Page 1

arXiv:1401.3806v1 [math.PR] 16 Jan 2014

Homogenization of Heat Equation with

Large Time-dependent Random Potential

Yu Gu∗

Guillaume Bal∗

January 17, 2014

Abstract

This paper concerns the homogenization problem of heat equation with large, time-dependent,

random potentials in high dimensions d ≥ 3. Depending on the competition between temporal

and spatial mixing of the randomness, the homogenization procedure turns to be different. We

characterize the difference by proving the corresponding weak convergence of Brownian motion

in random scenery. When the potential depends on the spatial variable macroscopically, we

prove a convergence to SPDE.

1Introduction

Small scales abound in equations of physical importance, where homogenization has become pop-

ular to analyze the asymptotics and reduce the complexity. For parabolic equation with random

coefficients, when the solution can be expressed as the average with respect to certain diffusion

process, homogenization may be recast as a problem of weak convergence of random motion in

random environment, see a good introduction in [12].

The equation we consider in this paper is of the form ∂tuε(t,x) = ∆uε(t,x) + iVε(t,x)uε(t,x).

The size of Vεis chosen large enough to produce some non-trivial effects on the asymptotic limit.

The imaginary unit brings stability, and saves the effort of controlling the unbounded exponential

function. Similar types of equations, including the ones with time-independent or real potentials,

are analyzed in [1, 2, 3, 14, 15, 8, 6], using analytic and probabilistic methods, see a review in [4].

The method we use here is probabilistic, i.e., a Feynman-Kac representation and weak convergence

approach.

By a Feynman-Kac formula, a key object to analyze is the so-called Brownian motion in random

scenery, i.e., the random process of the form?t

change of variables, we observe a threshold of α = 2, which separates the effects of the random

mixings generated by temporal and spatial variables. In other words, depending on whether α > 2

or α < 2, the averaging of

?t

(yg2254@columbia.edu; gb2030@columbia.edu)

0Vε(s,Bs)ds, which corresponds to Kesten’s model of

random walk in random scenery in the discrete setting [10]. If Vε(t,x) ∼ V (t/εα,x/ε), by a simple

0Vε(s,Bs)ds is induced by the temporal or spatial mixing of V ,

∗Department of Applied Physics & Applied Mathematics, Columbia University, New York, NY 10027

1

Page 2

respectively. As a result, the ways we prove weak convergence of

different correspondingly. When α > 2, it is a standard proof of central limit theorem for functions

of mixing processes [5, Chapter 4] after freezing the Brownian motion. When α ≤ 2, the spatial

mixing dominates. We make use of the Brownian motion by the Kipnis-Varadhan’s method [11],

i.e., constructing the corrector function and applying martingale decomposition. When α = 2, a

ergodicity suffices to pass to the limit; when α < 2, a quantitative martingale central limit theorem

[13] is applied. As α → ∞, the spatial mixing tends to zero, so heuristically the Brownian motion

remains in the weak convergence limit of?t

The rest of the paper is organized as follows. In Section 2, we introduce the problem setup and

present the main results. Section 3, 4 and 5 are devoted to the cases α ∈ (2,∞), α ∈ [0,2] and

α = ∞, respectively. Technical lemmas are left in the Appendix.

Here are notations used thoughout the paper. Since we have two independent random sources,

i.e., the random potential from the equation and the Brownian motion induced by the Feynman-Kac

formula, we are in the product probability space. E is used to denote the expectation with respect

to the random environment, and EBthe expectation with respect to the Brownian motion. Joint

expectation is denoted by EEB. We use a ∧ b = min(a,b) and a ∨ b = max(a,b). N(µ,σ2) denotes

the normal distribution with mean µ and variance σ2, and qt(x) is the density of N(0,t). We use

Idto denote the d × d identity matrix. a ? b stands for a ≤ Cb for some constant C independent

of ε.

?t

0Vε(s,Bs)ds are forced to be

0Vε(s,Bs)ds, leading to a stochastic equation.

2Problem setup and main results

We are interested in equations of the form

∂tuε(t,x) =1

2∆uε(t,x) + i1

εδV (t

εα,x

ε)uε(t,x),(2.1)

where α,δ > 0, the dimension d ≥ 3, and V (t,x) is a mean-zero, time-dependent, stationary

random potential. The initial condition uε(0,x) = f(x) ∈ Cb(Rd). Without lose of generality, we

assume f taking values in R. For fixed α, δ is chosen so that the large, highly oscillatory, random

potential generates non-trivial effects on uεas ε → 0. It turns out that for different values of α,

the asymptotic effects may be different, and the ways we prove convergence are different as well.

