Triviality of the 2D stochastic Allen-Cahn equation

El e c t ro nic J

o f

Pr

ob a bi l i t y

Electron. J. Probab.17(2012), no. 39, 1–14.

ISSN:1083-6489 DOI:10.1214/EJP.v17-1731

Triviality of the 2D stochastic Allen-Cahn equation

Martin Hairer

Marc D. Ryser

Hendrik Weber

Abstract

We consider the stochastic Allen-Cahn equation driven by mollified space-time white noise. We show that, as the mollifier is removed, the solutions converge weakly to 0, independently of the initial condition. If the intensity of the noise simultaneously converges to0at a sufficiently fast rate, then the solutions converge to those of the deterministic equation. At the critical rate, the limiting solution is still deterministic, but it exhibits an additional damping term.

Keywords:SPDEs; Allen-Cahn equation; white noise; stochastic quantisation.

AMS MSC 2010:60H15; 81T08.

Submitted to EJP on January 14, 2012, final version accepted on May 29, 2012.

SupersedesarXiv:1201.3089v1.

1 Introduction

We consider the following evolution equation on the two-dimensional torusT^2: du= ∆u+u−u³

dt+σdW , u(0) =u⁰. (Φ)

Hereu⁰is a suitably regular initial condition,σa positive constant, andW anL²(T²)- valued cylindrical Wiener process defined on a probability space (Ω,F,P). In other words, at least at a formal level, ^dW_dt is space-time white noise.

This equation and variants thereof have a long history. The deterministic part of the equation is theL²gradient flow of the Ginzburg-Landau free energy functional

Z

T²

1

2|∇u(x)|²+V(u(x)) dx ,

with the potential energyV given by the standard double-well functionV(u) =¹₄(u²−1)², see [14]. This provides a phenomenological model for the evolution of an order parameter describing phase coexistence in a system without preservation of mass. At large scales,

∗Supported by the Institute of Mathematical Statistics (IMS) and the Bernoulli Society.

†University of Warwick, UK. E-mail:[email protected]

‡Duke University, USA. E-mail:[email protected]

§University of Warwick, UK. E-mail:[email protected]

the dynamic of phase boundaries is known to converge to the mean curvature flow [1, 9, 12].

The noise termσdW accounts for thermal fluctuations at positive temperature. On a formal level the choice of space-time white noise is natural, because it satisfies the right fluctuation-dissipation relation. At least for finite-dimensional gradient flows it is natural to take the bilinear form that determines the mechanism of energy dissipation as covariance of the noise, as this guarantees the invariance of the right Gibbs measure un- der the dynamics. Naively extending this observation to the current infinite dimensional context yields (Φ).

White noise driven equations such as (Φ) are known to be ill-posed in space-dimension d≥2[17, 8]. Actually, the linearised version of (Φ) (simply remove the termu³) admits only distribution-valued solutions ford≥2. For anyκ >0these solutions take values in the Sobolev spaceH^{2−d2 −κ}, but they do not take values in H^2−d2 . In general, it is impossible to apply nonlinear functions to elements of these spaces and the standard approach to construct solutions of (Φ) [8, 11] fails.

In the present article, we introduce a cutoff at spatial lengths of orderεand we study the limit asε→0for finite noise strength for (Φ). More precisely, we set

Wε(t) = X

|k|≤1/

ekβk(t), ε >0,

where{ek}_k∈Z2 is the Fourier basis onT^{2, and}{βk}_k∈Z2 are complex Brownian motions that are i.i.d. except for the reality conditionβ¯_k=β_−k. We thus consider

duε= ∆uε+uε−u³_ε

dt+σ(ε)dWε, uε(0) =u⁰, (Φε) and study the weak limit ofuεasε→0.

The main result of this article can loosely be formulated as follows (a precise state- ment will be given in Theorems 2.1 and 2.2 below):

Theorem 1.1. Letσbe bounded and such thatlim_ε→0σ²(ε) log 1/ε=λ² ∈[0,+∞]. If λ²= +∞, thenu_εconverges weakly to0, in probability. Otherwise, it converges weakly in probability to the solutionwλof

∂_tw_λ= ∆w_λ− _8π³λ²−1

w_λ−w_λ³, w_λ(0) =u⁰. (Ψ_λ) Remark 1.2. The result for constantσwas conjectured in [16], based on numerical simulations.

Remark 1.3. The borderline caseλ 6= 0is particularly interesting as it provides an example of stochastic damping: in the limit asε→0, the stochastic forcing is converted into an additional deterministic damping term,−_8π³λ²wλ, to the Allen-Cahn equation. In particular, ifλ²>^8π₃, the zero-solution becomes globally attracting.

