Sharp estimates for metastable lifetimes in parabolic SPDEs: Kramers' law and beyond

We prove a Kramers-type law for metastable transition times for a class of one-dimensional parabolic stochastic partial differential equations (SPDEs) with bistable potential. The expected transition time between local minima of the potential energy depends exponentially on the energy barrier to overcome, with an explicit prefactor related to functional determinants. Our results cover situations where the functional determinants vanish owing to a bifurcation, thereby rigorously proving the results of formal computations announced in [Berglund and Gentz, J. Phys. A 42:052001 (2009)]. The proofs rely on a spectral Galerkin approximation of the SPDE by a finite-dimensional system, and on a potential-theoretic approach to the computation of transition times in finite dimension.

1 Introduction Metastability is a common physical phenomenon, in which a system quickly moved across a first-order phase transition line takes a long time to settle in its equilibrium state. This behaviour has been established rigorously in two main classes of mathematical models. The first class consists of lattice models with Markovian dynamics of Metropolis type, such as the Ising model with Glauber dynamics or the lattice gas with Kawasaki dynamics (see [dH04,OV05] for recent surveys). The second class of models consists of stochastic differential equations driven by weak Gaussian white noise. For dissipative drift, sample paths of such equations tend to spend long time spans near attractors of the system without noise, with occasional transitions between attractors. In the particular case where the drift term is given by minus the gradient of a potential, the attractors are local minima of the potential, and the mean transition time between local minima is governed by Kramers' law [Eyr35,Kra40]: In the small-noise limit, the transition time is exponentially large in the potential barrier height between the minima, with a multiplicative prefactor depending on the curvature of the potential at the local minimum the process starts in and at the highest saddle crossed during the transition. While the exponential asymptotics was proved to hold by Freidlin and Wentzell using the theory of large deviations [VF69,FW98], the first rigorous proof of Kamers' law, including the prefactor, was obtained more recently by Bovier, Eckhoff, Gayrard and Klein [BEGK04,BGK05] via a potential-theoretic approach. See [Ber11] for a recent review.
The aim of the present work is to extend Kramers' law to a class of parabolic stochastic partial differential equations of the form du t (x) = ∆u t (x) − U (u t (x)) dt + √ 2ε dW (t, x) , (1.1) where x belongs to an interval [0, L], u(x) is real-valued and W (t, x) denotes space-time white noise. If the potential U has several local minima u i , the deterministic limiting system admits several stable stationary solutions: these are simply the constant solutions, equal to u i everywhere. It is natural to expect that the transition time between these stable solutions is also governed by a formula of Kramers type. In the case of the doublewell potential U (u) = 1 4 u 4 − 1 2 u 2 , the exponential asymptotics of the transition time was determined and proved to hold by Faris and Jona-Lasinio [FJL82]. The prefactor was computed formally, by analogy with the finite-dimensional case, by Maier and Stein [MS01,MS03,Ste05], except for particular interval lengths L at which Kamers' formula breaks down because of a bifurcation. The behaviour near bifurcation values has been derived formally in [BG09].
In the present work, we provide a full proof for Kramers' law for SPDEs of the form (1.1), for a general class of double-well potentials U . The results cover all finite positive values of the interval length, and thus include bifurcation values. One of the main ingredients of the proof is a result by Blömker and Jentzen on spectral Galerkin approximations [BJ13], which allows us to reduce the system to a finite-dimensional one. This reduction requires some a priori bounds on moments of transition times, which we obtain by large-deviation techniques (though it might be possible to obtain them by other methods). Transition times for the finite-dimensional equation can be accurately estimated by the potential-theoretic approach of [BEGK04,BGK05], provided one can control capacities uniformly in the dimension. Such a control has been achieved in [BBM10] in a particular case, the so-called synchronised regime of a chain of coupled bistable particles introduced in [BFG07a,BFG07b]. Part of the work of the present paper consists in establishing such a control for a general class of systems. We note that although we limit ourselves to the one-dimensional case, there seems to be no fundamental obstruction to extending the technique to SPDEs in higher dimensions driven by a Q-Wiener process. Very recently, Barret has independently obtained an alternative proof of Kramers' law for nonbifurcating one-dimensional SPDEs, using a different approach based on approximations by finite differences [Bar12].
