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Skew Brownian Motion and Applications in Fluid Dispersion Ed Waymire Department of Mathematics Oregon State University Corvallis, OR 97331 * Based on joint work with Thilanka Appuhamillage, Vrushali Bokil, Enrique Thomann, Brian Wood, and

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SLIDE 1

Ed Waymire Department of Mathematics Oregon State University Corvallis, OR 97331 *Based on joint work with Thilanka Appuhamillage,

Vrushali Bokil, Enrique Thomann, Brian Wood, and Jorge Ramirez, and supported by a grant from the National Science Foundation.

Skew Brownian Motion and Applications in Fluid Dispersion

Workshop on Computational Methods with Applications in Finance, Insurance and the Life Sciences AND Stochastic Methods in Partial Differential Equations and Applications of Deterministic and Stochastic PDEs, November 17 - 21, 2008

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SLIDE 2

CONCENTRATION EQUATION DIVERGENCE FORM

∂c ∂t = 1 2∇(D∇c) − ∇ · (vc) c(t, y) =

  • Rk c0(x)p(t, x, y)dx

(D(y)∇yc)|I · ny = 0 c(y, 0) = c0(y)

(F-P)

Nash (1958)

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SLIDE 3

CONCENTRATION EQUATION DIVERGENCE FORM (//-interface)

D+ D− x y v → ∂c ∂t = 1 2∇(D∇c) − ∇ · (vc) c(t, y) =

  • Rk c0(x)p(t, x, y)dx

(D(y)∇yc)|I · ny = 0 c(y, 0) = c0(y) I

(F-P)

Nash (1958)

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SLIDE 4

CONCENTRATION EQUATION DIVERGENCE FORM (//-interface) ( -interface )

D+ D− D− D+ x y y x v → ∂c ∂t = 1 2∇(D∇c) − ∇ · (vc) c(t, y) =

  • Rk c0(x)p(t, x, y)dx

(D(y)∇yc)|I · ny = 0 c(y, 0) = c0(y) I I ⊥

(F-P)

Nash (1958)

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SLIDE 5

OUTLINE OF TALK

  • THE CLASSIC TAYLOR-ARIS PROBLEM AND

EXTENSION TO //-INTERACES -- THE ANSWER

  • SKEW BROWNIAN MOTION AND LOCAL TIME -- THE

REASON

  • RELATED STOCHASTIC PARTICLE TRACKING

QUESTIONS

  • ORTHOGONAL INTERFACES
  • (PSTN, LOCAL TIME, OCCUPATION TIME) AND

ELASTIC SKEW BROWNIAN MOTION

  • SOME RELATED FUTURE DIRECTIONS
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SLIDE 6

CLASSICAL TAYLOR-ARIS HOMOGENEOUS ! D

  • -- D

v → D = D + 8(b − a)2v2 945D v = b

a

v(y) dy b − a

No Interface

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SLIDE 7

(NONHOMOGENEOUS) // INTERFACE !

  • -- D

v →

D+

D−

D =?

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SLIDE 8

NONHOMOGENEOUS // INTERFACE !

  • -- D

v →

D+

D−

w/ J. Ramirez, E. Thomann,

  • R. Haggerty, B.Wood

SIAM Multiscale Modeling &Simulation

2006

D = Da + 8v2(b − a)2 945Dh

Da = D+ + D− 2

1 Dh = 1 D+ + 1 D−

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SLIDE 9

Stochastic Particle Motion (//-interface)

D+ D− x y

v →

∂c ∂t = 1 2∇(D∇c) − ∇ · (vc) c(y, 0) = c0(y) X = {(X(t), Y (t)) : t ≥ 0} dX(t) = v(Y (t))dt +

  • D(Y (t)dB1(t)

Nash (1958)

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SLIDE 10

Stochastic Particle Motion (//-interface)