Remark 1.4. Recently, there has been a lot of interest in (Φ)in the regime where the noise is small [13, 3, 5]. There, the authors studied(Φ)in arbitrary space dimension on the level of large deviation theory. As in (Φε)they consider a modified version of (Φ) where the noise termdW is replaced by a noise termdWεwith a finite spatial correlation lengthε. For this modified equation, solutions can be constructed in a standard way and a large deviation principle à la Freidlin-Wentzell can be obtained. One can then show that the rate functionals converge asε→0. The large deviation principle however is not uniform inε; this procedure corresponds to taking the amplitude of the noise much smaller thanε. The results obtained in this article quantify how small the noise should be as a function ofεin order for the solutions of (Φ)to be close to the deterministic equation.

Remark 1.5. We believe that the weak convergence to0 asε→ 0actually holds for a much larger class of potentials. Actually, one would expect it to be true whenever lim_|u|→∞V⁰⁰(u) = +∞. The proof given in this article does however depend crucially on the fact thatV(u)∼u⁴for large values ofu.

The main tools used in our proofs are provided by the theory of stochastic quantisation.

Actually, in the context of Euclidean Quantum Field Theory the question of existence of the formal invariant measure of (Φ) has been treated in the seventies (see e.g. [10]). Then, it had been observed that this measure, the so calledΦ⁴₂field, can be defined, but only if a logarithmically diverging lower order term is subtracted. The corresponding stochastic dynamical system (i.e. the renormalised version of (Φ)) has also been constructed [15, 2, 6]. Note that although this renormalised equation,

du= ∆u+u−:u³:

dt+σdW ,

formally resembles (Φ) it does not have a natural interpretation as a phase field model.

Our main argument is a modification of the construction provided in [6]. We present here a brief heuristic argument for the caseσ≡1. First, letC_ε>1and add and subtract the termC_εu_εto (Φ_ε) to get

duε= ∆uε−(Cε−1)uε−uε u²_ε−Cε

dt+σdWε. (1.1)

The key idea is to chooseCεin such a way that, for small values ofε, the termuε u²_ε−Cε is equal to the Wick product:u³_ε:with respect to the Gaussian structure given by the invariant measure of the linearised system (which itself depends onCε). Since, given the results in [6], one would expect:u³_ε:to at least remain bounded asε→0, it is not surprising that the additional strong damping term−C_εu_εcauses the solution to vanish in the limit.

2 Notations and Main Result

In order to formulate our results, we first introduce the class of Besov spaces that we will work with. As in [6] we choose to work in Besov spaces, because they satisfy the right multiplicative inequalities (see Lemma A.2). Denote by(·,·)theL²inner product, and by

ek(x) = _2π¹ e^ikx _k∈

Z² the corresponding orthonormal Fourier basis. Throughout the article, we work with periodic Besov spacesB^s_p,r(T²), wherep, r ≥ 1 ands ∈ R^. These spaces are defined as the closure ofC^∞(T²)under the norm

kuk_Bs

p,r(T²):=X^∞

q=0

2^qrs k∆quk^r_Lp(T²)

1/r

, (2.1)

where the∆q are the Littlewood-Paley projection operators given by∆0u= (e0, u)e0and

∆_qu= X

2^q−1≤|k|<2^q

(e_k, u)e_k, q≥1.

Regarding the exponents appearing in these Besov spaces, we will restrict ourselves throughout this article to exponentsp,randssuch that

p≥4, r≥1, −2

7p< s <0. (2.2)

We now reformulate Theorem 1.1 more precisely. The caseλ²= +∞is given by the following:

Theorem 2.1. Assumeu⁰ ∈ B_p,r^s such that (2.2) holds. Then for allε > 0andT >0, there exists a unique mild solutionu_ε. Ifσ(ε)is bounded uniformly inεand satisfies lim_ε→0σ²(ε) log(1/ε) = +∞then, for allδ∈(0, T),lim_ε→0kuεk_C([δ,T];Bs

p,r)= 0in probabil- ity.

On the other hand, whenσ²(ε) log(1/ε)converges to a finite limit, we have

Theorem 2.2. Assumeu⁰∈ B^s_p,rsuch that (2.2) holds. Iflim_ε→0σ²(ε) log(1/ε) =λ²∈R^, thenlimε→0kuε−wλk_C([0,T];Bs

p,r)= 0in probability, wherewλ is the unique solution to (Ψ_λ).

Remark 2.3. Ifσdecays sufficiently fast, for exampleσ(ε)∼ε^τ for someτ >0, then the conclusion of Theorem 2.2 actually holds in the space of space-time continuous functions.

To conclude this section, we introduce some concepts borrowed from the theory of stochastic quantization. Since we are not concerned with the dynamics of quantised fields, we only introduce the notions necessary for the proof techniques used below, and refer to [7] for a general introduction to the topic. Consider the linear version of equation (1.1), namely

dz_ε= (∆z_ε−(C_ε−1)z_ε)dt+σ(ε)dW_ε. (2.3) ForCε>1, this equation has a unique invariant measure onL²(T²), which we denote byµε. It isµε that will play the role of the “free field” in the present article. Under µε, the kth Fourier component ofzε is a centred complex Gaussian random variable with variance ^σ2₂^(ε) Cε−1 +|k|²−1

. Furthermore, distinct Fourier components are independent, except for the reality conditionzε(−k) =zε(k).