The remainder of this paper is organised as follows. Section 2 contains the precise definition of the model, an overview of needed properties of the deterministic system, and the statement of all results. Section 3 outlines the essential steps of the proofs. Technical details of the proofs are deferred to subsequent sections. Section 4 contains the needed estimates on the deterministic partial differential equation, including an infinitedimensional normal-form analysis of bifurcations. In Section 5 we derive the required a priori estimates for the stochastic system, mainly based on large-deviation principles. Section 6 contains the sharp estimates of capacities, while Section 7 combines the previous results to obtain precise estimates of expected transition times in finite dimension.

Parabolic SPDEs with bistable potential
Let L be a positive constant, and let E = C([0, L], R ) denote the Banach space of continuous functions u : [0, L] → R , equipped with the sup norm · L ∞ .
We consider the parabolic SPDE du t (x) = ∆u t (x) − U (u t (x)) dt + √ 2ε dW (t, x) , t ∈ R + , x ∈ [0, L] and initial condition u 0 ∈ E, satisfying the same boundary conditions. In (2.1), ∆ denotes the second derivative (the one-dimensional Laplacian), ε > 0 is a small parameter, and W (t, x) denotes space-time white noise, defined as the cylindrical Wiener process compatible with the b.c. The local potential U : R → R will be assumed to satisfy a certain number of properties, which are detailed below. When considering a general class of local potentials, it is useful to keep in mind the example U (u) = 1 4 u 4 − 1 2 u 2 . (2.4) Observe that U has two minima, located in u = −1 and u = +1, and a local maximum in u = 0. Furthermore, the quartic growth as u → ±∞ makes U a confining potential. As a result, for small ε, solutions of (2.1) will be localised with high probability, with a preference for staying near u = 1 or u = −1.
The bistable and confining nature of U are two essential features that we want to keep for all considered local potentials. A first set of assumptions on U is the following: Assumption 2.1 (Assumptions on the class of potentials U ). U1: U : R → R is of class C 3 . In some cases (namely, when L is close to π for Neumann b.c. and close to 2π for periodic b.c.), our results require U to be of class C 5 . 1 U2: U has exactly two local minima and one local maximum, and U is nonzero at all three stationary points. Without loss of generality, we may assume that the local maximum is in u = 0 and that U (0) = −1. The positions of the minima will be denoted by u − < 0 < u + . U3: There exist constants M 0 > 0 and p 0 2 such that the potential and its derivatives satisfy |U (j) (u)| M 0 (1 + |u| 2p 0 −j ) for j = 0, 1, 2, 3 and all u ∈ R . U4: There exist constants α ∈ R , β > 0 such that U (u) βu 2 − α for all u ∈ R . U5: For any γ > 0, there exists an M 1 (γ) such that U (u) 2 γu 2 − M 1 (γ) for all u ∈ R .
Remark 2.2. A sufficient condition for U3-U6 of Assumption 2.1 to hold is that the potential can be written as U (u) = p(u) + U 0 (u), where p is a polynomial of even degree 2p 0 4, with strictly positive leading coefficient, and U 0 is a Lipschitz continuous function, cf. [Cer99, Remark 2.6].
Let us recall the definition of a mild solution of (2.1). We denote by e ∆t the Markov semigroup of the heat equation ∂ t u = ∆u, defined by the convolution (e ∆t u)(x) = L 0 G t (x, y)u(y) dy . (2.7) Here G t (x, y) denotes the Green function of the Laplacian compatible with the considered boundary conditions. It can be written as where the e k form a complete orthonormal basis of eigenfunctions of the Laplacian, with eigenvalues −ν k . That is, • for periodic b.c., e k (x) = 1 √ L e 2kπ i x/L , k ∈ Z , and ν k = 2kπ L 2 ; (2.9) • for Neumann b.c., Here the stochastic integral can be represented as a series of one-dimensional Itô integrals where the W (k) t are independent standard Wiener processes (see for instance [Jet86]). It is known that for a confining local potential U , (2.1) admits a pathwise unique mild solution in E [DPZ92].

