Stochastic perturbations of the Allen–Cahn equation ∗

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Stochastic perturbations of the Allen–Cahn equation ^∗

Tony Shardlow

Abstract

Consider the Allen-Cahn equation with small diffusion ² perturbed by a space time white noise of intensityσ. In the limit, σ/² → 0, solutions converge to the noise free problem in theL2 norm. Under these conditions, asymptotic results for the evolution of phase boundaries in the deterministic setting are extended, to describe the behaviour of the stochastic Allen-Cahn PDE by a system of stochastic differential equations. Computations are described, which support the asymptotic derivation.

1 Introduction

Consider the Itˆo stochastic partial differential equation du=

h

²u_xx+f(u) i

dt+σ dW(t), u_x=0 atx= 0,1, u=g att= 0.

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where1, the system is gradientf =−∇F(u), andW is a space–time white noise. Thus, if e_i is an orthonormal basis forL₂(0,1) andβ_i are IID standard Brownian motions then

W(t) =X e_iβ_i(t).

A full introduction to space-time white noise and the theory of stochastic PDEs is given by [4]. The potential F will be a double well potential having wells of equal depth and minima ats_±. We have in mind particularly

F(u) = 1

8(1−u²)², f(u) :=−∇F(u) = 1

2(u−u³), (2)

∗Mathematics Subject Classifications: 60H15, 74N20, 45M05.

Key words: dynamics of phase-boundaries, stochastic partial differential equations, asymptotics.

c2000 Southwest Texas State University and University of North Texas.

Submitted April 18, 2000. Published June 15, 2000.

Work done at the IMA, University of Minnesota and OCIAM, Oxford University

1

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0 0.5

1

0 500 1000

−1

−0.5 0 0.5 1

space time

u

0 0.2 0.4 0.6 0.8 1

0 100 200 300 400 500 600 700 800 900

space

time

Figure 1: Solution of stochastic Allen–Cahn: = 0.08, σ= 0.00: surface and contour plots. Computed ∆t= 0.005 and ∆x= 0.008

which when substituted in (1) yields the Allen–Cahn equation and wheres_±=

±1.

Figures 1–5 show typical solutions of the Allen–Cahn equation with small noise and with Neumann conditions. The right hand figure gives a tracking of the interface position, defined as the contour u= 0. The solutions were computed using the backward Euler finite difference scheme described in [11]; in the figures ∆tdenotes time step and ∆xdenotes the grid spacing of the discretisa- tion. The initial condition consists of three regions, two taking value +1 and the third taking value−1. For the unperturbed equationσ= 0, eventually the two inner interfaces disappear, leaving a single region where the solution is approx- imately −1 away from the boundary. With homogeneous Dirichlet boundary conditions, there would be a boundary layer, where the solutions changes rapidly at the boundary, to satisfy the boundary condition.

There are many results for the equation in caseσ = 0. The equation was originally written down as a model of the evolution of the alignments in crys- tals [1]. Chafee-Infante [3] study the equation on a bounded domain as a bifurcation problem in the limit→0. The equation is shown to have only two stable equilibria for sufficiently small, corresponding to solutions of only one phase. New equilibria are created as→0, but all are unstable. The equation exhibits meta stability, meaning that solutions quickly move to a state whereu takes values near the minima ofF except at interfacial layers of width. These states are not equilibria, but do persist for exponentially long amounts of time.

The evolution of the meta stable states has been described as an ODE in the positions of the interface by a number of authors [12, 6, 2].

The effect of perturbations on the Allen-Cahn equation has been studied

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0 0.5

1

0 500 1000

−1.5

−1

−0.5 0 0.5 1 1.5

space time

u

0 0.2 0.4 0.6 0.8 1

0 100 200 300 400 500 600 700 800 900

space

time

Figure 2: Solution of stochastic Allen–Cahn: = 0.08, σ = 0.015: surface and contour plots. Computed with ∆t = 0.005 and ∆x = 0.008. The ratio σ/√

= 0.05.