As a consequence of translation invariance, one has the identity D²_ε:=

Z

L²

|φε(x)|²µε(dφε) = 1

2π 2Z

L²

kφεk²_L2µε(dφε) (2.4)

= 1 8π²

X

|k|≤1/ε

σ²(ε) Cε−1 +|k|² .

We thendefinethe Wick powers of any fieldu_εwith respect to the Gaussian structure given byµεby

:uⁿ_ε: =Dⁿ_εHn(uε/Dε),

whereH_n denotes thenth Hermite polynomial. In this article, we will only ever use the Wick powers forn≤3, for which one has the identities

:u¹_ε: =uε, :u²_ε: =u²_ε−D²_ε, :u³_ε: =u³_ε−3D²_εuε. (2.5) From now on, whenever we use the notation:uⁿ_ε:, (2.5) is what we refer to. For any two expressionsAandBdepending onε, we will throughout this article use the notation A.B to mean that there exists a constantCindependent ofε(and possibly of other relevant parameters clear from the respective contexts) such thatA≤C B.

3 Trivial limit for strong noise

In this section, we provide the proof of Theorem 2.1. First, in Subsection 3.1, we explain the “correct” choice of the renormalization constantC_εin (1.1). In Subsection 3.2, we then obtain bounds on the linearised equation, as well as its Wick powers. Finally, in Subsection 3.3, we obtain a bound on the remainder and we combine these results in order to conclude.

3.1 Fixing the renormalization constant ForCε>1, we rewrite (Φε) as

duε= Aεuε−uε(u²_ε−Cε)

dt+σ(ε)dWε, (3.1)

where the linear operatorAεis given byAε= ∆−(Cε−1). Motivated by the heuristic arguments provided in Section 1, the goal of this section is to determineC_εin such a way that the nonlinear termu_ε(u²_ε−C_ε)is equal to the Wick product:u³_ε:. It then follows from (2.4) and (2.5) thatCεis implicitly determined by the equation

Cε= 3D_ε²= 3 8π²

X

|k|≤1/ε

σ²(ε)

C_ε−1 +|k|² . (3.2)

To describe the behavior of the solution to (3.2), we shall use the notationA_ε∼B_εto meanlim_ε→0Aε/Bε= 1.

Lemma 3.1. For any values of the parameters, equation (3.2) has a unique solution Cε>1. Ifσis uniformly bounded and such thatlim_ε→0σ²(ε) log(1/ε) =∞, then one has

Cε∼ 3

4πσ²(ε) log1

ε . (3.3)

In particular,lim_ε→0Cε= +∞.

Before we proceed to the proof of this result, we state the following very useful result:

Lemma 3.2. Leta, R≥1. Then there exists a constantCsuch that the bound

X

|k|≤R

1

a+|k|²−πlog 1 + R²

a

≤ C

√a

1∧ R

√a

, (3.4)

holds. Here, the sum goes over elementsk∈Z^2. Proof. The second expression on the left is nothing butR

|k|≤R dk

a+|k|², so we want to bound the difference between the sum and the integral. Using the monotonicity and positivity of the functionx7→ _a+x¹₂ and restricting ourselves to one quadrant, we see that one has the bounds

X

|k|≤R ki>0

1

a+|k|² ≤ 1 4

Z

|k|≤R

dk

a+|k|² ≤ X

|k|≤R ki≥0

1 a+|k|² . As a consequence, the required error is bounded by

4

bRc

X

k=0

1 a+k² ≤ 4

a+ 4 Z R

0

dx a+x² .

The required bound follows at once, using the fact thataandRare bounded away from 0by assumption.

Proof of Lemma 3.1. Since the right hand side in (3.2) decreases from∞down to0as the left hand side grows from1 to∞, it follows immediately that (3.2) always has a unique solutionCε>1.

Since, by Lemma 3.2, one has P

|k|≤¹_ε 1

1+|k|² ∼ 2πlog¹_ε and since by assumption σ²(ε) log¹_ε → ∞, there existsε0such that

3 8π²

X

|k|≤1/ε

σ²(ε)

1 +|k|² ≥2, (3.5)

for allε < ε₀. As a consequence, we haveC_ε≥2for such values ofε, and we will use this bound from now on. On the other hand, if we know thatC_ε≥2, thenC_εis bounded from aboveby the left hand side of (3.5), so that

Cε≤Kσ²(ε) log1

ε , (3.6)

for some constantKand forεsmall enough.

It now follows from Lemma 3.2 that C_ε= 3σ²(ε)

8π log

1 + 1

ε²(C_ε−1)

+R_ε, (3.7)

for some remainderRεwhich is uniformly bounded asε→0. Since, by (3.6), the first term on the right hand side goes to∞, this shows thatRεis negligible in (3.7), so that

Cε∼ 3σ²(ε) 8π log

1 ε²C_ε

= 3σ²(ε) 8π

log 1

ε² −logCε

. SinceCεis negligible with respect to _ε¹2 by (3.6), the claim follows.

