Skew Brownian Motion and Applications in Fluid Dispersion

Transcription

1 Skew Brownian Motion and Applications in Fluid Dispersion Ed Waymire Department of Mathematics Oregon State University Corvallis, OR *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. 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, 28

2 CONCENTRATION EQUATION DIVERGENCE FORM c t = 1 (D c) (vc) 2 c(y, ) = c (y) (D(y) y c) I n y = c(t, y) = c (x)p(t, x, y)dx R k (F-P) Nash (1958)

3 (//-interface) CONCENTRATION EQUATION DIVERGENCE FORM c t = 1 (D c) (vc) 2 c(y, ) = c (y) (D(y) y c) I n y = c(t, y) = c (x)p(t, x, y)dx R k (F-P) Nash (1958) I y D + D x v

4 (//-interface) CONCENTRATION EQUATION DIVERGENCE FORM c t = 1 (D c) (vc) 2 c(y, ) = c (y) (D(y) y c) I n y = c(t, y) = c (x)p(t, x, y)dx R k (F-P) Nash (1958) I y D + D x v ( -interface ) y I D D + x

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

6 No Interface CLASSICAL TAYLOR-ARIS HOMOGENEOUS! D v --- D D = D + 8(b a)2 v 2 945D v = b a v(y) dy b a

7 (NONHOMOGENEOUS) // INTERFACE! D + v --- D D D =?

8 NONHOMOGENEOUS // INTERFACE! D + v --- D D w/ J. Ramirez, E. Thomann, R. Haggerty, B.Wood SIAM Multiscale Modeling &Simulation 26 D = D a + 8v2 (b a) 2 945D h D a = D+ + D 2 1 = 1 D h D D

9 c t = 1 (D c) (vc) c(y, 2 Nash (1958) Stochastic Particle Motion (//-interface) X = {(X(t), Y (t)) : t } dx(t) = v(y (t))dt + D(Y (t)db 1 (t) ) = c (y) y v D + D x

10 c t = 1 (D c) (vc) c(y, 2 Nash (1958) Stochastic Particle Motion (//-interface) ) = c (y) X = {(X(t), Y (t)) : t } dx(t) = v(y (t))dt + D(Y (t)db 1 (t) D α + Y (t) = f(b α (t)) = D+ + D y v D + D x B α denotes skew Brownian motion with parameter α

11 c t = 1 (D c) (vc) c(y, 2 Nash (1958) Stochastic Particle Motion (//-interface) ) = c (y) X = {(X(t), Y (t)) : t } dx(t) = v(y (t))dt + D(Y (t)db 1 (t) D α + Y (t) = f(b α (t)) = D+ + D y v D + D x B α denotes skew Brownian motion with parameter α Ito-Tanaka dy (t) = D+ D D + + D dl(, t) + D(Y (t))db 2 (t)

12 D + Y t = f(b α (t)) B α (t) = t 1 Jn (t)a n B(t) n=1 Ito-McKean (1963) D {

13 Y t = f(b α (t)) D + Ito-McKean (1963) t D { p (α) (y, y; t) = 1 2πt e (y y )2 2t 1 2πt e (y y )2 2t + (2α 1) 2πt e (y+y )2 2t if y >, y > (2α 1) 2πt e (y+y )2 2t if y <, y < 2α 2πt e (y y )2 2t if y, y > 2(1 α) 2πt e (y y )2 2t if y, y <. Walsh (1978)

14 Y t = f(b α (t)) D + Ito-McKean (1963) D { p (α) (y, y; t) = p (y, y; t) = 1 4πD+ t 1 4πD t 1 2πt e (y y )2 2t 1 2πt e (y y )2 2t t + (2α 1) 2πt e (y+y )2 2t if y >, y > (2α 1) 2πt e (y+y )2 2t if y <, y < 2α 2πt e (y y )2 2t if y, y > 2(1 α) 2πt e (y y )2 2t if y, y <. [ exp [ exp { (y y ) 2 4D + t { (y y ) 2 1 D+ + 1 D πt exp 1 D+ + 1 D πt exp Walsh (1978) } + D+ { }] D D + exp (y+y ) 2 D + 4D + t if y >, y > } D + { }] D D + exp (y+y ) 2 D + 4D t if y <, y < { (y } D y D+ ) 2 4D D + t if y, y > { (y } D + y D ) 2 if y, y <. 4D t 4D D + t

