Homogenization with stochastic differential equations

Transcription

1 Homogenization with stochastic differential equations Scott Hottovy University of Arizona Program in Applied Mathematics October 12, 2011

2 Modeling with SDE Use SDE to model system (e.g. particle displacesment x t R n at time t in viscous fluid) dx t = b(x t ) dt + σ(x t ) d α W t, x 0 = x. W t, m-dim. Wiener Process, b drift σ noise. This assumes zero correlation. t N σ(x t ) d α W t = lim σ(x t α N i, ω)(w ti W ti 1 ), 0 i=1 t α i = αt i + (1 α)t i 1,

3 Modeling with SDE Use SDE to model system (e.g. particle displacesment x t R n at time t in viscous fluid) dx t = b(x t ) dt + σ(x t ) d α W t, x 0 = x. W t, m-dim. Wiener Process, b drift σ noise. This assumes zero correlation. t N σ(x t ) d α W t = lim σ(x t α N i, ω)(w ti W ti 1 ), 0 i=1 t α i = αt i + (1 α)t i 1, Integral varies with α. Special cases α = 0 Itô, α = 1/2 Stratonovich, α = 1 anti-itô. t W s d α W s, = 1 ( ) 1 2 W2 t 2 α t 0

4 (courtesy Giovanni Volpe) Measure forces Experiment

5 General Model dxt m = vt m dt ( F (x dvt m m = t ) m γ(x t m ) ) m v t m dt + σ(x m t ) m dw t. For σ, γ positive and Lipschitz, then use property of stochastic integral: For f smooth (E[ f (s, ω) f (t, ω) 2 ] k s t 1+ɛ ), t 0 f (s, ω) d α W s = t 0 f (s, ω) dw s, for all α.

6 Approximated by the Smoluchowski-Kramers approximation as m 0, dx t = F (x t) γ(x t ) dt + σ(x t) γ(x t ) d αw t, Lost smoothness of x t. Stochastic integral varies with α

7 Approximated by the Smoluchowski-Kramers approximation as m 0, dx t = F (x t) γ(x t ) dt + σ(x t) γ(x t ) d αw t, Lost smoothness of x t. Stochastic integral varies with α [ F (xt ) dx t = γ(x t ) + ασ(x t) d γ(x t ) dx t ( )] σ(xt ) γ(x t ) dt + σ(x t) γ(x t ) dw t, The second drift term is called spurious (or noise induced) drift. What is α?

8 Connection to PDE dxt m = vt m dt mdvt m = F (xt m ) γ(xt m )v t dt + σ(xt m ) dw t, Density p m (x, v, t x, v, t) satisfies the backward Kolmogorov equation p m t = σ(x)2 2m = L x,v p m, 2 p m v 2 + v p m x + ( (F (x) γ(x)v) m ) pm v Also, for all (x, v ) R 2, p m satisfies the forward Kolmogorov (or Fokker-Planck) equation p m t = L x,v p m.

9 Homogenization Homogenization theory studies the effects of high-frequency oscillations in the coefficients upon solutions of PDE. [Evans 98]

10 Homogenization Homogenization theory studies the effects of high-frequency oscillations in the coefficients upon solutions of PDE. [Evans 98] Ex: Conductor composed of several materials with different conductivities.

11 Conductivity is a periodic function a(x). The temperature, u ɛ satisfies, Lu(x) = u ɛ (x) = 0, n i,j=1 ( ( x ) ) a ij ux ɛ ɛ i (x) = f (x) in U, x j in U. Assume the uniform ellipticity condition n a ij (y)ξ i ξ j θ ξ 2, i,j=1 for constant θ > 0, and all y, ξ R n.