3.2 Bounds on the linearised equation

We split the solution to (3.1) into two parts by introducing the stochastic convolution zε(t) :=σ(ε)

Z t

−∞

e^(t−s)AεdWε(s), (3.8)

and performing the change of variablesvε(t) :=uε(t)−zε(t). With these notations,vε

solves

∂tvε=Aεvε− v_ε³+ 3v_ε²zε+ 3vε:z_ε²: + :z_ε³:

(Φ^aux_ε ) v_ε(0) =u⁰−z_ε(0).

We thus split the original problem into two parts: first, we show that the stochastic convolution converges to0, then we show that the remaindervεalso converges to0.

By construction, the stochastic convolution (3.8) is a stationary process and its invari- ant measure is given byµ_ε. We first establish a general estimate for its renormalized powers:z_εⁿ:, which will be useful for boundingv_εlater on. Throughout this section, we assume thatlim_ε→0σ²(ε) log(1/ε) =∞and thatCεis given by (3.2). We then have:

Lemma 3.3. Letr, k, p≥1,s <0. Then, for alln∈N^{, we have}

ε→0lim Ek:z_εⁿ:k^k_Bs

p,r = 0. (3.9)

Proof. Following the calculations of the proof of [6, Lemma 3.2], we see that Ek:z_εⁿ:k^k_Bs

p,r .kγεk_H^kn2_βn , (3.10) whereβn = 1 +₂^rksnp and

γε(x) = X

|k|≤1/ε

σ²(ε)

C_ε−1 +|k|²ek(x). Since

kγεk²_Hβn = X

|k|≤1/ε

σ⁴(ε)(1 +|k|²)^βn (Cε−1 +|k|²)² ,

andβn<1, the claim follows from the boundedness ofσand the fact thatCε→ ∞.

Corollary 3.4. Letn, p, r≥1ands <0. Then:zⁿ_ε:∈L^p([0, T];B^s_p,r)P-a.s., for allε >0. In particular,

ε→0limEk:z_εⁿ:k_Lp(0,T;B^s_p,r)= 0. (3.11) Proof. This follows from the stationarity ofz_ε, Fubini’s theorem and Lemma 3.3.

We establish now the main result of this subsection.

Proposition 3.5. Consider the stochastic convolutionz_εdefined in (3.8) and letp, r≥1, s <0andT >0. Then

ε→0limEkz_εk_C([0,T];Bs

p,r)= 0. (3.12)

Proof. We begin by decomposing the stochastic convolution into two parts, z_ε(t) =e^tAεz_ε(0) +σ(ε)

Z t 0

e^(t−s)AεdW_ε(s).

The bound on the first term follows from Lemma 3.3 and Lemma A.3, so it remains to focus on the second term, which we denote hereafter asz¯ε(t). In order to bound it, we use thefactorization method, see [8, p. 128], as well as [11, p. 47] for a more detailed presentation. Recalling that

Z t σ

(t−s)^α−1(s−σ)^−αds= π sinπα, we fixα∈(0,¹₂)and rewritez¯εas

¯

zε(t) = sinπα π

Z t 0

e^(t−s)AεYε(s) (t−s)^α−1ds, (3.13) where

Yε(s) :=σ(ε) Z s

0

(s−σ)^−αe^(s−σ)AεdWε(σ).

Next, we introduce the mappingΓε:y7→Γεydefined by Γ_εy(t) := sinπα

π Z t

0

e^(t−s)Aεy(s) (t−s)^α−1ds,

and show thatΓ_ε : L^q([0, T];B^s_p,r) → C([0, T];B_p,r^s ) is a bounded mapping forq > 1/α. First, it is a consequence of the strong continuity ofe^tAε thatΓ_εy∈ C([0, T];B_p,r^s )for all y∈ C([0, T];B_p,r^s )such thaty(0) = 0[11, p. 48]. Next, observe thats7→(t−s)^α−1is in L^q¯([0, t])for allq¯∈[1,(1−α)⁻¹), and hence we can use Hölder’s inequality to deduce that for allq > _α¹,

sup

t∈[0,T]

kΓ_εy(t)k_Bs

p,r .kyk_Lq([0,T];B^s_p,r). (3.14) A standard density argument allows us to conclude thatΓε:L^q([0, T];B_p,r^s )→ C([0, T];B_p,r^s ) is indeed a bounded mapping forq >1/α.