0 0.5

1

0 500 1000

−1.5

−1

−0.5 0 0.5 1 1.5

space time

u

0 0.2 0.4 0.6 0.8 1

0 100 200 300 400 500 600 700 800 900

space

time

Figure 3: as in Figure 2, except a different realisation of the noise.

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0 0.5

1

0 50 100 150 200

−1.5

−1

−0.5 0 0.5 1 1.5

space time

u

0 0.2 0.4 0.6 0.8 1

0 20 40 60 80 100 120 140 160 180

space

time

Figure 4: Solution of stochastic Allen–Cahn: = 0.08, σ = 0.1: surface and contour plots. Computed ∆t= 0.005 and ∆x= 0.008. The ratioσ/√

= 0.35.

0 0.5

1

0 50 100 150 200

−2

−1.5

−1

−0.5 0 0.5 1 1.5

space time

u

0 0.2 0.4 0.6 0.8 1

0 20 40 60 80 100 120 140 160 180

space

time

Figure 5: Solution of stochastic Allen–Cahn: as in Figure 4 but a different realisation.

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previously by Laforgue-O’Malley [8, 7] and Reyna-Ward [10, 13]. These pa- pers discuss small deterministic perturbations of the operator and indicate that metastability is very sensitive to perturbation. The present work tackles stochastic perturbations, but brings out a similar result, that the exponential drift responsible for the metastability may be dominated by noise.

To accurately describe the nature of the ODE approximation to (1) with σ= 0, introduceU, the solution to the free space problem

U_xx+f(U) = 0, U(±∞) =s_±, U(0) = 0. (3) Leth= (h₁, . . . , h_N) denote the positions of the interfaces. Letα_i = (−1)ⁱα₀, where α₀ = ±1 indicates whether u(0) ≈ s_±. Ward [12] uses the following approximation to solutionsuof (1) whenis small

u^h=C₀+ XN i=1

n U

α_i(x−h_i)

−C_i o

, C_i = (

s₊, α_i= 1;

s₋, α_i=−1; (4) This is not the only way to define an approximationu^h, see for example [2] for a slightly different approach.

For convenience, fix h₀ = 0 and h_N₊₁ = 1 as the positions of the homogeneous Neumann boundaries. The ODE describing the evolution of his

dh_i dt = 2

kU⁰k² h

µ_i+1e^−σⁱ⁺¹^(1+δ^i,N⁾⁻¹^`ⁱ⁺¹−µ_ie^−σⁱ^(1+δ^i,1⁾⁻¹^`ⁱ i

, i= 1, . . . , N (5) where `_i :=h_i−h_i−1 denotes the distance between interfaces; δ_i,j is the Kro- necker delta function; µ_i andσ_i are positive constants described later in terms of F (in case F given by (2), µ_i = 4, kU⁰k² = 2/3, σ_i = 1 ). This equation holds upto the time of collapse of an interface (whenh_i+1−hi≤, somei) upto exponentially small terms. This result has been established rigorously in [2].

In this paper, the above results are extended somewhat to include the case whereσ >0. Equation (1) is well posed for all time; its existence and uniqueness properties are described in [5]. The simplest case is whenf is globally Lipschitz from L₂(0,1) to itself, in which case a mild solution exists taking values in L₂(0,1). In §2, we show rigorously for Dirichlet boundary conditions that in this case the basic structure of the problem is preserved when σ ^1/2; in particular, for initial data inL₂(0,1) and allT >0, there existsKsuch that

Eku_,0(t)−u_,σ(t)k²≤Kσ²/, 0≤t≤T, (6) whereu_,σis the solution of (1) with diffusion coefficient²and noise intensity σ. (The function f(u) =u−u³ is not Lipschitz as required, but experiments indicate the same phenomena hold). When σ ^1/2, the noise dominates the solution, a consequence of space–time white noise having being ill posed in L₂(0,1) (viz.,EkW(t)k²=∞).

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An SDE is formally derived in§3 to account for the motion of the interfaces whenσ^1/2. The interface positionshare defined as being the minimiser of kVk whereV :=u−u^hoverh∈R^N withh_i+1−h_i≥andN = 1,2, . . .. The SDE is

dh_i= kU⁰k²

1 1−A_i

h

dt +σ^1/2

kU⁰k dβ_i(t) +O(kVk²)dt,

whereA_i:=C^−1/2hei, Vi, for a constantC and a unit vectorei (to be defined later), andβ_i(·) are IID standard Brownian motions. The equation may become singular even when V is order ^1/2, as is expressed by the term 1/(1−A_i).