15 COMPUTATION OF EFFECTIVE DISPERSION RATE X(t) = x + t v(y (s))ds + t D(Y (s))db(s)

16 COMPUTATION OF EFFECTIVE DISPERSION RATE X(t) = x + t Bhattacharaya (1982) v(y (s))ds + t g(y) = U(y) U, g Ran(A). D(Y (s))db(s) [ 1 Var t t g(y) = v(y) v, ] g(y s )ds = 1 t t = 2 t = 2 t t t s t s g Ran(A) E [g(y s1 )g(y s2 )] ds 1 ds 2 E { g(y s s ) E [ g(y s ) Yu, u s s ]} ds ds E {g(y s s ) T s g(y s s )} ds ds. t 2 lim s = 2 g, s E {g(y s s ) T s g(y s s )} ds = 2 T s g ds = 2 g, h π. π g, T s g π ds

17 FINDING h Solving Poisson equation: g Ran(A) = 1, h(y) := Ah = g T s g ds. T t h(y) h(y) Ah(y) = lim t t 1 t t = lim = T t+s g(y) T s g(y) ds dt s g(y) ds (s )ds = lim s T s g(y) T g(y) = E π g g(y) = g(y) = g(y)

18 PARTICLE TRACKING EXPERIMENTS (MCMC) and THE HMYLA NUMERICAL SCHEME α -EXPERIMENT Hoteit, Mose, Younes, Lehmann, Ackerer (22) Q2: SINGLE PARTICLE MOTION? (``PARTICLE PATH PROBLEM )

19 PARTICLE TRACKING: Continuous Coefficients < inf x SPATIAL GRID D(x) > σ 2 = sup x D(x) TEMPORAL GRID ɛ = 2 σ 2 (BD) p i,i±1 = D(i )ɛ 2 2 p i,i = 1 D(i )ɛ 2 ± v(i )ɛ 2 D. Stroock, S.R. Varadhan (1997): Chapter 11

20 THEOREM (Harrison-Shepp, 1981, Ann. Probab.) (LeGall 1984, LNM, Springer) (``Random Walk Approximation ) Let < α < 1 DEFINE dy (t) = D(Y (t))db α (t) SPATIAL GRID > TEMPORAL GRID ɛ = 2 p i,i±1 = 1 2 if i p,1 = α = 1 p 1, THEN THUS Y ( ) α B (α) f(y ( ) α ) f(b (α ) ) Ramirez, J. (24) Seol, Y-S (28) ( f is continuous.)

21 c t = 1 (D c) (vc) c(y, 2 ) = c (y) Stochastic Particle Motion ( -interface) y v X = {(X(t), Y (t)) : t } dy (t) = D(X(t))dB 2 (t) D D + x dx(t) = vdt + dl(, t) + D(X(t))dB 1 (t)

22 Drift and local time removal transformations yield: c((x, y), t) = e where { } v2 2D(x) t e v D(x) x E x c (B α (t), y) exp{ cγ α + (t)} c = v2 2 ( 1 D + 1 D ) Γ α + (t) = meas{s t : B α (s) } NEXT ADVENTURE: CALCULATE JOINT PDF

23 Drift and local time removal transformations yield: c((x, y), t) = e where { } v2 2D(x) t e v D(x) x E x c (B α (t), y) exp{ cγ α + (t)} c = v2 2 ( 1 D + 1 D ) Γ α + (t) = meas{s t : B α (s) } NEXT ADVENTURE: CALCULATE JOINT PDF α = 1 2 Karatzas and Shreve (1981) ELASTIC BM JOINT LAPLACE TRANSFORM LT-INVERSION