To conclude the proof, we assume for the moment that there existK_ε>0such that sup

t∈[0,T]EkYε(t)k_Bs

p,r ≤Kε, lim

ε→0Kε= 0. (3.15)

From (3.15), it then follows that EkYεk_Lq([0,T];B^s_p,r)≤

T sup

t∈[0,T]EkYεk^q_Bs p,r

^1/q

.T^1/qK_ε, (3.16) where the first inequality is due to Jensen’s inequality and Fubini’s theorem, and the second inequality follows from (3.15) in conjunction with Fernique’s theorem. By (3.16), Y_ε ∈ L^q([0, T] ;B_p,r^s )P-a.s. and hencez¯_ε= Γ_εY_ε ∈ C([0, T];B_p,r^s )P-a.s. Furthermore, it follows from (3.14)–(3.16) that

E sup

t∈[0,T]

k¯zε(t)k_Bs

p,r .EkYεk_Lq([0,T];B_p,r^s ).Kε, so thatk¯zεk_C([0,T];Bs

p,r)→0in probability, as required.

It remains to establish (3.15). By definition of the Besov norm (2.1) and Jensen’s inequality,

EkYε(t)k_Bs

p,r ≤X^∞

q=0

2^qrsEk∆qYε(t)k^r_Lp

^1/r

. (3.17)

As a consequence, (3.15) follows if we can show that

Ek∆qY_ε(t)k^p_Lp≤K_ε2^qpτ , (3.18) for someτ <|s|and someK_ε→0.

Fix now q ∈ N. Thanks to Fubini’s theorem, the Gaussianity of ∆_qY_ε(t), and the independence of its different Fourier components,

Ek∆qYε(t)k^p_Lp= Z

T²E

X

2^q−1≤|k|<2^q

(Yε(t), ek)ek(ξ)

p

dξ

. Z

T²

E

X

2^q−1≤|k|<2^q

(Yε(t), ek)ek(ξ)

2^p/2

dξ (3.19)

. Z

T²

X

2^q−1≤|k|<2^q

E|(Yε(t), ek)|^2p/2 dξ.

Itô’s isometry and the definition ofAεyield E|(Yε(t), ek)|²≤σ²(ε)h

2

Cε−1 +|k|²i2α−1 Z ∞ 0

e^−ττ^−2αdτ .σ²(ε)

Cε−1 +|k|²2α−1

, (3.20)

where the last inequality is due to2α <1. Inserting (3.20) back into (3.19) we obtain the bound

Ek∆qYε(t)k^p_Lp.σ^p(ε)



 X

2^q−1≤|k|<2^q

1 Cε−1 +|k|²

!1−2α



p/2

.σ^p(ε)



 2^2qτ (Cε−1)^δ

X

2^q−1≤|k|<2^q

1

|k|^{2+2τ−4α−2δ}





p/2

,

which is valid for all τ > 0 and all δ ∈ (0,1−2α). Since we can make bothα andδ arbitrarily small, we can in particular choose them in such a way that2α+δ < τ <|s|, so that the exponent is strictly greater than2. This implies that the corresponding inverse power of|k|is summable over allk, so that (3.18) is satisfied.

3.3 Bounds on the remainder

First, we need a technical lemma for the mappingM^ε, defined as

(M^εy) (t) :=e^{t Aε} u⁰−z_ε(0) +

Z t 0

e^(t−τ)Aε

3

X

l=0

a_ly^l(τ) :z^3−l_ε (τ):dτ , (3.21)

where thealare some real-valued constants. In order to formulate the results of this section, we introduce the Banach space

E_T :=C([0, T];B_p,r^s )∩L^p([0, T];B^s_p,r^¯ ), equipped with the usual maximum norm

kxk_E

T := max

kxk_C([0,T];B^s_p,r),kxk_Lp([0,T];B^¯s_p,r)

. (3.22)

Regarding the parameters appearing inET, we shall usually assume that(p, r, s,s)¯ satisfy the bounds

p≥4, r≥1, ¯s= 2s+2

p, −2

7p < s <0. (3.23) Lemma 3.6. Fixε >0,T >0, and assume (3.23). Then there exist positive constantsδ andK_εwithlim_ε→0K_ε= 0such that

kM^εyk_E

T ≤ 1 +K_εT^δ u⁰−z_ε(0) B^s_p,r

+KεT^δ

3

X

l=0

:z_ε^3−l:

_Lp([0,T];Bs

p,r)kyk^l_E

T. (3.24)

Proof. The bound of the first term on the right-hand side of (3.21) is given in Proposi- tion A.4. Next, we split the second term into two parts,Ω¹_ε+ Ω²_ε, where

Ω¹_ε(t, y) = Z t

0

e^(t−τ)Aε

2

X

l=0

al:z_ε^3−l(τ):y^l(τ)dτ,

Ω²_ε(t, y) = Z t

0

e^(t−τ)Aεy³(τ)dτ .

We bound Ω¹_ε first. Since ((l+ 1)−1)s+ 1−2/p > 0 for l = 0,1,2, we can employ Lemma A.1 to find that there existδ >0andK_εas in the statement such that

Z t 0

e^(t−τ)Aε:z_ε^3−l(τ):y^l(τ)dτ _E

T

≤KεT^δ

:z_ε^3−l:y^l _Lp/(l+1)

([0,T];B^(2l+1)sp,r ).