The deterministic equation
Consider for a moment the deterministic partial differential equation Stationary solutions of (2.13) have to satisfy the second-order ordinary differential equation u (x) = U (u(x)) , (2.14) together with the boundary conditions. Note that this equation describes the motion of a particle of unit mass in the inverted potential −U . There are exactly three stationary solutions which do not depend on x, given by Depending on the boundary conditions and the value of L, there may be additional, nonconstant stationary solutions. They can be found by observing that (2.14) is a Hamiltonian system, with first integral Orbits of (2.14) belong to level sets of H ( Figure 1b). Bounded orbits only exist for Figure 1a). The periodic solution corresponding to H = E crosses the u-axis at u = u 2 (E) and u = u 3 (E), and has a period (2.18) The fact that U (0) = −1 implies that lim E→0 T (E) = 2π (in this limit, stationary solutions approach those of a harmonic oscillator with unit frequency). In addition, we Figure 2. Schematic representation of the deterministic bifurcation diagram. Nonconstant stationary solutions appear whenever L is a multiple of π for Neumann b.c., and of 2π for periodic b.c. For Neumann b.c., the stationary solutions u * n,± contain n kinks. For periodic b.c., all members of the family {u * n,ϕ , 0 ϕ < L} contain n kink-antikink pairs. The transition states (n = 1) are also called instantons [MS03].
We will make the following assumption, which imposes an additional condition on the local potential: Furthermore, a sufficient (but not necessary) condition for Assumption 2.3 to hold true is that . Note that this condition is satisfied for the particular potential (2.4).
Under Assumption 2.3, nonconstant stationary solutions satisfying periodic b.c. only exist for L > 2π, while stationary solutions satisfying Neumann b.c. only exist for L > π; they are obtained by taking the top or bottom half of a closed curve with constant H. Additional stationary solutions appear whenever L crosses a multiple of 2π or π. More precisely (Figure 2), • for periodic b.c., there exist n families of nonconstant stationary solutions whenever L ∈ (2nπ, 2(n + 1)π] for some n 1, where members of a same family are of the form u * n,ϕ (x) = u * n,0 (x + ϕ), 0 ϕ < L; • for Neumann b.c., there exist 2n nonconstant stationary solutions whenever L ∈ (nπ, (n + 1)π] for some n 1, where solutions occur in pairs u * n,± (x) related by the symmetry u * n,− (x) = u * n,+ (L − x). Next we examine the stability of these stationary solutions. Stability of a stationary solution u 0 is determined by the variational equation It follows that u * 0 is always unstable: it has one positive eigenvalue for L 2π, and the number of positive eigenvalues increases by 2 each time L crosses a multiple of 2π. The eigenvalues of Q[u * ± ] are given by −ν k − U (u ± ) and are always negative, implying that u * + and u * − are stable. • For Neumann b.c., the eigenvalues of Q[u * 0 ] are given by −λ k , where Again u * 0 is always unstable: it has one positive eigenvalue for L π, and the number of positive eigenvalues increases by 1 each time L crosses a multiple of π. As before, u * + and u * − are always stable. The problem of determining the stability of the nonconstant stationary solutions is equivalent to characterising the spectrum of a Schrödinger operator, and thus to solving a Sturm-Liouville problem. In general, there is no closed-form expression for the eigenvalues. However, a bifurcation analysis for L equal to multiples of 2π or π (cf. Section 4.3) shows that • for periodic b.c., the stationary solutions u * n,ϕ appearing at L = 2nπ have 2n − 1 positive eigenvalues and one eigenvalue equal to zero (associated with translation symmetry), the other eigenvalues being negative; • for Neumann b.c., the stationary solutions u * n,± appearing at L = nπ have n positive eigenvalues while the other eigenvalues are negative.