Before presenting the main result, we make some assumptions on V (t,x).

Let (Ω,F,P) be a random medium associated with a group of measure-preserving, ergodic

transformations {τ(t,x),t ∈ R,x ∈ Rd}. Let V ∈ L2(Ω) with?

By defining T(t,x)on L2(Ω) as T(t,x)f(ω) = f(τ(t,x)ω) and assuming it is strongly continuous in

L2(Ω), we obtain the spectral resolution

ΩV(ω)P(dω) = 0. Define V (t,x,ω) =

V(τ(t,x)ω). The inner product and norm of L2(Ω) are denoted by ?.,.? and ?.?, respectively.

T(t,x)=

?

Rdeiξ0t+iξ·xU(dξ0,dξ),(2.2)

where ξ = (ξ1,...,ξd) and U(dξ0,dξ) is the associated projection valued measure. Let {Dk,k =

0,...,d} be the L2(Ω) generator of T(t,x).

2

Page 3

LetˆR(ξ0,ξ) be the power spectrum associated with V, i.e.,ˆR(ξ0,ξ)dξ0dξ = (2π)d+1?U(dξ0,dξ)V,V?,

we obtain

R(t,x) = E{V (t,x)V (0,0)} = ?T(t,x)V,V? =

1

(2π)d+1

?

Rd+1eiξ0t+iξ·xˆR(ξ0,ξ)dξ0dξ.(2.3)

For any set S ⊆ Rd+1, define the σ−algebras: FS= σ(V (s,x) : (s,x) ∈ S), and we assume the

mixing property of V as follows.

Assumption 2.1 (Mixing property). V is uniformly bounded. There exists a function ϕ(r) :

[0,∞) → [0,∞) such that ∀n > 0, ϕ(r) ≤ Cn(1 ∧ r−n) for some Cn, and

sup

A∈FS1,B∈FS2,P(B)>0,dist(S1,S2)≥r

|P(A|B) − P(A)| ≤ ϕ(r),(2.4)

where dist(S1,S2) is the Euclidean distance between S1,S2.

In the following, we denote

σ(V (s,x) : s ≤ t,x ∈ Rd) = Ft,

σ(V (s,x) : s ≥ t,x ∈ Rd) = Ft.

(2.5)

(2.6)

By [5, Page 170, Lemma 1], the above mixing property implies that

|E{XY } − E{X}E{Y }| ≤ 2ϕ

1

2(r)?E{X2}E{Y2}?1

2,(2.7)

if X is FS1−measurable and Y is FS2−measurable with dist(S1,S2) ≥ r. For example, we have

|R(t,x)| ? 1 ∧ (|t|2+ |x|2)−nfor any n > 0.

The following is the main result.

Theorem 2.2 (α ∈ [0,∞): homogenization). Under Assumption 2.1, let

∂tuε(t,x)=

1

2∆uε(t,x) + i

1

2∆u0(t,x) − ρ(α)u0(t,x),

1

α

ε

2∨1V (t

εα,x

ε)uε(t,x),(2.8)

∂tu0(t,x)=

(2.9)

with initial condition uε(0,x) = u0(0,x) = f(x) and

ρ(α) =

?∞

?∞

0R(t,0)dt

?∞

α ∈ (2,∞),

α = 2,

α ∈ [0,2).

0EB{R(t,Bt)}dt

0EB{R(0,Bt)}dt

(2.10)

Then uε(t,x) → u0(t,x) in probability as ε → 0.

When α > 2, by a change of parameter εα?→ ε, we have

corresponds to the case when V has no micro-structure in the spatial variable, and the potential

is of the form

ε,x). We obtain a transition from homogenization to convergence to SPDE in

the following theorem.

1

α

ε

2V (t

εα,x

ε) ?→

1

√εV (t

ε,

x

1

ε

α), so α = ∞

1

√εV (t

3

Page 4

Theorem 2.3 (α = ∞: convergence to SPDE). Under Assumption 2.1, let

∂tuε(t,x)=

1

2∆uε(t,x) + i1

1

2∆u0(t,x) + i˙W(t,x) ◦ u0(t,x),

√εV (t

ε,x)uε(t,x), (2.11)

∂tu0(t,x)=

(2.12)

with initial condition uε(0,x) = u0(0,x) = f(x) and Gaussian noise˙W(t,x) of covariance structure

E{˙W(t,x)˙W(s,y)} = δ(t − s)?

Remark 2.4. The product ◦ in the limiting SPDE is in the Stratonovich’s sense.

