Stochastic Cahn–Hilliard equation in higher space dimensions: the motion of bubbles

We study the stochastic motion of a droplet in a stochastic Cahn–Hilliard equation in the sharp interface limit for sufficiently small noise. The key ingredient in the proof is a deterministic slow manifold, where we show its stability for long times under small stochastic perturbations. We also give a rigorous stochastic differential equation for the motion of the center of the droplet.


Introduction
In this work we consider the stochastic Cahn-Hilliard equation (also known as the Cahn-Hilliard-Cook equation [18]) posed on a two-dimensional bounded smooth domain Ω ⊂ R 2 : Here, ε is a small positive parameter measuring the relative importance of surface energy to the bulk free energy, and ∂ n denotes the exterior normal derivative to the boundary ∂Ω.F is assumed smooth with two equal nondegenerate minima, at u = ±1.A typical example is F (u) = 1 4 (u 2 − 1) 2 .We focus on this special case here, although most of the results hold for a very general class of nonlinearities.Only the precise formulation of the stability result and the condition on the noise strength there will change depending on the growth of F at ∞.The forcing is given by an additive white in time noise ∂ t W .As we rely for simplicity of presentation on Itô's formula, we assume that the Wiener process is sufficiently smooth in space, and moreover sufficiently small in ε, so that it does not destroy the typical patterns in the solutions.The existence and uniqueness of solutions is well-studied (see e.g.[19,16]) and we always assume that we have a unique solution.Moreover, as we assume the noise to be smooth in space, the solution should be regular in space, too.The deterministic Cahn-Hilliard equation is a gradient flow in the H −1topology for the following energy In order to minimize this energy, one can expect that, for 0 < ε ≪ 1, solutions of (1.1) stay mostly near u = −1 and u = +1, the stable minima of F (u).Moreover, the gradient can be of order ε −1 , so we expect small transition layers with thickness of order ε.Because of this, we can think of Ω as split into subdomains on which u ε (•, t) takes approximately the constant values −1 and 1, with boundaries ε-localized about an interface Γ ε (t).
The interface is expected to move according to a Hele-Shaw or Mullins-Sekerka problem, where circular shaped droplets are stable stationary solutions of the dynamics.In [7] formal derivation suggested a stochastic Hele-Shaw problem in the limit in case the noise is of order ε.There it was also shown that for very small noise, the dynamics is well approximated by a deterministic Hele-Shaw problem, see also [9].Also in [23] (or [17] in the deterministic case) the dynamics of the interface in the sharp interface limit was studied, but without obtaining an equation on the interface.A rigorous discussion of the sharp interface limit in the deterministic case can be found in [4].
In our result we focus on the almost final stage where the interface is already a single spherical droplet in the domain, and thus the only possible dynamics is given by the translation of the droplet, at least as long as the droplet stays away from the boundary.The deterministic case was studied in [3,6] and it was shown that the droplet moves (in ε) exponentially slow.Due to noise, we expect here a dominant stochastic motion of the droplet on a faster time-scale than exponentially slow.
As we want to study a single small droplet, the average mass of the solution is close to ±1.In this regime an initially constant solution is locally stable, and one has to wait for a large deviation event, that leads to the nucleation of droplets.See for example [10,11,13,15,14].Let us finally remark, that although the result in [3,6] holds also for three spatial dimensions, we focus here on the case of dimension d = 2 only.With our method presented it is straightforward to treat the three dimensional case, only the technical details will change.Details will be provided in [22].
Moreover, the case of the mass-conservative Allen-Cahn equation is similar.
See also [7] for the motion of a droplet along the boundary, or [5,12] for the deterministic case.For the one dimensional case see [8].