The termA_i is large whenV has a considerable component in U⁰⁰((x−h_i)/), which essentially describes the direction of a branching interface as depicted in Figure 6. When the equation makes sense, the termA_i has a negligible effect on the dynamics ofhas it is multiplied by exponentially small terms.

The precise relation between the Brownian motions β_i and the white noise W is described in §3. However, when β_i and W are considered independent, we expect that the trajectories of the interfaceshgiven by the stochastic PDE and stochastic ODE should converge weakly asV becomes small. By (6), for an initial conditionu₀=u^h, EkVk² is orderσ²/. Thus, let ˜hbe a solution of

dh_i= kU⁰k²

h

dt+σ^1/2 kU⁰k dβ_i(t),

(7) for initial condition h = h0 (that is, we neglect 1/(1−A_i) and the error O(kVk²) ). Let h minimise ku−u^hk where u solves (2) with u₀ = u^h⁰. We would like

EG(˜h(t))−EG(h(t))→0 asσ/², →0 (8) for smooth test functionalsG:R^N →Rwhere the expectation is taken over all h˜ (resp.,h) which have dimensionN at timet.

The last section of this paper, §4, covers numerical experiments that support (8). The experiments compute the mean and variance of the deviation of the interface position from its initial position for the asymptotic SDE (7) and for (1) for a single interface initial condition. Thus, we take the first steps to examine (8) forg(x) = (x−h₀) andg(x) = (x−x)ˆ ², where ˆx=Eh. The computations indicate agreement between the two dynamical systems forσ/^1/2= 0.1 and 0.035.

2 Finite time limits as σ → 0

The finite time limits in σ and of the stochastic Allen–Cahn equations (1) with homogeneous Dirichlet boundary conditions are studied. Throughout this

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−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

Figure 6: Plot of U (dashed) and U +hV,eiiei (solid) when h_i = 0, A_i = 1, F(u) = (1−u²)²/8, and= 0.08.

section we take the nonlinearityf to be globally Lipschitz fromL₂(0,1) to itself.

Denote the solution of the Allen–Cahn equation (1) by u_,σ, and letu:=u_,0, the solution to the noise free problem.

The space–time white noise W may be considered in terms of its Fourier expansion. If e_i is a complete orthonormal system for L₂(0,1), and β_i is a sequence of independent standard Brownian motions, the processW(·) may be thought of as

W(t) = X∞ i=1

e_iβ_i(t).

It is clear thatW(·) does not converge inL₂(0,1). However, stochastic integrals can be defined with respect to an operator that smoothes the processW(·) [4].

This is made explicit by the Itˆo isometry. The Itˆo isometry in infinite dimensions states that, for a linear operator Φ mapping H to H,

EZ _t

0 Φ(s)dW(s)²= Z _t

0 kΦ(s)k²_HSds. (9) (k · kHS is the Hilbert-Schmidt norm, see [4]). It will be important to estimate this quantity when Φ(s) =e⁻²^A(t−s).

Lemma 2.1 For allt >0, there existsC_t>1such that C_t⁻¹

≤

Z _t

0 ke⁻²^A(t−s)k²_HSds≤C_t

, 0< ≤1.

Proof This result is proved for Awith homogeneous Dirichlet conditions.

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(lower bound) WhenA is defined with Dirichlet conditions, its eigenvalues arek²π² fork= 1,2, . . .. Hence, from (9),

Z _t

0 ke⁻²^A(t−s)k²_HSds= X∞ k=1

1

2²k²π²(1−e⁻²²^k²^tπ²)

≥ X^∞

k=b1/c

1

2²k²π²(1−e^−2tπ²)

≥ 1

2²π²(1−e^−2tπ²) Z _∞

b1/c

1 s²ds

≥ 1

2π²(1−e^−2tπ²).