24 Drift and local time removal transformations yield: c((x, y), t) = e where { } v2 2D(x) t e v D(x) x E x c (B α (t), y) exp{ cγ α + (t)} c = v2 2 ( 1 D + 1 D ) Γ α + (t) = meas{s t : B α+ (s) } NEXT ADVENTURE: CALCULATE JOINT PDF α 1 2 ELASTIC SKEW BM

25 ELASTIC SKEW BROWNIAN MOTION + FEYNMAN-KAC + SUFFICIENT ALGEBRAIC MIRACLES JOINT TRIVARIATE LAPLACE TRANSFORM + LT INVERSION MIRACLES JOINT DENSITIES

26 Elastic Killing Time: P (m > t B) = e γl(α) (,t) u(a) = E a e λt e γlα (,t) g(b α (t))dt, λ, γ >, g C(, ) + Feynman-Kac SkEBVP λu 1 2 u = g αu () (1 α)u ( ) = γu() (Skewed-elastic boundary) EXPLOITS THE DEFINITION: B α (t) = n=1 1 Jn (t)a n B(t)

27 Solve SkEBVP to obtain LT of joint distribution E 1 [b, ) (B α (t))exp { λt βγ α t γl α (, t)} dt = { 2α exp b } 2(λ + β) [ 2(λ + β) γ + (1 α) 2λ + α ] 2(λ + β)

28 LT-INVERSION: t = [ e λt βτ γl 2α(1 α)l 2π(t τ) 3/2 τ 1/2 exp ((1 α)l)2 { 2(t τ) { 2α exp b } 2(λ + β) [ 2(λ + β) γ + (1 α) 2λ + α ] 2(λ + β) (b + αl)2 2τ }] dldτ dt Corollaries: P x (B α t P x (Γ α + dr) dz, Γ α + dr) etc. P x (l α (, t) dτ, Γ α +(t) dγ)

29 LT-INVERSION: t = [ e λt βτ γl 2α(1 α)l 2π(t τ) 3/2 τ 1/2 exp ((1 α)l)2 { 2(t τ) { 2α exp b } 2(λ + β) [ 2(λ + β) γ + (1 α) 2λ + α ] 2(λ + β) (b + αl)2 2τ }] dldτ dt Corollaries: P x (B α t P x (Γ α + dr) c((x, y), t) = e dz, Γ α + dr) etc. v2 2D(x) t e v D(x) x P x (l α (, t) dτ, Γ α +(t) dγ) c (z, y)e cr P x (B α t dz, Γ α + (t) dr)

30 A FEW ILLUSTRATIVE DENSITY COROLLARIES P (B (α) t > b, l α (, t) dl, Γ (α) + (t) dτ) = 2α(1 α)l 2π(t τ) 3/2 τ exp{ (1 α)2 l 2 + αl)2 (b }dldτ 1/2 2(t τ) 2τ P (Γ (α) + (t) dτ) = 2α(1 α)t π(t τ) 1/2 τ 1/2 [(1 α) 2 τ + α 2 (t τ)] dτ P x (B α t db, l α (, t) dl) = 2α(l + b) 2πt 3 exp { (l + b)2 2t } dbdl, b, l >

31 APPLICATIONS c((x, y), t) = e v2 2D(x) t e v D(x) x c (z, y)e cr P x (B α t dz, Γ α + (t) dr) BREAKTHROUGH CURVES: From the explicit formula for the concentration one immediately observes the exponential decay rate given by the D + D MCMC + NUMERICAL TESTING TAYLOR-ARIS D = D + + 8(b a)2 v 2 945D +

32 OTHER SKEW DISPERSION GEOMETRIES L 1 L 2 D + 1 D + 2 L 3 D v (T) (OSU ECOSYSTEMS INFORMATICS)

33 Additional References M. Barlow, J. Pitman, M. Yor (1989): LNM, Springer K. Burdzy, Z-Q Chen (21): Ann.Prob. Barlow, M. Burdzy, H. Kaspi, A. Mandelbaum (2): Elect. Comm. Probab. Lejay, A., M. Martinez (26): AoAP Ramirez, J., E. Thomann, E.Waymire J. Chastenet, B. Wood (28): WRR Ramirez, J. (27): OSU Phd Thesis