Assumptions on spaces and noise
We fix the underlying space H −1 (Ω) with scalar product •, • and norm • .
The standard scalar-product in L 2 (Ω) is denoted by (•, •) or •, • L 2 .Moreover, we use • ∞ for the supremum norm in C 0 or L ∞ .As the Cahn-Hilliard equation preserves mass, we also consider the subspace H −1 0 (Ω) of the Sobolev space H −1 (Ω) with zero average.Recall, that the inner product in H −1 0 (Ω) is given by where −∆ is the self-adjoint positive operator defined in Let W be a Q-Wiener process in the underlying Hilbert space H −1 (Ω), where Q is a symmetric operator and (e k ) k∈N forms a complete H −1 (Ω)-orthonormal basis of eigenfunctions of Q with corresponding non-negative eigenvalues α 2 k , i.e.
Qe k = α 2 k e k .It is well known that W is given as a Fourier series in H −1 for a sequence of independent standard real-valued Brownian motions {β k (t)} k∈N , cf.DaPrato and Zabzcyck [20].
In order to guarantee mass-conservation of solutions to (1.1), the process W is supposed to take values in H −1 0 only, i.e. it satisfies In order to simplify the presentation, we rely on Itô's formula.Thus we have to assume that the trace of the operator Note that one always has We assume that the Wiener process and thus Q depends on ε, and thus the noise strength is defined by either η 0 or η 1 .
In the sequel for results in L 2 -spaces, we need also higher regularity of Q.
For this purpose define the trace of Note that e k was normalized in H −1 and not in L 2 .

Outline and main result
In our main results we rely on the existence of a deterministic slow manifold.This was already studied in detail in [3] or [6], where a deterministic manifold of approximate solutions was constructed that consists of translations of a droplet state, see section 2 for details.Crucial points are the spectral properties of linearized operators that allows to show that the manifold is locally attracting.
In the deterministic case solutions are attracted to an exponentially small neighborhood of the manifold and follow the manifold until the droplet hits the boundary.Moreover, the motion of the interface is given by an ordinary differential equation.In the stochastic case this is quite different.
In section 3 we derive the motion along the manifold by projecting the dynamics of the stochastic Cahn-Hilliard equation to the manifold.This is a rigorous description of the motion that involves no approximation.We will see that sufficiently close to the manifold, the dynamics is in first approximation given by the projection of the Wiener process onto the slow manifold, which is a stochastic equation for the motion of the center of the droplet.
In section 4 we consider the stochastic stability of the slow manifold first in H −1 and then in L 2 .This heavily relies on the deterministic stability and on small noise, but as both the equation and the noise strength depends on ε we cannot use standard large deviation results.We use a technical Lemma from [7] in order to show that with overwhelmingly high probability one stays close to the slow manifold for very long times.Due to the stochastic forcing, we cannot exclude the possibility of rare events that will destroy the droplet or nucleate a second droplet.Also the stability of the manifold holds for any polynomial time scale in ε −1 , which is much larger than the time scale in which the droplet moves.So we expect the droplet to hit the boundary at a specific polynomial time scale.The final section 5 collects technical estimates used throughout the paper.

The slow manifold
Our stochastic motion of the droplet is based on the slow manifold constructed in [3] in the deterministic case.In this section we collect some important results from [3] which we need throughout this work.We start with constructing the slow manifold Mρ ε consisting of translations of a single droplet with radius ρ > 0 and discuss the spectrum of the linearized Cahn-Hilliard and Allen-Cahn operator afterwards.These spectral properties are crucial in showing the stochastic stability of the slow manifold.