Using Lemma A.2 and adding up the respective contributions yields the terms with l= 0,1,2on the right-hand side of (3.24).

We now boundΩ²_ε. Sincey ∈ ET ands <¯s, the embeddingB_p,r^¯s ,→ B^s_p,r implies that y ∈L^p([0, T];B_p,r^s ). From Lemma A.1 withn= 3and Lemma A.2 withl = 2, it follows again that there existδandK_εsuch that

Z t 0

e^(t−τ)Aεy³(τ)dτ ET

≤KεT^δ y y²

_Lp/3

([0,T];B_p,r^5s) (3.25)

≤KεT^δ kyk³_Lp([0,T];B^¯s_p,r), which is the term withl= 3on the right-hand side of (3.24).

Lemma 3.7. Letε >0, assume (3.23) and consider(Φ^aux_ε )withu⁰∈ B_p,r^s . Then for all T >0, there existsP-a.s. a unique mild solutionv_ε∈ E_T.

Proof. The existence of unique local solutions to (Φ^aux_ε ) follows from (3.24) and is shown in detail in [6, Prop. 4.4]. Furthermore, a fixed point argument in a weighted supremum norm shows thatvε(T^∗)∈ C(T²). Since, forC(T²)-valued initial datum, (Φ^aux_ε ) admits a unique global solution inC([0, T];C(T²))∩ C((0,∞),C^∞(T²)), see e.g. [11, Thm. 6.4; Prop.

6.23], the claim follows from the fact that this space is a subspace ofE_T.

Before we state the main result of this section, we introduce the Banach space E_T^δ :=C([δ, T];B^s_p,r)∩L^p([0, T];B_p,r^¯s ), δ∈[0, T),

equipped with the normkxk_Eδ

T :=kxk_C([δ,T];Bs

p,r)+kxk_Lp([0,T];B^s_p,r^¯ ). With this notation, we have:

Proposition 3.8. Assume (3.23) and consider the sequence of regularized problems (Φ^aux_ε )with fixed initial conditionu⁰∈ B^s_p,r. For allT >0, the unique global solutionvε∈

ET from Lemma 3.7 converges to zero in sense that, for everyδ∈(0, T),limε→0kvεk_Eδ

T =

0in probability.

Proof. We introduce the stopping timeτε,δas τ_ε,δ:=T∧infn

t≥δ:kvεk_Eδ t ≥1o

, (3.26)

with the convention thatτε,δ=T if the set is empty. Next, we establish the limit

ε→0limEkv_εk_Eδ

τε,δ = 0. (3.27)

Recalling thatvεsolves the fixed point equation M^εvε =vε, we can use Lemma 3.6, combined with

sup

t∈[δ,T]

e^tAε u⁰−zε(0) _Bs

p,r

≤e^{−δ Cε}

u⁰−zε(0) _Bs

p,r

, (3.28)

to show that there existsγ >0andK_εwithlim_ε→0K_ε= 0such that kvεk_Eδ

τε,δ ≤K_ε(1 +T^γ)

u⁰−z_ε(0) B_p,r^s

+KεT^γ

3

X

l=0

kvεk^l_Eδ τε,δ

:z_ε^3−l: _Lp([0,τδ

ε];B^s_p,r). Sincekvεk_Eδ

τε,δ ≤1by construction, the claim (3.27) then follows from Lemma 3.3 and Corollary 3.4. Since, by the definition ofτε,δ, this implies thatlimε→0P(τε,δ< T) = 0, the claim follows.

Proof of Theorem 2.1. Sinceuε=zε+vε, the claim follows from Propositions 3.5 and 3.8, in conjunction with the embeddingB^s_p,r^¯_¯ ,→ B^s_p,r, which holds ifs¯≥sandp¯≥p.

4 Deterministic limit for weak noise

In this section, we give the proof of Theorem 2.2. The technique of proof is almost identical to the previous section, but we define objects in a slightly different way. This time, we define an operatorA= ∆−1, and we set

zε(t) :=σ(ε) Z t

−∞

e^(t−s)AdWε(s). (4.1)

We furthermore define all of our Wick products with respect to the lawµ_εofz_ε, so that all throughout this section (2.5) holds, but withD_εgiven by

D²_ε= 1 8π²

X

|k|≤1/ε

σ²(ε) 1 +|k|² . Note that, by Lemma 3.2, one has

ε→0limD²_ε= λ² 8π .

As before, we rewrite the solution to (Φε) asuε=vε+zε, wherevεis solution to

∂tvε=Avε+ (2−3D²_ε) vε+zε

+

3

X

l=0

alv^l_ε:z^3−l_ε :, (4.2)

with initial conditionvε(0) =u⁰−zε(0)and suitable constantsal. Note first that one has the following result:

Proposition 4.1. Letzεbe defined as in (4.1). Then, for everyT >0and everyn >0, the limits

ε→0limkzεk_C([0,T];Bs

p,r)= 0, lim

ε→0k:z_εⁿ:k_Lp([0,T];B^s_p,r)= 0, hold in probability.