A last important object for the analysis is the potential energy For u + v satisfying the b.c., the Fréchet derivative of V at u in the direction v is given by Thus stationary solutions of the deterministic equation (2.13) are also stationary points of the potential energy. A similar computation shows that the second-order Fréchet derivative of V at u is the bilinear map (2.26) Hence the eigenvalues of the second derivative coincide, up to their sign, with those of the Sturm-Liouville problem for the variational equation (2.21). In particular, the stable stationary solutions u * + and u * − are local minima of the potential energy. We call transition states between u * + and u * − the stationary points of V at which ∇ 2 V has one and only one negative eigenvalue. Thus • for periodic b.c., u * 0 is the only transition state for L 2π, while for L > 2π, all members of the family u * 1,ϕ are transition states; • for Neumann b.c., u * 0 is the only transition state for L π, while for L > π, the transition states are the two stationary solutions u * 1,± . Note that for given L and given b.c., V has the same value at all transition states. Transition states are characterised by the following property: Consider all continuous paths γ in E connecting u * − to u * + . For each of these paths, determine the maximal value of V along the path, and call critical those paths for which that value is the smallest possible. Then for any critical path, the maximal value of V is assumed on a transition state.

Main results
We can now state the main results of this work. We start with the case of Neumann b.c. We fix parameters r, ρ > 0 and an initial condition u 0 such that We are interested in sharp estimates of the expected first-hitting time E u 0 {τ + } for small values of ε.
Recall from (2.23) that the eigenvalues of the variational equation at u * 0 ≡ 0 are given by −λ k where λ k = (kπ/L) 2 − 1. Those at u * − are given by −ν − k where (2.28) When L > π, we denote the eigenvalues at the transition states u * 1,± by −µ k where We further introduce two functions Ψ ± : R + → R + , which play a rôle for the behaviour of E u 0 {τ + } when L is close to π. They are given by where I ±1/4 and K 1/4 denote modified Bessel functions of first and second kind. The functions Ψ ± are bounded below and above by positive constants, and satisfy (2.33) See Figure 3 for plots of these functions.
Theorem 2.5 (Neumann boundary conditions). For Neumann b.c. and sufficiently small r, ρ and ε, the following holds true.
1. If L < π and L is bounded away from π, then 2. If L > π and L is bounded away from π, then 3. If L π and L is in a neighbourhood of π, then 4. If L π and L is in a neighbourhood of π, then where C is given by (2.37), and the remainder R − is of the same order as R + . Note that (2.33) (together with the fact that µ k (L) → λ k (π) as L → π + ) shows that E u 0 {τ + } is indeed continuous at L = π. In a neighbourhood of order √ ε of L = π, the prefactor of the transition time is of order ε 1/4 , while it is constant to leading order when L is bounded away from π.
We have written here the different expressions for the expected transition time in a generic way, in terms of eigenvalues and potential-energy differences. Note however that several quantities appearing in the theorem admit more explicit expressions: is determined by solving (2.14) with the help of the first integral (2.16). For the symmetric double-well potential (2.4), it can be expressed explicitly in terms of elliptic integrals.
• The two identities imply that the prefactor in (2.34) is given by (2.41) • Since there is no closed-form expression for the eigenvalues µ k , it might seem impossible to compute the prefactor appearing in (2.35). In fact, techniques developed for the computation of Feynman integrals allow to compute the product of such ratios of eigenvalues, also called a ratio of functional determinants, see [For87,MT95,CdV99,MS01,MS03] .