12 Two length scales, macro O(1), and micro O(ɛ). Assume solution u ɛ (x) u(x) as ɛ 0. Goal is to find the PDE that u(x) satisfies. u ɛ (x) = u 0 (x, x/ɛ) + ɛu 1 (x, x/ɛ) + ɛ 2 (x, x/ɛ) +... u : U Q R, where Q is the unit cube in R n. Plug this solution into the original PDE

13 result Lu ɛ = 1 ɛ 2 L 1u ɛ (L 1u 1 +L 2 u 0 )+(L 1 u 2 +L 2 u 1 +L 3 u 0 )+O(ɛ) = f. Then Solve the system of PDE O ( ) 1 ɛ 2 L1 u 0 = 0, O ( ) 1 ɛ L 1 u 1 + L 2 u 0 = 0, O (1) L 1 u 2 + L2u 1 + L 3 u 0 = f,

14 Colored Noise Example For a physical process x t that satisfies dx t = b(x t ) + σ(x t ) dw t, x t is continuous, but not differentiable.

15 Colored Noise Example For a physical process x t that satisfies dx t = b(x t ) + σ(x t ) dw t, x t is continuous, but not differentiable. Physically more realistic to replace W t by differentiable process ξt ɛ = 1 ( ) (t s) k ɛ ɛ 2 W s ds, for smooth kernel k. ξ ɛ t is called colored noise. Now xt ɛ is smooth (dependent on smoothness of k) and satisfies ( ) dxt ɛ = b(xt ɛ ) + σ(xt ɛ ) dξɛ t dt dt

16 Since ξt ɛ W t expect ( ) dxt ɛ = b(xt ɛ ) + σ(xt ɛ ) dξɛ t dt dt

17 Since ξ ɛ t W t expect dx t = b(x t ) dt + σ(x t ) dw t

18 Since ξ ɛ t W t expect dx t = b(x t ) dt + σ(x t ) dw t Wong, Zakai (1965) The solution to dx ɛ t = converges to the solution of ( ) b(xt ɛ ) + σ(xt ɛ ) dξɛ t dt dt dx t = b(x t ) dt + σ(x t )d α=1/2 W t = b(x t ) σ(x t) dσ(x t) dt + σ(x t )d 0 W t. dx

19 Colored Noise Approximation Ornstein-Ulhenbeck SP to model colored noise dy t = ay t 2a ɛ 2 dt + ɛ 2 dw t, y 0 = 0.

20 Colored Noise Approximation Ornstein-Ulhenbeck SP to model colored noise dy t = ay t 2a ɛ 2 dt + ɛ 2 dw t, y 0 = 0. 2a t { a } y t = ɛ 2 exp 0 ɛ 2 (s t) dw (s) Mean zero Gaussian random variable with covariance { E[y t1 y t2 ] = exp a } ɛ 2 t 1 t 2.

21 Use homogenization theory to derive Wong and Zakai result. The generator for (x ɛ t, y t ) { dx ɛ t = b(xt ɛ ) + σ(xɛ t )yt ɛ dt dy t = ayt 2a dt + dw ɛ 2 ɛ 2 t, L ɛ = b(x) x + σ(x)y ɛ = L ɛ L ɛ 2 L ỹ. x ay ɛ 2 y + a 2 ɛ 2 y 2 Ansatz: Solution to the backward Kolmogorov equation p m (x, y) = p 0 (x, y) + ɛp 1 (x, y) + ɛ 2 p 2 (x, y) +...

22 BK equation p 0 t = (L ỹ p 2 + L 1 p 1 + L 2 p 0 ) + 1 ɛ 2 L ỹ p ɛ (L ỹ p 1 + L 1 p 0 )

23 BK equation p 0 t = (L ỹ p 2 + L 1 p 1 + L 2 p 0 ) + 1 ɛ 2 L ỹ p ɛ (L ỹ p 1 + L 1 p 0 ) O(ɛ 2 ), ay p 0 (x, y) ɛ 2 + a 2 p 0 (x, y) y ɛ 2 y 2 = 0. Only solution is p 0 (x, y) = p 0 (x).

