Homogenization with stochastic differential equationsshottovy/presentationMP11F.pdfModeling with SDE...

Homogenization with stochastic differential equations

Scott [email protected]

University of Arizona Program in Applied Mathematics

October 12, 2011

Modeling with SDE

• Use SDE to model system (e.g. particle displacesmentxt ∈ Rn at time t in viscous fluid)

dxt = b(xt) dt + σ(xt) dαWt , x0 = x.

• Wt , m-dim. Wiener Process, b drift σ noise. This assumeszero correlation.∫ t

0σ(xt) dαWt = lim

N→∞

N∑i=1

σ(xtαi , ω)(Wti −Wti−1),

tαi = αti + (1− α)ti−1,

• Integral varies with α. Special cases α = 0 Ito, α = 1/2Stratonovich, α = 1 anti-Ito.∫ t

0Ws dαWs ,=

1

2W2

t −(

1

2− α

)t

Experiment

(courtesy Giovanni Volpe)Measure forces

General Model

dxmt = vmt dt

dvmt =

(F (xmt )

m− γ(xmt )

mvmt

)dt +

σ(xmt )

mdWt .

• For σ, γ positive and Lipschitz, then use property ofstochastic integral:

• For f smooth (E [|f (s, ω)− f (t, ω)|2] ≤ k |s − t|1+ε),∫ t

0f (s, ω) dαWs =

∫ t

0f (s, ω) dWs , for all α.

• Approximated by the Smoluchowski-Kramers approximation asm→ 0,

dxt =F (xt)

γ(xt)dt +

σ(xt)

γ(xt)dαWt ,

• Lost smoothness of xt . Stochastic integral varies with α

dxt =

[F (xt)

γ(xt)+ α

σ(xt)

γ(xt)

d

dxt

(σ(xt)

γ(xt)

)]dt +

σ(xt)

γ(xt)dWt ,

• The second drift term is called spurious (or noise induced)drift.

• What is α?

Connection to PDE

dxmt = vmt dt

mdvmt = F (xmt )− γ(xmt )vt dt + σ(xmt ) dWt ,

Density pm(x ′, v ′, t ′|x , v , t) satisfies the backward Kolmogorovequation

∂pm∂t

=σ(x)2

2m

∂2pm∂v2

+ v∂pm∂x

+

((F (x)− γ(x)v)

m

)∂pm∂v

= Lx ,vpm,

Also, for all (x ′, v ′) ∈ R2, pm satisfies the forward Kolmogorov (orFokker-Planck) equation

∂pm∂t ′

= L∗x ′,v ′pm.

Homogenization

• “Homogenization theory studies the effects of high-frequencyoscillations in the coefficients upon solutions of PDE.” [Evans98]

• Ex: Conductor composed of several materials with differentconductivities.

http://www.sciencephoto.com/media/8839

• Conductivity is a periodic function a(x).The temperature, uε satisfies,

Lu(x) = −n∑

i ,j=1

(aij

(xε

)uεxi (x)

)xj

= f (x) in U,

uε(x) = 0, in ∂U.

• Assume the uniform ellipticity condition

n∑i ,j=1

aij(y)ξiξj ≥ θ|ξ|2,

for constant θ > 0, and all y , ξ ∈ Rn.

• Two length scales, macro O(1), and micro O(ε).

• Assume solution uε(x)→ u(x) as ε→ 0.

• Goal is to find the PDE that u(x) satisfies.

uε(x) = u0(x , x/ε) + εu1(x , x/ε) + ε2(x , x/ε) + ...

• u : U × Q → R, where Q is the unit cube in Rn.

• Plug this solution into the original PDE

• result

Luε =1

ε2L1u0+

1

ε(L1u1+L2u0)+(L1u2+L2u1+L3u0)+O(ε) = f .

• Then Solve the system of PDEO(1ε2

)L1u0 = 0,

O(1ε

)L1u1 + L2u0 = 0,

O (1) L1u2 + L2u1 + L3u0 = f ,

Colored Noise Example

• For a physical process xt that satisfies

dxt = b(xt) + σ(xt) dWt ,

• xt is continuous, but not differentiable.

• Physically more realistic to replace Wt by differentiableprocess

ξεt =1

ε

∫k

((t − s)

ε2

)Ws ds,

• for smooth kernel k . ξεt is called colored noise.

• Now xεt is smooth (dependent on smoothness of k) andsatisfies

dxεt =

(b(xεt ) + σ(xεt )

dξεtdt

)dt

• Since ξεt →Wt expect

dxεt =


dξεtdt

)dt

• Since ξεt →Wt expect

dxt = b(xt) dt + σ(xt) dWt

Wong, Zakai (1965)

The solution to

dxεt =


dξεtdt

)dt

converges to the solution of

dxt = b(xt) dt + σ(xt)dα=1/2Wt

= b(xt) +1

2σ(xt)

dσ(xt)

dxdt + σ(xt)d0Wt .