We now turn to the case of periodic b.c. In that case, the eigenvalues of the variational equation at u * 0 ≡ 0 are given by −λ k where λ k = (2kπ/L) 2 − 1. Those at u * − are given by (2.42) When L > 2π, we denote the eigenvalues at the family of transition states u * 1,ϕ by −µ k where µ 0 < µ −1 = 0 < µ 1 < µ 2 , µ −2 < . . . (2.43) We further introduce two functions Θ ± : R + → R + , which play a rôle for the behaviour of E u 0 {τ + } when L is close to 2π. They are given by where Φ(x) = (2π) −1/2 x −∞ e −t 2 /2 dt denotes the distribution function of a standard Gaussian random variable. The functions Θ ± are bounded below and above by positive constants, and satisfy (2.47) See Figure 4 for plots of these functions.
Theorem 2.6 (Periodic boundary conditions). For periodic b.c. and sufficiently small r, ρ and ε, the following holds true.
1. If L < 2π and L is bounded away from 2π, then 2. If L 2π and L is in a neighbourhood of 2π, then (2.50) and the remainder R + satisfies (2.38). 3. If L 2π and L is in a neighbourhood of 2π, then where C is given by (2.50), and the remainder R − is of the same order as R + . 4. If L > 2π and L is bounded away from 2π, then (2.52) Note that for L 2π − O( √ ε), the prefactor of E u 0 {τ + } is proportional to √ ε/L. This is due to the existence of the continuous family of transition states owing to translation symmetry. The quantity plays the rôle of the "length of the saddle". One shows (cf. Section 6.2) that for L close to 2π, µ 1 is close to −2λ 1 and which implies shows that (2.51) and (2.52) are indeed compatible. As in the case of Neumann b.c., several of the above quantities admit more explicit expressions. For instance, the identities (2.40) imply that the prefactor in (2.48) is given by 2π See [Ste04] for an explicit expression of the prefactor for L > 2π, for a particular class of double-well potentials.
3 Outline of the proof

Potential theory
A first key ingredient of the proof is the potential-theoretic approach to metastability of finite-dimensional SDEs developed in [BEGK04,BGK05]. Given a confining potential V : R d → R , consider the diffusion defined by where W t denotes d-dimensional Brownian motion. The diffusion is reversible with respect to the invariant measure where Z is the normalisation. This follows from the fact that its infinitesimal generator Let A, B, C ⊂ R d be measurable sets which are regular (that is, their complement is a region with continuously differentiable boundary). We are interested in the expected first-hitting time Dynkin's formula shows that w A (x) solves the Poisson problem The solution of (3.5) can be expressed in terms of the Green function G A c (x, y) as Reversibility implies that the Green function satisfies the symmetry Another important quantity is the equilibrium potential It satisfies the Dirichlet problem The key observation is that the relations (3.10), (3.7) and (3.6) can be combined to yield The approach used in [BEGK04] is to take C to be a ball of radius ε centred in x, and to use Harnack inequalities to show that w A (z) w A (x) on C. It then follows from (3.11) that The left-hand side can be estimated using a priori bounds on the equilibrium potential.
Thus a sufficiently precise estimate of the capacity cap C (A) will yield a good estimate for . Now it follows from Green's identities that the capacity can also be expressed as a Dirichlet form evaluated at the equilibrium potential: (3.14) Even more useful is the variational representation where H A,B denotes the set of twice weakly differentiable functions satisfying the boundary conditions in (3.9). Indeed, inserting a sufficiently good guess for the equilibrium potential on the right-hand side immediately yields a good upper bound. A matching lower bound can be obtained by a slightly more involved argument. Several difficulties prevent us from applying the same strategy directly to the infinitedimensional equation (2.1). It is possible, however, to approximate (2.1) by a finitedimensional system, using a spectral Galerkin method, to estimate first-hitting times for the finite-dimensional system using the above ideas, and then to pass to the limit.