D+ D− x y

v →

∂c ∂t = 1 2∇(D∇c) − ∇ · (vc) c(y, 0) = c0(y) X = {(X(t), Y (t)) : t ≥ 0} dX(t) = v(Y (t))dt +

  • D(Y (t)dB1(t)

α∗ = √ D+ √ D+ + √ D− Bα denotes skew Brownian motion with parameter α Y (t) = f(Bα∗(t))

Nash (1958)

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SLIDE 11

Stochastic Particle Motion (//-interface)

D+ D− x y

v →

∂c ∂t = 1 2∇(D∇c) − ∇ · (vc) c(y, 0) = c0(y) X = {(X(t), Y (t)) : t ≥ 0} dX(t) = v(Y (t))dt +

  • D(Y (t)dB1(t)

α∗ = √ D+ √ D+ + √ D− Bα denotes skew Brownian motion with parameter α dY (t) = D+ − D− D+ + D− d(0, t) +

  • D(Y (t))dB2(t)

Y (t) = f(Bα∗(t))

Ito-Tanaka ⇒

Nash (1958)

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SLIDE 12

Yt t D+ D{

Ito-McKean (1963)

= f(Bα∗(t))

Bα(t) =

∞

  • n=1

1Jn(t)An|B(t)|

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SLIDE 13

Walsh (1978)

Yt t D+ D{

Ito-McKean (1963)

= f(Bα∗(t))

p(α)(y0, y; t) =               

1 √ 2πte− (y−y0)2

2t

+ (2α−1)

√ 2πt e− (y+y0)2

2t

if y0 > 0, y > 0

1 √ 2πte− (y−y0)2

2t

− (2α−1)

√ 2πt e− (y+y0)2

2t

if y0 < 0, y < 0

2α √ 2πte− (y−y0)2

2t

if y0 ≤ 0, y > 0

2(1−α) √ 2πt e− (y−y0)2

2t

if y0 ≥ 0, y < 0.

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SLIDE 14

Walsh (1978)

Yt t D+ D{

Ito-McKean (1963)

= f(Bα∗(t))

p(α)(y0, y; t) =               

1 √ 2πte− (y−y0)2

2t

+ (2α−1)

√ 2πt e− (y+y0)2

2t

if y0 > 0, y > 0

1 √ 2πte− (y−y0)2

2t

− (2α−1)

√ 2πt e− (y+y0)2

2t

if y0 < 0, y < 0

2α √ 2πte− (y−y0)2

2t

if y0 ≤ 0, y > 0

2(1−α) √ 2πt e− (y−y0)2

2t

if y0 ≥ 0, y < 0.

p∗(y0, y; t) =               

1 √ 4πD+t

  • exp
  • − (y−y0)2

4D+t

  • +

√ D+− √ D− √ D−+ √ D+ exp

  • − (y+y0)2

4D+t

  • if y0 > 0, y > 0

1 √ 4πD−t

  • exp
  • − (y−y0)2

4D−t

  • −

√ D+− √ D− √ D−+ √ D+ exp

  • − (y+y0)2

4D−t

  • if y0 < 0, y < 0

1 √ D++ √ D− 1 √ πt exp

  • − (y

√ D−−y0 √ D+)2 4D−D+t

  • if y0 ≤ 0, y > 0

1 √ D++ √ D− 1 √ πt exp

  • − (y

√ D+−y0 √ D−)2 4D−D+t

  • if y0 ≥ 0, y < 0.
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SLIDE 15

COMPUTATION OF EFFECTIVE DISPERSION RATE

X(t) = x0 + t v(Y (s))ds + t

  • D(Y (s))dB(s)
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SLIDE 16

g(y) = U(y) − U, g ∈ Ran(A). ∴ Var 1 √ t t g(Ys)ds

  • =

1 t t t E [g(Ys1)g(Ys2)] ds1 ds2 = 2 t t s E

  • g(Ys−s) E
  • g(Ys)
  • Yu, u s − s

ds ds = 2 t t s E {g(Ys−s) Tsg(Ys−s)} ds ds. − − − − →

t→∞

2 lim

s→∞

s E {g(Ys−s) Tsg(Ys−s)} ds = 2 ∞ g, Tsgπds = 2

  • g,

∞ Tsg ds

  • π

= 2g, hπ.