Construction of the bubble
We use a bounded radially symmetric stationary solution to the Cahn-Hilliard equation on the whole space R 2 .As this solution (and all its derivatives) decays exponentially fast away from the droplet, its translations serve as good approximations for droplets inside the bounded domain.A function u ∈ C 2 (R 2 ) is such a solution if, and only if, it is radial and satisfies for some constant σ.We also need some condition on monotonicity, in order to ensure that u is a single droplet centered at the origin.
The following proposition, cf.[3] Thm. 2.1, concerns the existence of such radial solutions of the rescaled PDE Proposition 2.1.There exists a number ρ > 0 and smooth functions σ : where C > 0 is a constant and α(ρ) denotes the root close to 1 of the equation Then there exists a constant C > 0 such that where V is a bounded function such that Here we used the usual O-notation, that a term is O(g(ρ)) if there exists a constant such that the term is bounded by Cg(ρ) for small ρ > 0.
For a fixed radius ρ > 0 of the droplets and a fixed distance δ > 0 from the boundary of the domain, Proposition 2.1 assures that we can associate with each center ξ ∈ Ω ρ+δ = {ξ : d(ξ, ∂Ω) > ρ + δ} a droplet, which is a function u ξ : Ω → R with the following properties: a) It is an almost stationary solution of the Cahn-Hilliard equation in the sense that it fails to satisfy the equation, or the boundary conditions, by terms which are of the order O(e −c/ε ) (including their derivatives) b) It jumps from near −1 to near 1 in a thin layer with thickness of order ε around the circle of radius ρ and center ξ.
For ε ≪ 1 we define the droplet state where the number a ξ is chosen to be zero at some fixed ξ 0 ∈ Ω ρ+δ and is determined for generic ξ ∈ Ω ρ+δ by imposing that the mass of u ξ is constant on Ω ρ+δ , i.e., For example, we choose ξ 0 to be a point of maximal distance from the boundary ∂Ω.We could also fix a small mass and then determine the radius ρ > 0 such that the droplet centered at ξ 0 has exactly that mass.An easy argument based on Proposition 2.1 v) shows (cf.Lemma 3.1 in [3]) that with similar estimates for the derivatives of a ξ with respect to ξ i .

The Quasi-Invariant Manifold and Equilibria
In this section we state the construction of a manifold Mε ρ of droplets of the form ξ → u ξ + v ξ , where v ξ is a tiny perturbation, such that Mε ρ is an approximate invariant manifold for equation (1.1).The construction of Mε ρ is made in such a way that stationary solutions to (1.1) with approximately circular interface are in Mε ρ and can be detected by the vanishing of a vector field ξ → c ξ .Here we follow [3].
Theorem 2.2.Assume that ρ > 0 is such that Ω ρ = {ξ ∈ Ω : d(ξ, ∂Ω) > ρ} is non-empty and let δ > 0 be a fixed small number.Then there is an ε 0 > 0 such that, for any 0 < ε < ε 0 there exist C 1 functions defined in Ω ρ+δ and such that Ω v ξ dx = 0, for which (iii) Similar estimates with C replaced by Cε −k , with k the order of differentiation, hold for the derivatives of v ξ , c ξ with respect to x, ξ.
(iv) The function ũξ = u ξ + v ξ satisfies the boundary conditions in (1.1) and where Then there is a sufficiently small η > 0 such that u ∈ Ñη is an equilibrium of (1.1) if and only if u = ũξ , c ξ = 0 for some ξ ∈ Ω ρ+δ .

Spectral estimates for the linearized operators
An essential point in the stochastic stability are the spectral properties of the linearized Cahn-Hilliard and Allen-Cahn operator.We consider the linearization around any droplet state in our slow manifold, and it is crucial that eigenfunctions not tangential to the manifold have negative eigenvalues uniformly bounded away from zero, while all other eigenvalues have eigenfunctions tangential to the manifold.

The Cahn-Hilliard operator on
We study the linearized Cahn-Hilliard operator in more detail.We consider L ξ as an operator on H −1 0 (Ω) and cite a theorem of [2] below.As we have exponentially small terms, we use the following definition: Definition 2.3.We say that a term is of order O(exp) if it is asymptotically exponentially small as ε → 0, i.e. of order O(e −c/ε ) for some positve constant c.
(i) The operator L ξ can be extended to a self-adjoint operator on H −1 0 , the subspace of the Sobolev space H −1 consisting of functions with zero average.L ξ is bounded from below.
and let δ > 0 be fixed.Then there is ε 0 > 0 and constants c, C, C ′ > 0 independent of ε such that, for 0 < ε < ε 0 and ξ ∈ Ω δ , the following estimates hold: (iii) In the two-dimensional subspace U ξ corresponding to the small eigenvalues λ ξ 1 , λ ξ 2 there is an orthonormal basis (in where the matrix (a ξ ij ) is nonsingular and a smooth function of ξ and ũξ j is the derivative of ũξ with respect to ξ j .Moreover ψ ξ i is a smooth function of ξ and where ψ i,j is the derivative of ψ i with respect to ξ j .
As we will need the statement in more detail later, we will comment on the proof of (iii).The main ingredient is the following theorem.For its proof we refer to [21].
Theorem 2.5.Let A be a selfadjoint operator on a Hilbert space H, I a compact interval in R, {ψ 1 , . . ., ψ N } linearly independent normalized elements in D(A).We assume (i)