(upper bound) For allt >0, there exists a constantK_tso that (1−e^−2λtπ²)≤K_tλ, for 0≤λ≤1.

Hence, Z _t

0 ke⁻²^A(t−s)k²_HSds≤

b1/cX

k=1

1

2²k²π²(1−e⁻²²^k²^tπ²) +

X∞ k=1+b1/c

1

2²k²π²(1−e⁻²²^k²^tπ²)

≤

b1/cX

k=1

K_t²k² 2π²²k² +

X∞ k=b1/c+1

1 2²k²π²

≤ K_t 2π²+ 1

2²π² X∞ k=1+b1/c

1 k²

≤ K_t 2π²+

2²π²,

as required. ♦

Lemma 2.2 Consider t >0; for a constantC_γ depending on γ, kA^−γ(I−e^−At)k ≤C_γ t^γ, 0< γ≤1;

kA^γe^−Atk ≤C_γt^−γ, γ >0.

Proof This is a standard result on fractional powers of sectorial operators [9].

♦

Theorem 2.3 Fix T >0. There are three limits as, σ→0:

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1. In the limit , σ→0with σ/^1/2→0, E sup

0≤t≤Tku,σ(t)−u(t)k²→0;

2. Suppose further that f is globally Lipschitz fromH^−r(0,1) toH^−r(0,1).

For each r > 1/2, there exists a process u(·) taking values in H^−r(0,1) such that in the limit, σ→0 withσ/^1/2→ν,

E sup

0≤t≤Tku,σ(t)−u(t)k²_H−r(0,1)→0;

3. In the limit σ/^1/2→ ∞, E sup

0≤t≤Tku_,σ(t)k²→ ∞.

Proof (i) Clearly,

u_,σ(t)−u(t) = Z _t

0 e⁻²^A(t−s) h

f(u_,σ(s))−f(u(s)) i

ds +σ

Z _t

0 e⁻²^A(t−s)dW(s).

The stochastic integral may be bounded as follows: by the Itˆo Isometry (9) and for 0≤t≤T,

Ek Z _t

0

e⁻²^A(t−s)dW(s)k²= Z _t

0 ke⁻²^A(t−s)k²_HSds (by Lemma 2.1)

≤C_T .

Therefore, denoting the Lipschitz constant off byK, we have for 0≤t≤T, Eku,σ(t)−u(t)k²)^1/2≤

Z _t

0 K(Eku,σ(s)−u(s)k²)^1/2ds+C_T^1/2 σ ^1/2. By applying Gronwall’s lemma, we have proved

E sup

0≤t≤Tku,σ(t)−u(t)k²_1/2

≤ σ

^1/2e^KtC_T^1/2→0, as σ/^1/2→0. (10)

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(ii) Consider a sequence (σ_n, _n) withσ_n →0 and σ_n/^1/2_n →ν as n→ ∞.

Forn, m∈N, the Variation of Constants formula gives u_n_,σ_n(t)−u_m_,σ_m(t) =

Z _t

0 (e⁻²ⁿ^A(t−s)−e⁻²^m^A(t−s))f(u_n_,σ_n(s))ds +

Z _t

0 e⁻²^m^A(t−s) h

f(u_n_,σ_n(s))−f(u_m_,σ_m(s)) i

ds + (σ_n−σ_m)

Z _t

0 e⁻²ⁿ^A(t−s)dW(s) +σ_m

Z _t

0 (e⁻²ⁿ^A(t−s)−e⁻²^m^A(t−s))dW(s).

Each term can be bounded inH^−2r(0,1) forr≥1/4. Recall that in the Dirichlet case thatk · k_−r:=kA^−r· kis equivalent to theH^−2r(0,1) norm. This norm is used here to gain the necessary inequalities.

Consider the first term: By Lemma 2.2 (without loss take_m> _n), Ek

Z _t

0 (e⁻²ⁿ^A(t−s)−e⁻²^m^A(t−s))f(u_n_,σ_n(s))dsk²_−r_1/2

≤ Z _t

0 kA^−r(I−e⁻⁽²^m⁻²ⁿ^)A(t−s))k · ke⁻²ⁿ^A(t−s)k ·(Ekf(u_n_,σ_n)k²)^1/2ds (by Lemma 2.2)

≤C Z _t

0 (t−s)^r(²_m−²_n)^rK(Ekun,σnk²)^1/2ds.