COMPUTATION OF EFFECTIVE DISPERSION RATE Bhattacharaya (1982)

X(t) = x0 + t v(Y (s))ds + t

  • D(Y (s))dB(s)

g(y) = v(y) − v, g ∈ Ran(A)

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SLIDE 17

Solving Poisson equation: Ah = −g g ∈ Ran(A) = 1⊥, h(y) := ∞ Tsg ds. Ah(y) = lim

t→0

Tth(y) − h(y) t = lim

t→0

1 t ∞ Tt+sg(y) − Tsg(y) ds = ∞ dTsg(y) ds (s)ds = lim

s→0 Tsg(y) − T0g(y)

= Eπg − g(y) = 0 − g(y) = −g(y)

lishers B.V., isterdam, The Physica A 168 (1990) 677-696 North-Holland 1\02139, USA 9 I 405 Orsay, I Avenue, Los Utrecht, The ,TAYLOR DISPERSION REVISITED 12215, USA -
  • C. VAN DEN BROECK*
:shire ~ Park rlington lIer Department of Chemistry, University of California at San Diego, B-040, La Jolla, San Diego, CA 92093, USA Received 21 May 1990 ,vile neiro Jtrecht ¡, Leiden \1urray Hil Review The theory and applications of Taylor dispersion are reviewed. The connection between problems of this type and the theory of stochastic processes and kinetic theory is clarified.
  • 1. Introduction
the currency

In 1954, Taylor (1) investigated, both theoretically and experimentally, the

motion of particles suspended in a fluid in Poiseuille flow through a cylindrical
  • tube. He observed that, while the particles are being carried downstream (that

is, along the tube's axis x) with an average velocity u equal to that of the fluid, they are also being dispersed in a diffusion-like manner, around this average

  • motion. More precisely, he proved that the probability density P(x, t) for
  • bserving a suspended particle at position x at time t, obeys the following
convection-diffusion (or Fokker-Planck) equation, for (asymptotically) large times t: terdam, The ELSEVIER; :ed at a total :ed at a total :ed at a total :ed at a total lumes in all) II, 1000 AE a,p(x, t) + u axP(X, t) = K a;p(x, t) . (1) ace Airlifted idia, Brazil, are available To his own surprise, Taylor found that the effective diffusion coefficient K is inversely proportional to the molecular diffusion coeffcient D of the suspended . i #1

partic es ,

pply missing transmitted in isher, Elsevier
  • 2 2
U a

K = 48D '

(2) ~ article to the considerations j in the Royal ion therewith. ~n for copying pays through 08 of the U.S. ~nter. Inc.. 27 ission to copy yable through as for general Lined from the * Permanent address: L.V.C., B-36lO Diepenbeek, Belgium. #1 In his derivation, Taylor neglectëd the molecular diffusive motion of the suspended particles in the direction of the flow. The correct result for K reads: K= u2a2/48D + D. For particles suspended in fluids, the second term is usually negligible as compared to the first one. 0378-4371/90/$03.50 (Q 1990 - Elsevier Science Publishers BV (North-Holland) negligence or

FINDING h

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SLIDE 18

Hoteit, Mose, Younes, Lehmann, Ackerer (2002)

α∗-EXPERIMENT

PARTICLE TRACKING EXPERIMENTS (MCMC) and THE HMYLA NUMERICAL SCHEME

Q2: SINGLE PARTICLE MOTION ?

(``PARTICLE PATH PROBLEM’’)

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SLIDE 19

σ2

0 = sup x D(x)

∆ > 0

SPATIAL GRID TEMPORAL GRID

pi,i±1 = D(i∆) 2∆2 ± v(i∆) 2∆ pi,i = 1 − D(i∆) ∆2 = ∆2 σ2

PARTICLE TRACKING: Continuous Coefficients

0 < inf

x D(x) ≤

(BD)

  • D. Stroock, S.R.