The solution to (2.1) is written by Feynman-Kac formula as

RR(t,x − y)dt. Then uε(t,x) ⇒ u0(t,x) in distribution as ε → 0.

uε(t,x) = EB{f(x + Bt)exp(i1

εδ

?t

0

V (t − s

εα,x + Bs

ε

)ds)}. (2.13)

Since Theorem 2.2 and 2.3 are both results for fixed (t,x), by stationarity of V , uε(t,x) has the

same distribution as

˜ uε(t,x) =EB{f(x + Bt)exp(i1

=EB{f(x + Bt)exp(i1

εδ

?t

?t

0

V (−s

εα,x + Bs

˜V (s

ε

)ds)}

εδ

0

εα,x + Bs

ε

)ds)},

(2.14)

where˜V (s,x) := V (−s,x). Since˜V and V has the same covariance function and mixing property

we need, from now on we will write our solution to (2.1) for simplicity as

uε(t,x) = EB{f(x + Bt)exp(i1

εδ

?t

0

V (s

εα,x + Bs

ε

)ds)}.(2.15)

2.1 Remarks on low dimensional cases

The case d = 1 with a real potential is addressed in [15, 8] using probabilistic and analytic ap-

proaches respectively. Their result shows that α ∈ (0,∞) leads to homogenization while α = 0,∞

leads to SPDE. When α > 2, our proof is similar to that in [15]. When α ∈ [0,2], we follow the

approach in [6], which relies on the fact d ≥ 3.

For d = 2, while the case of α > 2 can be analyzed in the same way, our approach for α ∈

[0,2] does not necessarily work. For the time-independent potential, weak convergence results are

obtained in [16, 7].

3

α ∈ (2,∞): temporal mixing and homogenization

When α > 2, δ =α

2. By Feynman-Kac formula, the solution is written as

uε(t,x) = EB{f(x + Bt)exp(iε−α/2

?t

0

V (s/εα,(x + Bs)/ε)ds)}. (3.1)

4

Page 5

By the scaling property of Brownian motion, stationarity of V , and a change of parameter εα?→ ε2,

we have

uε(t,x) = EB{f(x + εBt/ε2)exp(iε

?t/ε2

0

V (s,εβBs)ds)},(3.2)

with β = 1−2

is small.

α∈ (0,1). Since β > 0, the spatial mixing from V for the process ε?t/ε2

?

probability with u0satisfying

0

V (s,εβBs)ds

Let σ =

2?∞

0R(t,0)dt.The goal in this section is to prove that uε(t,x) → u0(t,x) in

∂tu0(t,x) =1

2∆u0(t,x) −1

2σ2u0(t,x).(3.3)

The result comes from the following two propositions.

Proposition 3.1.

(εBt/ε2,ε

?t/ε2

0

V (s,εβBs)ds) ⇒ (N1

t,σN2

t),

where N1

sense.

t∼ N(0,tId), independent from N2

t∼ N(0,t). The weak convergence ⇒ is in the annealed

Proposition 3.2. For independent Brownian motions B1

t,B2

t,

(εB1

t/ε2,εB2

t/ε2,ε

?t/ε2

0

V (s,εβB1

s)ds − ε

?t/ε2

0

V (s,εβB2

s)ds) ⇒ (N1

t,N2

t,√2σN3

t),

where N1

the annealed sense.

t,N2

t∼ N(0,tId), N3

t∼ N(0,t), and they are independent. The weak convergence ⇒ is in

By Proposition 3.1, we have E{uε(t,x)} → u0(t,x); by Proposition 3.2, we have E{|uε(t,x)|2} →

|u0(t,x)|2. So uε(t,x) → u0(t,x) in L2(Ω).

We only prove Proposition 3.2. The proof of Proposition 3.1 is similar with some simplifications.

Proof of Proposition 3.2. The goal is to show that for any a,b ∈ Rd,c ∈ R, as ε → 0

t/ε2+ic(ε?t/ε2

0

EEB{eia·εB1

t/ε2+ib·εB2

V (s,εβB1

s)ds−ε?t/ε2

0

V (s,εβB2

s)ds)} → e−1

2|a|2t−1

2|b|2t−c2σ2t.(3.4)

We first consider the average with respect to the random environment. Let

Xε(t) = ε

?t/ε2

0

V (s,εβB1

s)ds − ε

?t/ε2

0

V (s,εβB2

s)ds,

∆t = ε−γ1+ε−γ2, 0 < γ2< γ1< 2 to be determined, and N = [

t

ε2∆t] ∼ tεγ1−2. Define the intervals

5