Proof. It follows from [6, Lem. 3.2] that Ek:z_εⁿ:k_Bs

p,r .σⁿ(ε)→0,

asε→0. The proof thatzεalso converges to0inC([0, T];B^s_p,r)is virtually identical to the proof of Proposition 3.5, so we omit it.

It remains to establish thatlim_ε→0kvε−w_λk_C([0,T];Bs

p,r)= 0in probability, which is the content of the following result:

Proposition 4.2. Assume (3.23) and let u0 ∈ B^s_p,r. Let vε ∈ ET be the unique mild solution to (4.2), andw_λ∈ ET the unique solution to(Ψ_λ). Then

ε→0limkvε−wλk_E

T = 0 in probability.

Proof. Settingδε= 3D_ε²−^3λ_8π² andaλ= 1−^3λ_8π², we can rewrite the equations forvεand wλas

∂_tv_ε= ∆v_ε+a_λv_ε−v_ε³−δ_εv_ε+ (2−3D²_ε)z_ε+

2

X

l=0

a_lv_ε^l:z_ε^3−l:,

∂twλ= ∆wλ+aλwλ−w_λ³.

Settingρ_ε=v_ε−w_λ, we see thatρ_εsolves the following evolution equation:

∂tρε= ∆ +aλ

ρε−ρε v_ε²+vεwλ+w²_λ + (2−3D²_ε)z_ε−δ_εv_ε+

2

X

l=0

a_lv^l_ε:z^3−l_ε :.

SettingAˆ= ∆ +a_λ, we have the mild formulation ρ_ε(t) =e^Atˆρ_ε(0)−

Z t 0

e^A(t−s)ˆ ρ_ε(s) v²_ε+v_εw_λ+w²_λ (s)ds + (2−3D_ε²)

Z t 0

e^A(t−s)ˆ zε(s)ds−δε

Z t 0

e^A(t−s)ˆ vε(s)ds +

2

X

l=0

a_l Z t

0

e^A(t−s)ˆ v_ε^l(s) :z_ε^3−l:(s)ds .

It then follows from Lemmas A.1 and A.2 that

kρεkE_T .kρε(0)kB^s_p,r+T^δkρεkE_T kvεk²_Lp([0,T];B^¯s_p,r)+kwλk²_Lp([0,T];B^s_p,r^¯ )

+δεkvεk_ET +T^δ

2

X

l=0

(1 +kvεk^l_E

T)k:z_ε^3−l:k_Lp([0,T];B^s_p,r). (4.3) We now use the fact that there existsKsuch that the deterministic solutionwλsatisfies kwλkET ≤K. Settingτ_ε= ¯T∧inf{t:kρεk_E

t ≥1}for someT¯≤T such thatT¯^δ((K+ 1)²+ K²)≤ ¹₂, it follows from (4.3) that

kρεk_Eτε,δ .kρε(0)k_B^s_p,r+ ¯T^δ

2

X

l=0

(1 +K)^l δε+k:z_ε^3−l:k_Lp([0,T];B^s_p,r)

.

This bound can easily be iterated, and the claim then follows similarly to the proof of Proposition 3.8.

A Technical results

In this appendix, we collect a few technical results.

Lemma A.1. LetA= ∆−ΛforΛ≥1and letf ∈L^p/n([0, T];B^(2n−1)sp,r )withp > n≥1, s <0ands¯= 2/p+ 2ssuch that

(n−1)s+ 1−n

p >0. (A.1)

Then there existsδ >0such that

Z t 0

e^(t−τ)Af(τ)dτ _E

T

≤K(Λ)T^δ kfk_Lp/n([0,T];B(2n−1)s p,r ), with a constantK(Λ)such thatlim_Λ→∞ K(Λ) = 0.

Proof. Modulo straightforward modifications yieldingK(Λ)→0, the proof is identical to the proof of [6, Lem. 3.6].

Lemma A.2. Let n, p, r ≥ 1, s < 0, s¯ = 2/p+ 2s such that |s| < _p(2n+1)² and l < n. Assume thatgi∈L^p([0, T];B_p,r^¯s )fori= 1, . . . , landh∈L^p([0, T];B_p,r^s ). Then, there exists a constantC >0such that

kh g1· · ·glk_{Lp/(l+1)([0,T];B}(2l+1)s

p,r )≤Ckhk_Lp([0,T];B^s_p,r) l

Y

j=1

kgjk_Lp([0,T];B_p,r^¯s ). (A.2)

Proof. This is a straightforward modification of [6, Cor. 3.5].

Lemma A.3. Letp, r≥1ands < s¯ . Then, there exists a constantC >0such that e^t∆x

B^s_p,r ≤Ct^¯s−s2 kxk_B¯s

p,r ∀x∈ B^¯s_p,r.