, N
(ii) There is a number a > 0 such that I is a-isolated in the spectrum of A: where In our case we take E = span{ , and a = ε 2 .According to theorem 2.4(ii) the spectral gap is of order ε and therefore I is a-isolated.Let us now discuss that the eigenvectors corresponding to the smallest eigenvalues approximate well the tangent space of the slow manifold.First, the droplet state is an approximate solution, so for its derivative ũξ j (which is a tangent vector) we have Since the matrix ũξ i , ũξ j approaches a nonsingular limit as ε → 0 (see e.g.(5.1)), we also have |λ min | > C > 0. For i ∈ {1, 2} we denote the associated eigenvector to λ ξ i by ψ ξ i and define Thus, ψ ξ i ∈ E + O(exp) and one can write By definition of the distance d we have ũξ Noting ũξ j ≤ Cρ we get by multiplying It remains to show that the matrix B(ξ) defined by B jk (ξ) = ũξ j , ψ ξ k is invertible.This can be seen as follows: Therefore invertibility of B is equivalent to the invertibility of the matrix defined by ũξ i , ũξ j which is already proven.Remark 2.6.Note that Theorem 2.4 is restricted to the two-dimensional case.While the construction of an orthonormal basis as in (iii) is the same, thus far, for d = 3 it can be shown that the spectral gap is only of order O(ε 2 ).This heavily influences our analysis of stochastic stability and any improvement of this result will yield a better region of stability in the threedimensional setting.

2.3.2
The mass-conserving Allen-Cahn operator on L 2 0 (Ω).Next, we collect some results on the eigenvalue problem for the mass-conserving Allen-Cahn equation linearized around ũξ , for small 0 < ε ≪ 1, on L 2 (Ω).Here as defined previously ũξ is the bubble state, which is an element of the slow manifold.
Theorem 2.7.Let ũξ ∈ Mε ρ and let µ 1 ≤ µ 2 ≤ µ 3 ≤ . . .be the eigenvalues of (2.12).Then there is ε 0 such that for ε < ε 0 (2.14) The two-dimensional space W ξ spanned by the eigenfunctions corresponding to the eigenvalues µ 1 , µ 2 can be represented by This result can be found in [1] with ũξ replaced by u ξ .As v = ũξ − u ξ is exponentially small the theorem follows from an easy perturbation argument.Also note that for the eigenfunctions of Cahn-Hilliard we thus have by (2.15) and (2.11) Remark 2.8.Defining the projection Therefore, for all v ⊥ H −1 ψ ξ i we have L 2 , which is crucial for establishing stability.

Motion along the slow manifold : The dynamics of bubbles
Here we follow the approach to split the dynamics into the motion along the manifold and othogonal to it.

The new coordinate system
We will use the standard projection onto the manifold.A minor technical difficulty is that the eigenfunctions ψ ξ 1 and ψ ξ 2 of the linearization do not span the tangent space at a given point ũξ on the slow manifold.But as the difference to the true tangent space, which is spanned by the partial derivatives ∂ ξ 1 ũξ and ∂ ξ 2 ũξ , is exponentially small, we can use them as an approximate tangent space to project onto the manifold.The following proposition concerns the existence of a small tubular neighborhood of Mε ρ where the projection is well-defined, see [3].
Proposition 3.1.Let ũξ , Mε ρ , Ω ρ be as in Theorem 2.2; then, for η > 1, the condition inf implies the existence of a unique pair where ψ ξ 1 , ψ ξ 2 form a basis of the two-dimensional subspace corresponding to the two smallest eigenvalues of the linearized operator L ξ and are given by theorem 2.4 (iii).Moreover, the map u → (ξ, v) defined by (3.2) is a smooth map together with its inverse.
Let u(t) be a solution of (1.1).We will call the coordinates v and ξ defined in proposition 3.1 the Fermi coordinates of u(t).