By (10),Eku_n_,σ_nk²may be bounded uniformly in limits_n, σ_n →0 subject to σ_n/^1/2_n being bounded. Hence, there exists a constantC₁ with

Ek Z _t

0 (e⁻²ⁿ^A(t−s)−e⁻²^m^A(t−s))f(u_n_,σ_n(s))dsk²_−r_1/2

≤C₁(²_m−²_n)^r Z _t

0 (t−s)^rds.

Consider the second term:

Ek Z _t

0e⁻²^m^A(t−s) h

f(u_n_,σ_n(s))−f(u_m_,σ_m(s)) i

dsk²_−r_1/2

≤K Z _t

0

Eku_n_,σ_n(s)−u_m_,σ_m(s)k²_−r_1/2 ds.

Consider the third term: By the Itˆo isometry, Ek

Z _t

0 e⁻²ⁿ^A(t−s)dW(s)k²_−r= Z _t

0 kA^−re⁻²ⁿ^A(t−s)k²_HSds

= Z _t

0

X∞ k=1

1

(k²π²)^2re⁻²²ⁿ^k²^π²^(t−s)ds,

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which is finite forr >1/4.

Consider the fourth term: by Lemma 2.2 and the Itˆo isometry, we have for 0≤t≤T,

Ek Z _t

0 (e⁻²ⁿ^A(t−s)−e⁻²^m^A(t−s))dW(s)k²_−r

≤ Z _t

0 kA^−r(I−e⁽²ⁿ⁻²^m^)A(t−s))k²ke⁻²ⁿ^A(t−s)k²_HSds

≤kA^−r(I−e⁽²ⁿ⁻²^m^)At)k² Z _t

0 ke⁻²ⁿ^A(t−s)k²_HSds

≤C²(²_n−²_m)^2rt^2rC_T _n

≤C²C_Tt^2r

_n (²_n−²_m)^2r

Thus, taking a limit (σ, )→0 withσ²/bounded above, there exists a constant C₂ such that for 0≤t≤T,

Eku_n_,σ_n(t)−u_m_,σ_m(t)k²_−r_1/2

≤C₂((²_n−²_m)^r+ (σ_n−σ_m)) +

Z _t

0 K

Eku_n_,σ_n(s)−u_m_,σ_m(s)k²_−r_1/2 ds.

Gronwall’s inequality now gives, for a constantC₃ E sup

0≤t≤Tku_n_,σ_n(t)−u_m_,σ_m_(t)k²_−r_1/2

≤C((e²_m−e²_n)^r+ (σ_n−σ_m))e^KT. If_n, σ_n are Cauchy, the sequencesu_n_,σ_n are Cauchy with respect to

0≤t≤Tsup Ek · k²_−r_1/2

and thus a limiting process exists. The above formula also gives uniqueness for ifu_n_,σ_n→u₁ andu_m_,σ_m →u₂ where (_n, σ_n) and (_m, σ_m) are both Cauchy, then, by the above,

E sup

0≤t≤Tku₁(t)−u₂(t)k²_−r_1/2

≤ E sup

0≤t≤Tku_n_,σ_n(t)−u_m_,σ_m(t)k²_−r_1/2 +

E sup

0≤t≤Tkun,σn(t)−u₁(t)k²_−r_1/2 +

E sup

0≤t≤Tku2(t)−u_m_,σ_m(t)k²_−r_1/2

→0.

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(iii) Suppose thatEku,σ(t)k²<∞uniformly asσ/^1/2→ ∞for 0≤t≤T. For simplicity takeu₀= 0. Then, argue for a contradiction as follows: from the Variation of Constants formula and the Itˆo isometry,

Eku_,σ(t)k²=Eh k

Z _t

0 e⁻²^A(t−s)f(u_,σ(s))dsk²i + 2σEhD Z t

0 e⁻²^A(t−s)f(u_,σ(s))ds, Z _t

0 e⁻²^A(t−s)dW(s) Ei

+σ²Eh Z t

0 ke⁻²^A(t−s)k²_HSds i

.