Proof. The estimate follows from [4, Lem. 2.4] and the definition of the Besov norm (2.1).

Corollary A.4. Lets <0,r, p≥1, and¯s= 2s+²_p. Define the operatorA= ∆−Λand recall theET-norm as defined in (3.22). Then there existsδ >0such that for allΛ>1,

e^tAx

ET ≤ 1 +C(Λ)T^δ kxk_Bs

p,r, ∀x∈ B_p,r^s , wherelim_Λ→∞C(Λ) = 0.

Proof. The bound on theC [0, T] ;B^s_p,r

norm is trivial. Using Proposition A.3, we obtain for arbitraryγ >0

e^tAx

_Lp([0,T];Bs

p,r)≤ K Λ^γ/p

Z T 0

1 t^γ

e^t∆x _Bs

p,r

dt

!1/p

≤ K Λ^γ/pkxk_Bs

p,r

Z T 0

t^{p(s−¯s)/2−γ}dt

!1/p

≤ K Λ^γ/pkxk_Bs

p,rT^|s|/2−γ/p. Choosingγ < ^p₂|s|, the claim follows.

References

[1] S. Allen and J. Cahn. A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall.27, no. 6, (1979), 1085–1095.

[2] S. Albeverio and M. Röckner. Stochastic differential equations in infinite dimensions: so- lutions via Dirichlet forms. Probab. Theory Related Fields 89, no. 3, (1991), 347–386.

MR-1113223

[3] F. Barret, A. Bovier, and S. Méléard. Uniform estimates for metastable transition times in a coupled bistable system. Electron. J. Probab.15, no. 12, (2010), 323–345. MR-2609590 [4] H. Bahouri, J. Chemin, and R. Danchin. Fourier analysis and nonlinear partial differential

equations, vol. 343 ofGrundlehren der mathematischen Wissenschaften Series. Springer Verlag, 2010.

[5] S. Cerrai, and M. Freidlin. Approximation of quasi-potentials and exit problems for mul- tidimensional RDE’s with noise. Trans. Amer. Math. Soc.363, no. 7, (2011), 3853–3892.

MR-2775830

[6] G. Da Prato and A. Debussche. Strong solutions to the stochastic quantization equations.

Ann. Probab.31, no. 4, (2003), 1900–1916. MR-2016604

[7] G. Da Prato and L. Tubaro. Wick powers in stochastic PDEs: an introduction. Technical Report UTM 711, University of Trento(2007).

[8] G. Da Prato and J. Zabczyk.Stochastic equations in infinite dimensions, vol. 45 ofEncyclopedia of mathematics and its applications. Cambridge University Press, 1992. MR-1207136 [9] L.C. EVANS, H.M Soner, and P.E. Souganidis. Phase transitions and generalized motion by

mean curvature. Commun. Pure Appl. Math.45, no. 9, (1992), 1097–1123. MR-1177477 [10] J. Glimm and A. Jaffe. Quantum Physics: A Functional Integral Point of View. Second ed.,

Springer-Verlag, New York, 1987. MR-0887102

[11] M. Hairer. An introduction to stochastic PDEs. http://www.hairer.org/Teaching.html, 2009. Unpublished lecture notes.

[12] T. Ilmanen. Convergence of the Allen-Cahn equation to Brakke’s motion by mean curvature.

J. Differential. Geom.38, no. 2, (1993), 417–461. MR-1237490

[13] R. Kohn, F. Otto, M. Reznikoff, and E. Vanden-Eijnden. Action minimization and sharp- interface limits for the stochastic Allen-Cahn equation. Commun. Pure Appl. Math.60, no. 3, (2007), 393–438. MR-2284215

[14] L. Landau, and L. Ginzburg. On the theory of superconductivity. J. Expt. Theor. Phys.20, (1950), 1064–1082.

[15] G. Parisi, and Y.S. Wu. Perturbation theory without gauge fixing.Sci. Sinica24, no. 4, (1981), 483–496. MR-0626795

[16] M.D. Ryser, N. Nigam, and P.F. Tupper. On the well-posedness of the stochastic Allen-Cahn equation in two dimensions.J. Comp. Phys.231, no. 6, (2012), 2537–2550.

[17] J. Walsh. An introduction to stochastic partial differential equations.École d’Été de Probabil- ités de Saint Flour XIV-1984265–439. MR-0876085

Acknowledgments. MH acknowledges financial support by the EPSRC trough grant EP/D071593/1 and the Royal Society through a Wolfson Research Merit Award. Both MH and HW were supported by the Leverhulme Trust through a Philip Leverhulme Prize.

MDR is grateful to P.F. Tupper and N. Nigam for fruitful discussions, and acknowledges financial support from a Hydro-Québec Doctoral Fellowship. We would like to thank F.

Otto for suggesting to also consider the caseσ(ε)→0. All three authors are grateful for the relaxed atmosphere at the Newton institute, where this collaboration was initiated during the 2010 “Stochastic PDEs” programme.