Spectral Galerkin approximation
Let P d : E → E be the projection operator defined by where the e k are the basis vectors compatible with the boundary conditions, given by (2.9) or (2.10). We denote by E d the finite-dimensional image of E under P d . Let u t (x) be the mild solution of the SPDE (2.1) and let u called Galerkin approximation of order d. It is known (see, for instance, [Jet86]) that (3.17) is equivalent to the finite-dimensional system of SDEs where the W k (t) are independent standard Brownian motions, and the potential is given by (3.19) We will need an estimate of the deviation of the Galerkin approximation u (d) t from u t . Such estimates are available in the numerical analysis literature. For instance, [Liu03] provides an estimate for the Sobolev norm u 2 H r 0 = k (1 + k 2 ) r |y k | 2 , with r < 1/2. We shall use the more precise results in [BJ13], which allow for a control in the (stronger) sup norm. Namely, we have the following result: Theorem 3.1. Fix a T > 0. Let U be locally Lipschitz, and assume for all ω ∈ Ω. Then, for any γ ∈ (0, 1/2), there exists an almost surely finite random variable Z : Ω → R + such that for all ω ∈ Ω.
Proof: The result follows directly from [BJ13, Theorem 3.1], provided we verify the validity of four assumptions given in [BJ13, Section 2].
• Assumption 1 concerns the regularity of the semigroup e ∆t associated with the heat kernel, and is satisfied as shown in [BJ13, Lemma 4.1]. • Assumption 2 is the local Lipschitz condition on U .
• Assumption 3 concerns the deviation of P d W (t, x) from W (t, x) and is satisfied according to [BJ13, Proposition 4.2]. • Assumption 4 is (3.20).

Proof of the main result
For r, ρ > 0 sufficiently small constants we define the balls If u * ts stands for a transition state between u * − and u * + , we denote by the communication height from u * − to u * + . We fix an initial condition u 0 ∈ A, and write u (3.25) We first need some a priori bounds on moments of these hitting times. They are stated in the following result, which is proved in Section 5.
Proposition 3.2 (A priori bound on moments of hitting times). For any η > 0, there exist constants ε 0 = ε 0 (η), The next result applies to all finite-dimensional Galerkin approximations, and is based on the potential-theoretic approach. The detailed proof is given in Sections 6 and 7.

Proposition 3.3 (Bounds on expected hitting times in finite dimension). There exists
where the quantities C(d, ε), H(d) and R ± d,B (ε) are explicitly known. They satisfy • lim d→∞ C(d, ε) =: C(∞, ε) exists and is finite; Then we have the following result.
Proposition 3.4 (Averaged bounds on the expected first-hitting time in infinite dimension). Pick a δ ∈ (0, ρ). There exist ε 0 > 0 and probability measures ν + and ν − on ∂A such that for 0 < ε < ε 0 , (3.28) Proof: To ease notation, we write B = B(ρ), B ± = B(ρ ± δ) and T Kr = C(∞, ε) e H 0 /ε . For given v 0 ∈ ∂A and K > 0, define the event where v t and v (d) t denote the solutions of the original and the projected equation with respective initial conditions v 0 and P d v 0 . Theorem 3.1 and Markov's inequality imply Choosing ε 0 and d 0 (ε) such that Proposition 3.2 applies, the last summand can be bounded using (3.26) and the Cauchy-Schwarz inequality. This yields We decompose (3.32) In order to estimate the first summand, we note that by definition of B, B + and B − , The second summand in (3.32) can be bounded by Cauchy-Schwarz: This shows that Inserting (3.37) and (3.38) in (3.36) shows that lim sup Taking K sufficiently large, the third summand can be made smaller than the second one. The upper bound in (3.28) then follows with ν + = ν d,B − for d sufficiently large. The proof of the lower bound is analogous.
To finish the proof of the main result, we need Theorem 3.5. There exist constants ε 0 , κ, t 0 , m, c > 0 such that such that for all ε < ε 0 , Here u t and v t denote the mild solutions of the SPDE (2.1) with respective initial conditions u 0 and v 0 .