The third term is positive and order σ²/ by Lemma 2.1; the first term is positive; thus, to gain a contradiction, we show the second has lower order than σ²/.Indeed, for 0≤t≤T

σEhD Z t

0 e⁻²^A(t−s)f(u_,σ(s))ds, Z _t

0 e⁻²^A(t−s)dW(s) Ei

≤σEh Z t

0 e⁻²^A(t−s)f(u(s))²ds i_1/2

Eh Z t

0 ke⁻²²^A(t−s)k²_HSds i_1/2

,

≤σC_T^1/2 ^1/2K sup

0≤t≤T

Eku(t)k²_1/2 ,

which is clearly orderσ/^1/2. ♦

3 Formal derivation of an SDE

The positions of the interfacesh_iare well defined in the deterministic caseσ= 0 as the contours of u= (s₊+s₋)/2. In the case σ > 0, the interface may be wrinkled, making the contour ill defined. We choosehby solving the following minimisation problem: lethminimise

ku−u^hk (11)

overh∈R^N with|h_i+1−h_i| ≥and overN = 1,2, . . .. In this case, letting V :=u−u^h, φ_i :=α_i

U⁰

α_i(x−h_i)

, we have by differentiating (11) with respect toh_i

hφ_i, Vi= 0.

We’ll need the following asymptotic properties as we go along [12]:

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1. The solutionU of (3) satisfies

U(x) =s₊−a₊e^−σ⁺^x, x→ ∞; U(x) =s₋+a₋e^σ⁻^x, x→ −∞. wheres±are the zeros of f;σ_± = (−f⁰(s_±))^1/2;

loga_± = log(±s±) + Z _s_±

0

s_±

(2F(η))^1/2 + 1 η−s_±

dη.

2.

kU⁰k²≈ Z _∞

−∞U⁰(x)²dx= Z _s₊

s−

(2F(x))^1/2dx.

3. For casef(u) = ¹₂(u−u³); these quantities evaluate tos_± =±1;kU⁰k²= 2/3;a_±= 2;σ_± = 1.

Assume thathobeys the Itˆo equation

dh=ψ(h, t, ω)dt+ Θ(h, t, ω)dβ(t), (12) where Θ = diag(θ₁, . . . , θ_N) andψ = (ψ₁, . . . , ψ_N)^T and β(t) is a vector of N Brownian motions, to be specified later in terms ofW(t).

Apply the Itˆo formula tou=u^h+V using (4) and (12), du=−X

i

φ_idh_i−¹₂X

i

φ_ixθ_i²dt+ dV

=− X

i

φ_iψ_i+¹₂φ_ixθ²_i

dt+X

i

φ_iθ_idβ_i(t) +dV, where φ_ix= (φ_i)_x. Take the inner product withφ_i:

hφ_i, dui

=

n−ψ_ikφ_ik²+X

i

12hφ_ix, φ_iio

dt− kφ_ikθ_idβ_i(t) +X

i6=j

θ_jhφ_i, φ_jidβ_j(t).

Note that

hφix, φ_ii= φ²_i₁

0.

This quantity is very small and is neglected as the asymptotics of U show that φ_i is exponentially small away from the layers. Similarly, hφ_i, φ_ji is negligible fori6=j. Hence, we’ll work with

hφ_i, dui=−ψ_ikφ_ik²dt− kφ_ikθ_idβ_i(t) (13)

(14)

To compare, multiply (1) byφ_i:

hφ_i, dui=hφ_i, ²u_xx+f(u)idt+σhφ_i, dW(t)i. (14) Let

β_i(t) := 1 kφ_ik

Z _t

0 hφ_i, dW(s)i.

Theβ_i(t) are continuous martingales with variance Eβ_i(t)²= 1

kφ_ik² Z _t

0 khφ_i,·ik²_HSds= 1 kφ_ik²

Z _t

0 kφ_ik²ds=t.