The exact stochastic equation for the droplet
In the remainder of this section we adopt the approach of [8] and assume that the center ξ of the bubble ũξ defines a multidimensional diffusion process which is given by for some given vector field f : R 2 → R 2 and some variance σ : R 2 → H 2 .We proceed with deriving explicit formulas for f and σ, which still depend on the distance v to the manifold.
We use the Itô formula, in order to differentiate (3.2) with respect to t, and get du = dv Taking the inner product in the Hilbert space H −1 with ψ ξ k yields for any k ψ ξ k , du = ψ ξ k , dv On the other hand taking the scalar-product of (1.1) with ψ ξ k we derive Now (3.5) and (3.6) together imply , where we also used that w, dW g, dW = Qw, g dt.In order to eliminate dv, we apply the Itô formula to the orthogonality condition ψ ξ k , v = 0 and arrive at dv, ψ Now we use that dv = du − dũ ξ and the fact that dtdt = 0 and dW dt = 0 and get This yields together with (3.7) Define the matrix (A kj (ξ) By theorem 2.4 (iii) we have ψ ξ i,j = O(ε −1 ).Therefore,as long as inf In the comment to the proof of 2.4 we have seen that the matrix ψ ξ k , ũξ j is nonsingular and approaches a constant as ε → 0. As a consequence we observe that the matrix A(ξ) is invertible in a tube Γ around Mε ρ .This proof is straightforward.The details are similar to Lemma 3.3 and the tube Γ has radius ε η for any fixed η > 1.We denote the entries of the inverse matrix by A −1 kj (ξ).From (3.8) we derive Using the invertibility of A(ξ) we finally get formulas for f and σ: and (3.12)

Verification of the SDE
In the derivation, we made the assumption that ξ is a semimartingale with respect to the Wiener process W .We now prove that this assumption is indeed true.At least we find one splitting u = ũξ + v, where ξ is a semimartingale given by our derived SDE for ξ.
Lemma 3.2.Consider the pair of functions (ξ, v) as solutions of the system given by (4.4) and the ansatz (3.3),where σ and f are given by (3.11) and (3.12).Suppose that initially ψ Proof.We first prove that u = ũξ + v solves (1.1).
The orthogonality condition follows from d v, ψ ξ k = 0 since v(0) ⊥ T ũξ(0) M. We have At first we look at the dW -terms: = 0.
Next we consider the drift term: This completes the proof that ξ is indeed a semimartingale.