24 BK equation p 0 t = (L ỹ p 2 + L 1 p 1 + L 2 p 0 ) + 1 ɛ 2 L ỹ p ɛ (L ỹ p 1 + L 1 p 0 ) O(ɛ 1 ), ay p 1 (x, y) ɛ 2 + a 2 p 1 (x, y) y ɛ 2 y 2 = σ(x)y p 0(x) x. Set p 1 (x, y) = Φ(x, y) p 0 x + Φ 1(x). (Cell Problem) Set Φ 1 (x) = 0.. Φ(x, y) = 1 a σ(x)y

25 Cell problem always solvable? O(1) p 0 t = (L ỹ p 2 + L 1 p 1 + L 2 p 0 )

26 Cell problem always solvable? O(1) Lỹ p 2 = p ( 0 t + L 1 Φ(x, y) p ) 0 + L 2 p 0 x Use the Fredholm alternative to find a solvability condition. The r.h.s. integrated against the solution of, must hold. L ỹ ρ (x, y) = 0

27 This is the stationary Fokker-Planck (forward Kolmogorov equation) for the OU process,, Stationary density, Solvability condition, dỹ t = aỹ t dt + 2adW t ρ (y) = 1 2π exp ( y 2 2 ). ( p ( 0 t + L 1 Φ(x, y) p ) ) 0 + L 2 p 0 ρ (y) dy = 0 x

28 Diffusion process, ( p 0 t = b(x) + 1 ) p0 a σ (x)σ(x) x + 1 p 0 a σ(x) 2 x 2, Yields the SDE, dx t = ( b(x t ) + 1 ) a σ (x t )σ(x t ) dt + 2 a σ(x) d W t x ɛ t x t in distribution, as ɛ 0 (with much more work).

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30 For the Newton equation with u t = mv t, dx t = u t m dt du t = F (x t) m γ(x t)v t m dt + σ(x t) m dw t.

31 For the Newton equation with u t = mv t, dx t = u t m dt Solve the system for p 0 O ( ) 1 ( m O 1 O(1) du t = F (x t) m γ(x t)v t m dt + σ(x t) m dw t. m ) ( σ(x) 2 ) Lũp 0 = 2 2 γ(x)u u 2 u p 0 = 0, Lũp 1 = L 1 p 0 = ( u x + F (x) ) u p0, p 0 t = Lũp 2 + L 1 p 1,

32 For Fredholm alternative arg. need solution to the stationary FP of, dũ t = γ(x)ũ t dt + σ(x) dw t. Need σ, γ independent of u. OU process with parameter x. Solution is a Gaussian density. { } γ(x)u 2 ρ(u; x) = C(x) exp σ(x) 2.

33 the BK equation satisfied by p 0. σ(x) 2 2 ( ) p 0 F (x) 2γ(x) 2 x 2 + γ(x) σ(x)2 d(γ(x)) p0 2γ(x) 3 dx x = p 0 t. Compare to the PDE of SK σ(x) 2 2 ( [ p F (x) 2γ(x) 2 x 2 + γ(x) + α σ(x) ]) γ(x) 3 γ (x) σ(x)2 p γ(x) 2 σ (x) x = p t.

34 the BK equation satisfied by p 0. σ(x) 2 2 ( ) p 0 F (x) 2γ(x) 2 x 2 + γ(x) σ(x)2 d(γ(x)) p0 2γ(x) 3 dx x = p 0 t. Compare to the PDE of SK σ(x) 2 2 ( [ p F (x) 2γ(x) 2 x 2 + γ(x) + α σ(x) ]) γ(x) 3 γ (x) σ(x)2 p γ(x) 2 σ (x) x = p t. Effective SDE for x t, ( F (xt ) dx t = γ(x t ) σ(x t) 2 ) 2γ(x t ) 3 γ (x t ) dt + σ(x t) γ(x t ) d W t, W t is an arbitrary Wiener process. (From studying the inf. operator)

35 α = α(x) = α constant iff γ(x) = cσ(x) λ. Equation for α γ (x)σ(x) 2(γ (x)σ(x) γ(x)σ (x)). λ α = 2(λ 1) λ = α α 1 2

36 α vs λ

37 α vs λ Next Austin studies SDE systems with colored noise and delay.