Thereforeβ_i(t) are standard Brownian motions. Moreover, the processesβ_i(t) are independent (upto exponentially small terms), because

Ehβi(t), β_j(t)i=t hφ_i, φ_ji kφik · kφjk. Thus (14) becomes

hφi, dui=hφi, ²u_xx+f(u)idt+σ dβ_i(t). (15) Equate coefficients in (13) and (15):

−ψikφik²=hφi, ²u_xx+f(u)i, (16)

−θ_ikφ_ik²=σkφ_ik. (17) Expand the RHS of (16):

hφ_i, ²u_xx+f(u)i=hφ_i, ²u^h_xx+f(u^h)i+hφ_i, L^hVi+O(kVk²), (18) whereL^hu=²u_xx+df(u^h)u. Write the first term

hφ_i, ²u^h_xx+f(u^h)i= D

φ_i, ²u^h_xx+ XN i=1

f

U

α_i(x−h_i)

E +

D φ_i, E

E , (19) where

E:=f(u^h)−X

i

f(U(α_i(x−h_i)/)).

Because U solves (3), the first term is zero and, by the asymptotic analysis in [12], the quantity

hφ_i, Ei ≈2

˜

µ_i+1e^−σⁱ⁺¹⁻¹^`ⁱ⁺¹−µ˜_ie^−σⁱ⁻¹^`ⁱ

, i= 1, . . . , N (20)

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where `_i := h_i−h_i−1 (recall h₀ := 0 andh_N+1 := 1) and ˜µ_i := (a_iσ_i)² and

˜

µ₁= ˜µ_N₊₁= 0 and a_i=

(

a₊, ifα_i= 1;

a₋, ifα_i=−1;, σ_i= (

σ₋, ifα_i= 1;

σ₊, ifα_i=−1;. Consider the second term in (18): letL(u) =²u_xx+f(u) so that

hφi, L^hVi=hL^hφ_i, Vi+B_i=hL(u^h)_h_i, Vi+B_i (21) where B_i are boundary terms, which may be neglected for internal layers [12].

We wish to computeL(u^h)_h_i. First note that L(u^h) =X

i

hφ_i,L(u^h)i φ_i

kφ_ik²+ lower order terms.

Consequently, from (19)

hL(u^h)_h_i, Vi=hφi,L(u^h)ihφ_ix, Vi

kφik² =hφi,L(u^h)iAi, (22) where A_i:=hφix, Vi/kφik².

Collecting (16), (18), and (22), we have

−ψ_ikφ_ik²=hφ_i, Ei+hφ_i, EiA_i, and so

ψ_i= 1 1−A_i

hφ_i, Ei

kφ_ik² +O(kVk²). (23) Finally, from (20), (23), and (17), the SDE is

dh_i = 1 1−A_i

2 kφ_ik²

˜

µ_i+1e^−σⁱ⁺¹⁻¹^`ⁱ⁺¹−µ˜_ie^−σⁱ⁻¹^`ⁱ

dt+ σ

kφ_ikdβ_i(t).

The termkφikis independent ofi (upto exponentially small terms) and hence we write kφ_ik=^−1/2kU⁰kgiving

dh_i= 1 1−A_i

2 kU⁰k²

˜

µ_i+1e^−σⁱ⁺¹⁻¹^`ⁱ⁺¹−µ˜_ie^−σⁱ⁻¹^`ⁱ

dt+σ^1/2 kU⁰k dβ_i(t).

Similarly,A_i may be better written A_i = C

^1/2he_i, Vi, where

C:= kU⁰⁰k

kU⁰k², e_i(x) = φ_ix

kφixk ≈ U⁰⁰((x−h_i)/) kU⁰⁰((x−h_i)/)k.

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In the case whereh_i is a neighbour of the boundary (i= 1 ori=N), one term drops out (viz., ˜µ₁= 0 or ˜µ_N₊₁= 0) and the boundary termsB_iin (21) should be evaluated to give the lowest contribution. [12] computes the contribution fromB_i and this contribution is not effected by the stochastic perturbation: let µ_i= (a_iσ_i)² andδ_i,j denote the Kronecker delta, then for i= 1, . . . , N

dh_i = 1 1−A_i

2 kU⁰k²

µ_i+1e^−σⁱ⁺¹^(1+δ^i,N⁾⁻¹^`ⁱ⁺¹−µ_ie^−σⁱ^(1+δ^i,1⁾⁻¹^`ⁱ

dt +σ^1/2

kU⁰k dβ_i(t).