Approximate stochastic ODE for the droplet's motion
In this section we want to analyze the exact equation for the droplet's motion and its approximation in terms of ε.We start with splitting the ansatz (3.3) into its deterministic part and extra stochastic terms given by a process where due to the definitions (3.11) and (3.12) the stochastic processes A (r) t are given by dA (r) In this section let us first show that the ξ i are driven by a noise term of the type ũξ i , dW , which means that we project the Wiener process to the slow manifold.We also give bounds on the drift f (ξ) and the diffusion σ(ξ).In view of theorem 2.5 we have ũξ where ψ ξ k denotes the eigenfunctions corresponding to the small eigenvalues of L ξ .(see theorem 2.4).Using ũξ i ≤ Cρ we get by multiplying By rotating the eigenfunctions ψ ξ k with an orthonormal matrix Q we can introduce a new coordinate system ψξ k of eigenfunctions in such a way that ũξ 1 ψξ 1 and the corresponding matrix defined by bij = ũξ i , ψξ j is an almost diagonal matrix and the same holds true for its inverse.Hereby, Q will be uniquely defined by rotating the rows of B such that With respect to the new coordinate system we then have Lemma 3.3.Consider the matrix Ā(ξ) ∈ R 2×2 given by Then, as long as v ≤ Cε 1+κ for some κ > 0 and 0 < ε < ε 0 , Ā(ξ) is invertible and its inverse Ā−1 (ξ) can be estimated by Note that the same statement holds without the bar also for the matrix A(ξ).
Proof.By [6] we have and therefore ũξ i , ũξ j defines for small ρ an almost diagonal, invertible matrix of order O(1).Moreover, C 2 0 ρ 2 = ũξ k 2 .In the comment to theorem 2.4 we proved the link where we only needed that the basis ψ ξ i is orthonormal.Since the orthonormal transformation Q does not change this property, we similarly obtain such that invertibility of Z0 can be derived from the invertibility of ũξ i , ũξ j i,j .
On the other hand we have v, ψξ i,j ≤ C v ψ ξ i,j ≤ Cε κ .From this, we see directly that Ā(ξ) is invertible.
Using the form (3.15) of the matrix Z0 and relations (3.16) and (3.17) we see that where we neglected higher order terms.Next, we consider the decomposition where I denotes the identity matrix and E is a small perturbation thereof of order O(ρ).Then, one has by Taylor expansion . With this the lemma is proved.Proof.Immediate consequence of the definition where we changed the underlying coordinate system, and the previous lemma.Moreover, we know that Ā is for small ρ approximately a diagonal matrix, so we can replace ψξ r by ũξ r .Next, we estimate the magnitude of the drift term f in terms of ε.Lemma 3.5.Under the assumptions of Lemma 3.3 we have Proof.We need to estimate all dt-terms in the definition (3.14).Using Lemma 3.4 for estimating the variance σ we derive where we used the estimates ũξ ), which will be derived in section 5, cf.Lemma 5.1.Combining this with the estimate of Ā−1 ri (ξ) from Lemma 3.3 shows that the estimate holds true.Remark 3.6.(Itô-Stratonovich-correction) Let us take a closer look at (3.14).After some calculation, basically redoing the computation that led to (3.11) and (3.12) in the Stratonovich sense and thereby leaving out Itô corrections, one can show that with Stratonovich differentials Thus, we can solve for • dξ j and obtain also for the Itô differential which is (up to some exponentially small error) the projection of the Wiener process W onto the slow manifold Mε ρ of droplets.

Stochastic Stability
For the stochastic stability we derive bounds for the distance from the slow manifold given by v. First we give a result in H −1 and then extend it to L 2 .

H −1 -bounds
Recall that we splitted the solution via Fermi coordinates with the orthogonality condition v(t) ⊥ ψ ξ i (t) in H −1 (Ω) for i = 1, 2. In the following we always assume that we are working on times such that ξ(t) ∈ Ω ρ+δ so that everything is well defined.Writing (1.1) in the form du = L(u) dt + dW and expanding gives and on the other hand we have Here we used the definitions In the case From Theorem 2.2 (iv) we have for the residual Using the notations ∂ ξ ũξ = max ũξ i and σ = max σ ξ i we have where we used that We start with deriving a bound for the nonlinear term N (ũ ξ , v), v by using spectral information for the linearized Cahn-Hilliard operator L ξ in H −1 (Ω).
Here, it is useful that the spectral theory of the Cahn-Hilliard equation in H −1 coincides with the Allen-Cahn operator in L 2 (Remark 2.8).
Lemma 4.2.For u = u ξ +v with v H −1 (Ω) < c 0 ε 4 for some fixed sufficiently small c 0 > 0 we have Proof.Let γ 1 , γ 2 , γ 3 ≥ 0 with i γ i = 1.First, we notice that we have where we performed integration by parts.Together with the spectral information of Theorems 2.4 and 2.7 for the linearized Cahn-Hilliard operator in H −1 and the linearized non-local Allen-Cahn operator in L 2 we derive where we fixed γ 3 ≈ ε 2 and absorbed the positive L 2 -term into its negative counterpart.
As long as v H −1 ≤ c 0 ε 4 we have Here, we used H 1/3 (Ω) ֒→ L 3 (Ω) by Sobolev embedding and interpolation of H 1/3 between H −1 and H 1 .Combined with (4.7) we get by choosing c 0 sufficiently small compared to the other constants We need to control the terms of (4.6) containing inner products with first derivatives of u ξ and ũξ , respectively.As ũξ i can be seen as approximation of the eigenfunctions ψ ξ i together with the orthogonality condition (3.2), we may assume that up to some exponentially small error v ⊥ ∂u ξ ∂ξ i .Lemma 4.3.Let v be as in Proposition 3.1.Then we have and the same holds true for u ξ replaced by ũξ .
Proof.From 2.4 and 2.5 we see that the distance of With ũξ j − u ξ j = O(exp) the lemma is derived.
Finally, we can continue with estimating v, dv .By Lemmata 4.2 and 4.3 together with the estimate for the second derivatives of ũξ we derive Here, we also used that the drift term of dξ is of order O(ε −1 ) which we proved in Lemma 3.5.Thereby with lemma 4.3, the term j ũξ j , v dξ j remains exponentially small.We summarize the H −1 estimate in the following theorem: where Proof.By (4.5) and (4.8) we have As η 1 ≤ η 0 we obtain and thereby the claim.