Colored Noise Approximation

• Ornstein-Ulhenbeck SP to model colored noise

dyt = −aytε2

dt +

√2a

ε2dWt , y0 = 0.

yt =

√2a

ε2

∫ t

0exp

{ a

ε2(s − t)

}dW (s)

• Mean zero Gaussian random variable with covariance

E [yt1yt2 ] = exp{− a

ε2|t1 − t2|

}.

• Use homogenization theory to derive Wong and Zakai result.{dxεt = b(xεt ) + σ(xεt )yt

ε dt

dyt = −aytε2

dt +√

2aε2dWt ,

• The generator for (xεt , yt)

Lε = b(x)∂

∂x+σ(x)y

ε

∂

∂x− ay

ε2∂

∂y+

a

ε2∂2

∂y2

= L2 +1

εL1 +

1

ε2Ly .

• Ansatz: Solution to the backward Kolmogorov equationpm(x , y) = p0(x , y) + εp1(x , y) + ε2p2(x , y) + ....

• BK equation

∂p0∂t

= (Lyp2 + L1p1 + L2p0) +1

ε2Lyp0 +

1

ε(Lyp1 + L1p0)

• O(ε−2),

−ay

ε2∂p0(x , y)

∂y+

a

ε2∂2p0(x , y)

∂y2= 0

.Only solution is p0(x , y) = p0(x).

• BK equation

∂p0∂t

= (Lyp2 + L1p1 + L2p0) +1

ε2Lyp0 +

1

ε(Lyp1 + L1p0)

• O(ε−1),

−ay

ε2∂p1(x , y)

∂y+

a

ε2∂2p1(x , y)

∂y2= −σ(x)y

∂p0(x)

∂x.Set p1(x , y) = Φ(x , y)∂p0∂x + Φ1(x). (Cell Problem)

• Set Φ1(x) = 0.

Φ(x , y) =1

aσ(x)y

.

• Cell problem always solvable?

• O(1)

∂p0∂t

= (Lyp2 + L1p1 + L2p0)

• Cell problem always solvable?

• O(1)

−Lyp2 = −∂p0∂t

+ L1

(Φ(x , y)

∂p0∂x

)+ L2p0

• Use the Fredholm alternative to find a solvability condition.

• The r.h.s. integrated against the solution of

L∗yρ∞(x , y) = 0

,must hold.

• This is the stationary Fokker-Planck (forward Kolmogorovequation) for the OU process,

dyt = −ayt dt +√

2adWt

,

• Stationary density,

ρ∞(y) =1

2πexp

(−y2

2

).

• Solvability condition,

∫ ∞−∞

(−∂p0∂t

+ L1

(Φ(x , y)

∂p0∂x

)+ L2p0

)ρ∞(y) dy = 0

• Diffusion process,

∂p0∂t

=

(b(x) +

1

aσ′(x)σ(x)

)∂p0∂x

+1

aσ(x)

∂2p0∂x2

,

• Yields the SDE,

dxt =

(b(xt) +

1

aσ′(xt)σ(xt)

)dt +

√2

aσ(x) dWt

• xεt → xt in distribution, as ε→ 0 (with much more work).

• For the Newton equation with ut =√mvt ,

dxt =ut√m

dt

dut =F (xt)√

m− γ(xt)vt

mdt +

σ(xt)√m

dWt .

• Solve the system for p0

O(1m

)Lup0 =

(σ(x)2

2∂2

∂u2− γ(x)u ∂

∂u

)p0 = 0,

O(

1√m

)Lup1 = −L1p0 = −

(u ∂∂x + F (x) ∂

∂u

)p0,

O(1) ∂p0∂t = Lup2 + L1p1,

• For Fredholm alternative arg. need solution to the stationaryFP of,

dut = −γ(x)ut dt + σ(x) dWt .

• Need σ, γ independent of u.

• OU process with parameter x .

• Solution is a Gaussian density.

ρ(u; x) = C (x) exp

{−γ(x)u2

σ(x)2

}.

• the BK equation satisfied by p0.

σ(x)2

2γ(x)2∂2p0∂x2

+

(F (x)

γ(x)− σ(x)2

2γ(x)3d(γ(x))

dx

)∂p0∂x

=∂p0∂t

.

• Compare to the PDE of SK

σ(x)2

2γ(x)2∂2p

∂x2+

(F (x)

γ(x)+ α

[− σ(x)

γ(x)3γ′(x)− σ(x)2

γ(x)2σ′(x)

])∂p

∂x=∂p

∂t.

• Effective SDE for xt ,

dxt =

(F (xt)

γ(xt)− σ(xt)

2

2γ(xt)3γ′(xt)

)dt +

σ(xt)

γ(xt)dWt ,

• Wt is an arbitrary Wiener process. (From studying the inf.operator)

Equation for α

α = α(x) =γ′(x)σ(x)

2(γ′(x)σ(x)− γ(x)σ′(x)).

α constant iff γ(x) = cσ(x)λ.

α =λ

2(λ− 1)

λ =α

α− 12

α vs λ

• Next Austin studies SDE systems with colored noise and delay.

Homogenization with stochastic differential equationsshottovy/presentationMP11F.pdfModeling with SDE...

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