4 Numerical Experiments

We would like to show that the trajectories of the interfaces described by (2) and (7) converges weakly on a finite time interval in the smallσ/^1/2limit. To this end, consider an initial condition u₀ =u^h where h= (0.4). We compute the mean and variance of the deviation of the interface position fromx= 0.4 for both (2) (with initial conditionu₀) and (7) (with initial conditionh₀). Clearly, the average at timetis taken over realisations where the single interface persists at timet. The diagrams show the mean and variance on a time interval [0,200]

for parameter values (, σ) = (0.08,0.01) and (0.08,0.03). The trajectory of the interface for the noise free problem (= 0.08,σ= 0) is shown for reference.

References

[1] S. Allen and J. W. Cahn,A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsing, Acta Metallur- gica, (1979), pp. 1084–1095.

[2] J. Carr and R. L. Pego,Metastable patterns in solutions ofu_t=²u_xx− f(u), Comm. Pure Appl. Math., 42 (1989), pp. 523–576.

[3] N. Chafee and E. F. Infante, A bifurcation problem for a nonlinear partial differential equation of parabolic type, Applicable Anal., 4 (1974/75), pp. 17–37.

[4] G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, vol. 44 of Encyclopedia of Mathematics and its Applications, Cam- bridge University Press, Cambridge, 1992.

[5] G. Da Prato and J. Zabczyk,Ergodicity for infinite-dimensional systems, vol. 229 of London Mathematical Society Lecture Note Series, Cam- bridge University Press, Cambridge, 1996.

[6] G. Fusco and J. K. Hale, Slow-motion manifolds, dormant instability, and singular perturbations, J. Dynamics Differential Equations, 1 (1989), pp. 75–94.

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0 50 100 150 200 0

0.01 0.02 0.03 0.04 0.05 0.06 0.07

time

variance

0 50 100 150 200

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02 0

time

mean

Figure 7: For σ = 0.03, plots of mean and variance of the position of the interface relative to its initial position given by (2) and (7); 18500 trials are taken to compute for (2) and 20,000 for (7). The dashed line represents the motion of the interface withσ= 0.

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0 50 100 150 200 0

1 2 3 4 5 6 7 8 9x 10⁻³

time

variance

0 50 100 150 200

−0.03

−0.02

−0.01 0

time

mean

Figure 8: For σ = 0.01, plots of mean and variance of the position of the interface relative from its initial position given by (2) and (7); 500 trials are taken in both case. The dashed line represents the motion of the solution with σ= 0.

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[7] J. G. L. Laforgue and R. E. O’Malley, Jr.,On the motion of viscous shocks and the supersensitivity of their steady-state limits, Methods Appl.

Anal., 1 (1994), pp. 465–487.

[8] ,Viscous shock motion for advection-diffusion equations, Stud. Appl.

Math., 95 (1995), pp. 147–170.

[9] A. Pazy, Semi groups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer–

Verlag, 1983.

[10] L. G. Reyna and M. J. Ward, On exponential ill-conditioning and internal layer behavior, Numer. Funct. Anal. Optim., 16 (1995), pp. 475–500.

[11] T. Shardlow,Numerical methods for stochastic parabolic PDEs, Numer.

Funct. Anal. Optim., 20 (1999), pp. 121–145.

[12] M. J. Ward, Metastable patterns, layer collapses, and coarsening for a one-dimensional Ginzburg-Landau equation, Stud. Appl. Math., 91 (1994), pp. 51–93.

[13] M. J. Ward and L. G. Reyna,Internal layers, small eigenvalues, and the sensitivity of metastable motion, SIAM J. Appl. Math., 55 (1995), pp. 425–

445. Perturbation methods in physical mathematics (Troy, NY, 1993).

Tony Shardlow

Dept. Computer Science, University of Manchester, Oxford Road, Manchester M13 9PL, England

e-amil: [email protected]