Varadhan (1997): Chapter 11

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SLIDE 20

∆ > 0

SPATIAL GRID TEMPORAL GRID THEOREM (Harrison-Shepp, 1981, Ann. Probab.)

Let 0 < α < 1 = ∆2 pi,i±1 = 1 2 if i = 0

p0,1 = α = 1 − p1,0

(``Random Walk Approximation’’)

Y (∆)

α

⇒ B(α)

THEN DEFINE

Seol, Y-S (2008)

dY (t) =

  • D(Y (t))dBα∗(t)

f(Y (∆)

α∗ ) ⇒ f(B(α∗))

THUS ( is continuous.)

f

(LeGall 1984, LNM, Springer)

Ramirez, J. (2004)

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SLIDE 21

Stochastic Particle Motion ( -interface)

D+ D− x y

v →

∂c ∂t = 1 2∇(D∇c) − ∇ · (vc) c(y, 0) = c0(y) X = {(X(t), Y (t)) : t ≥ 0} ⊥

dY (t) =

  • D(X(t))dB2(t)

dX(t) = vdt + d(0, t) +

  • D(X(t))dB1(t)
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SLIDE 22

Drift and local time removal transformations yield: where NEXT ADVENTURE: CALCULATE JOINT PDF

c = v2 2 ( 1 D+ − 1 D− )

c((x, y), t) = e−

v2 2D(x) te v D(x) xEx

  • c0(Bα∗(t), y) exp{−cΓα∗

+ (t)}

  • Γα∗

+ (t) = meas{s ≤ t : Bα∗(s) ≥ 0}

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SLIDE 23

Drift and local time removal transformations yield: where NEXT ADVENTURE: CALCULATE JOINT PDF Karatzas and Shreve (1981)

c = v2 2 ( 1 D+ − 1 D− )

ELASTIC BM JOINT LAPLACE TRANSFORM LT

  • INVERSION

α = 1 2

⇒

⇒

c((x, y), t) = e−

v2 2D(x) te v D(x) xEx

  • c0(Bα∗(t), y) exp{−cΓα∗

+ (t)}

  • Γα∗

+ (t) = meas{s ≤ t : Bα∗(s) ≥ 0}

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SLIDE 24

Drift and local time removal transformations yield: where NEXT ADVENTURE: CALCULATE JOINT PDF ELASTIC SKEW BM

Γα∗

+ (t) = meas{s ≤ t : Bα+(s) ≥ 0}

⇒ c = v2 2 ( 1 D+ − 1 D− ) α = 1 2

c((x, y), t) = e−

v2 2D(x) te v D(x) xEx

  • c0(Bα∗(t), y) exp{−cΓα∗

+ (t)}

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SLIDE 25

ELASTIC SKEW BROWNIAN MOTION + FEYNMAN-KAC

⇒

JOINT TRIVARIATE LAPLACE TRANSFORM + SUFFICIENT ALGEBRAIC MIRACLES + LT INVERSION MIRACLES

⇒

JOINT DENSITIES

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SLIDE 26

P(m∞ > t|B) = e−γ(α)(0,t)

Elastic Killing Time:

λu − 1 2u = g αu(0) − (1 − α)u(0−) = γu(0)

EXPLOITS THE DEFINITION:

Bα(t) =

∞

  • n=1

1Jn(t)An|B(t)|

+ Feynman-Kac

⇒

(Skewed-elastic boundary) SkEBVP

u(a) = Ea ∞ e−λte−γα(0,t)g(Bα(t))dt, λ, γ > 0, g ∈ C(−∞, ∞)