Long-time stability in H −1
We follow a method used in [7] for the stochastic Allen-Cahn equation to show the long-time stability with respect to the H −1 norm.Define the stopping time τ ⋆ as the exit time from a neighborhood of the slow manifold before time T Note that we neglect the case that ξ(t) ∈ Ω δ+ρ at some point.We only need to cut with another stopping time to take care of this.
We showed in Theorem 4.4 that v satisfies a differential inequality of the form for all t ≤ τ ⋆ , provided that B ≤ c 0 ε 4 .From [7] using optimal stopping of martingales, we obtain from (4.10) and We define now q and assume the following Via an induction argument we derive as C ε ≤ aq.Chebychev's inequality finally yields With this, we can prove the following theorem: Theorem 4.5.For a solution u = u ξ + v with ξ ∈ Ω ρ+δ and v ⊥ ψ ξ j consider the exit time Also, assume that the noise strength satisfies for some k > 0 very small.Then the probability P (τ ⋆ < T ε ) is smaller than any power of ε, as ε tends to 0.
And thus for very large time scales with high probability the solution stays close to the slow manifold Mε ρ .Unless the droplet gets close to the boundary, i.e. ξ(t) ∈ Ω δ+ρ .Remark 4.6.In Remark 3.6 we saw that for η 0 being polynomial in ε the position ξ of the droplet is moving like a diffusion process driven by a Wiener process of strength √ η 0 which is multiplied by a diffusion coefficient of order O(1).Thus due to scaling, we would expect that the droplet hits the boundary of the domain after time scales of order larger that 1/η 0 .Thus the stability result tells us that with overwhelming probability the solution moves along the deterministic slow manifold until it hits the boundary of the domain.
Proof.The statement follows directly from (4.14) if q B 2 = O(ε k).Indeed, using the definition of C ε , a = O(ε) and B = O(ε 4 ), we have since η 1 ≤ η 0 .And therefore we finally get We can also treat smaller neighboorhoods of the slow manifold, by making the size of the noise even smaller.We can take the radius B = ε m and the noise strength η 0 = ε 2m+1+κ .If m > 4, then we can follow exactly the same proof, as all estimates needed just B ≤ c 0 ε 4 .We obtain: Theorem 4.7.For a solution u = u ξ + v with ξ ∈ Ω ρ+δ and v ⊥ ψ ξ j consider the exit time Also, assume that the noise strength satisfies for any κ > 0 small.Then the probability P (τ ⋆ < T ε ) is smaller than any power of ε, as ε tends to 0.