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SLIDE 27

E0 ∞ 1[b,∞)(Bα(t))exp {−λt − βΓα

t − γlα(0, t)} dt

= 2α exp

  • −b
  • 2(λ + β)
  • 2(λ + β)
  • γ + (1 − α)

√ 2λ + α

  • 2(λ + β)
  • Solve SkEBVP to obtain LT of joint distribution

⇒

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SLIDE 28

= 2α exp

  • −b
  • 2(λ + β)
  • 2(λ + β)
  • γ + (1 − α)

√ 2λ + α

  • 2(λ + β)
  • ∞

t ∞ e−λt−βτ−γl

  • 2α(1 − α)l

2π(t − τ)3/2τ 1/2 exp

  • −((1 − α)l)2

2(t − τ) − (b + αl)2 2τ

  • dldτdt

Corollaries: Px(Bα

t ∈ dz, Γα + ∈ dr)

Px(Γα

+ ∈ dr)

LT

  • INVERSION:

Px(α(0, t) ∈ dτ, Γα

+(t) ∈ dγ)

etc.

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SLIDE 29

= 2α exp

  • −b
  • 2(λ + β)
  • 2(λ + β)
  • γ + (1 − α)

√ 2λ + α

  • 2(λ + β)
  • ∞

t ∞ e−λt−βτ−γl

  • 2α(1 − α)l

2π(t − τ)3/2τ 1/2 exp

  • −((1 − α)l)2

2(t − τ) − (b + αl)2 2τ

  • dldτdt

Corollaries: Px(Bα

t ∈ dz, Γα + ∈ dr)

Px(Γα

+ ∈ dr)

LT

  • INVERSION:

Px(α(0, t) ∈ dτ, Γα

+(t) ∈ dγ)

c((x, y), t) = e−

v2 2D(x) te v D(x) x

∞

−∞

∞ c0(z, y)e−crPx(Bα∗

t

∈ dz, Γα∗

+ (t) ∈ dr)

etc.

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SLIDE 30

P(B(α)

t

> b, α(0, t) ∈ d, Γ(α)

+ (t) ∈ dτ) =

2α(1 − α) 2π(t − τ)3/2τ 1/2 exp{−(1 − α)22 2(t − τ) −(b + α)2 2τ }ddτ

P(Γ(α)

+ (t) ∈ dτ) =

2α(1 − α)t π(t − τ)1/2τ 1/2[(1 − α)2τ + α2(t − τ)]dτ

Px(Bα

t ∈ db, α(0, t) ∈ d) = 2α( + b)

√ 2πt3 exp

  • −( + b)2

2t

  • dbd,

b, > 0

A FEW ILLUSTRATIVE DENSITY COROLLARIES

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SLIDE 31
  • BREAKTHROUGH CURVES: From the

explicit formula for the concentration one immediately observes the exponential decay rate given by the

  • MCMC + NUMERICAL TESTING
  • TAYLOR-ARIS

APPLICATIONS

D+ ∨ D−

D = D+ + 8(b − a)2v2 945D+

c((x, y), t) = e−

v2 2D(x) te v D(x) x

∞

−∞

∞ c0(z, y)e−crPx(Bα∗

t

∈ dz, Γα∗

+ (t) ∈ dr)

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SLIDE 32

OTHER SKEW DISPERSION GEOMETRIES

D− D+

1

D+

2

(T)

L1

L2

L3

v ↓

(OSU ECOSYSTEMS INFORMATICS)

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SLIDE 33

Additional References

  • M. Barlow, J. Pitman, M.

Yor (1989): LNM, Springer

  • K. Burdzy, Z-Q Chen (2001): Ann.Prob.
  • Barlow, M. Burdzy, H. Kaspi, A. Mandelbaum

(2000): Elect. Comm. Probab.

  • Lejay, A., M. Martinez (2006): AoAP
  • Ramirez, J., E. Thomann, E.Waymire
  • J. Chastenet, B. Wood (2008): WRR
  • Ramirez, J. (2007): OSU Phd Thesis