Estimates in L 2 -norm
We want to extend the stability result to the L 2 -norm.As there are no bounds of the linearized Cahn-Hilliard operator in L 2 , we will rely on the results of the previous section.Recall (4.4), where As our object of interest is the L 2 -norm of v we consider the relation Recall that we denote the L 2 inner product by (•, •) and the H −1 inner product by •, • .By series expansion of W we obtain where we used the H −1 estimate of σ from the previous section and ũξ j L 2 = O(ε −1/2 ), as the derivative ũξ j is O(ε −1 ) on a set of order O(ε).Thus, for the Itô correction term we have Next, we study the mixed term (v, dv).By (4.4) we have For the martingale term we see that where the O-terms are all bounded in H −1 .
For T 4 we have c is by definition exponentially small and we established in section 3.3 that the drift term b is of order O(ε −1 η 1 ).Thus, we have It remains to estimate the term T 3 involving the nonlinearity.Integration by parts immediately yields which is a good term for the estimate.We continue with the other terms in For the higher order powers we obtain by Sobolev embedding and interpolation inequalities By choosing γ = 1/2 we finally derived The crucial term is the quadratic term in v, here we have to use the bound in H −1 .By interpolation and Young inequality Combining all estimates we have Recall that in the preceding section we established an optimal radius with respect to the H −1 -norm of order O(ε 4 ).We will add a condition on the L 2 -radius such that in the last estimate of the nonlinearity the leading order of the H 2 -terms is O(ε 2 ).
Definition 4.8.For k > 0 and m > 4 and some given large time T ε we define the stopping time Obviously, we set τ ε = T ε if none of the above conditions are fulfilled.Again, we assume that the solution is well-defined up to T ε .
Later, as we establish stability, we will need to refine the parameter k defining the L 2 -radius.For now, up to the stopping time τ ε , we have shown that for small ε T 3 ≤ −cε 2 v 2 H 2 + Cε 2m−4 .Next, we use that by Poincare v L 2 ≤ ∆v L 2 and η 1 ≤ η 0 to finally get the following estimate for d v 2 L 2 .Lemma 4.9.If k ≥ 0 and t ≤ τ ε , with τ ε given by (4.16), then for some c > 0 the following relation holds true where and

.18)
As in the H −1 case we will derive higher moments in the subsequent section and show stability.

Long-time stability in L 2
Under the assumptions of Lemma 4.9 we estimate for any p > 1 the pth moment of v 2 L 2 .Here we follow again the method used in [7] closely and therefore spare the reader some of the details of the derivation.By Itô calculus we obtain We briefly comment on estimating the Itô correction.Using (4.17) yields and by series expansion we see that Therefore, by Cauchy-Schwarz, we derived where A p is defined as For the sake of simplicity we define and assume that the noise strength is small enough such that a ε < 1.Note that by the definition of K ε we thus also need Cε 2m−6 < 1, which is true by assumption.
Applying Lemma 4.10 inductively we obtain Note that by (4.17) we have for t ≤ τ ε Hence, we derive for C a constant depending on p.
Note that in the previous Lemma, if v(0) 2 L 2 > Cε k+1 then τ ε = 0. Proof.By Lemma 4.10 and (4.22) we have With help of Lemma 4.11 we can finally prove stability in L 2 .
Proof.In Section 3 of [6]  and the norm on H −1 is given by Therefore, with (5.2) and choosing f j = ∂u ξ ∂x i , we have where the L 2 estimate will be established in Lemma 5.2.The same argument yields ψ ξ i,jk ≤ ψ ξ i,j L 2 .In light of Theorem 2.4 (iii) we compute Finally, by the definition in Theorem 2.4 we derive where we used that the matrix (a ξ ki ) does depend smoothly on ξ and is nonsingular.
We conclude with the estimates with respect to L 2 which were needed for section 4.3.Lemma 5.2.Under the same assumptions as in Lemma 5.1 the following estimates hold true Proof.First, we observe that by Theorem 2.2 it suffices to analyse the partial derivatives of u ξ as the correction term v ξ and all its derivatives are exponentially small.By Lemma 2.1 and 2.5 we have where we defined r = |x − ξ|.We use the radial geometry of the problem and the fact that U ′ localizes around the boundary of the bubble.For some small δ > 0 we consider the ring Ω δ = {x : ||x − ξ| − ρ| ≤ δ} .We compute On the set Ω \ Ω δ we utilize |U ′ (η)| ≤ ce −c|η| and derive Combined with (5.4) this shows ũξ j L 2 = O(ε −1/2 ).Estimating the second order derivatives can be carried out analogously.Definition 2.9, Lemma 5.1 and the L 2 -estimate of ũξ j directly yield ψ ξ i L 2 = O(ε −1 ).The bound for the second derivatives was established in (5.3).
by the negative Laplacian with Neumann boundary conditions.