This result has been proved in [MOS89, Corollary 3.1] in the case of Dirichlet b.c., under the condition L = π. The reason for this restriction is that for L = π, the Hessian at the potential minimum has a zero eigenvalue. For Neumann and periodic b.c., this difficulty does not occur, because the potential minima always have only strictly negative eigenvalues.
Proposition 3.6 (Main result). Pick δ ∈ (0, ρ/2). There exists ε 0 > 0 such that for all ε < ε 0 , (3.41) Proof: As before we write B = B(ρ), B ± = B(ρ ± δ). Given a constant T t 0 , consider the event Then Theorem 3.5 and the standard large-deviation estimate Corollary 5.10 show that for any T > 0, there exist constants ε 0 , κ 1 > 0 such that for ε < ε 0 , provided T is large enough that rc e −mT δ/2. In order to prove the upper bound, we start by observing that which can be bounded above with Proposition 3.4. Furthermore, by Cauchy-Schwarz, we have For the lower bound, we use the decomposition The first term on the right-hand side can be bounded below with Proposition 3.4, while the second one is bounded above by This concludes the proof, provided we choose η < κ 1 /2 when applying Proposition 3.4.

Deterministic system
This section gathers a number of needed results on the deterministic partial differential equation (2.13). Some general properties of the equation are discussed e.g. in [CI75,Jol89]. In Section 4.1 we introduce various function spaces and inequalities required in the analysis. In Section 4.2, we establish some general bounds on the potential energy V and its derivative. Section 4.3 analyses the behaviour of the potential energy at bifurcation points, and Section 4.4 contains a result on the relation between V and its restrictions to finite-dimensional subspaces.

Function spaces
We introduce two scales of function spaces that will play a rôle in the sequel. Let I denote either a compact interval [0, L] ⊂ R or the circle T 1 = R /(2πZ ).
We denote by C 0 = C 0 (I) the space of all continuous functions u : I → R . Note that I is compact so that the functions from C 0 are bounded. When equipped with the sup norm For s 0, we define the Sobolev norm and denote by H s = H s (T 1 ) = {u ∈ L 2 (T 1 ) : u H s < ∞} the fractional Sobolev space (also called Bessel potential space). Note that H s is a Hilbert space, and H 0 = L 2 . The norm u H 1 can be equivalently defined by u 2 Lemma 4.1. For any α 0 and s > α + 1/2, there exists a constant C = C(α, s) such that As a consequence, we have H s (T 1 ) ⊂ C α (T 1 ).
Remark 4.2. In the particular case s = 1, (4.4) can be strengthened to Morrey's inequality Let p, q satisfy 1 p 2 q ∞ and 1 p + 1 q = 1. Then the Hausdorff-Young inequalities [DS88] state that there exist constants C 1 (p) and C 2 (p) such that We consider now some properties of convolutions y * z defined by Young's inequality states that for 1 p, q, r ∞ such that 1 p + 1 q = 1 r + 1, y * z r y p z q . (4.8) Lemma 4.3. Let r, s, t ∈ (0, 1/2) be such that t < r +s−1/2. Then there exists a constant (4.10) Splitting the sum at −|k|, 0, |k|/2, |k| and 2|k|, and bounding each sum by an integral, one easily shows that 1 (4.11) (4.12) By the Cauchy-Schwarz inequality, (4.13) Ifỹ 2 ,z 2 denote the vectors with componentsỹ 2 l andz 2 l , it follows from (4.11) that by Young's inequality, and the results follows since Finally, the following estimate allows to bound the usual r -norm in terms of Sobolev norms.

Bounds on the potential energy
In this subsection, we derive some bounds involving the potential energy and its gradient. Periodic and Neumann boundary conditions can be treated in a unified way by writing the Fourier series as where b = 1 and z k = z −k for Neumann b.c., and b = 2 for periodic b.c. The value of the potential expressed in Fourier variables becomes Lemma 4.5 (Bounds on V ). There exist constants α , β , M 0 > 0 such that (4.20) Proof: By Assumption 2.1 (U3), we have which implies the upper bound. The lower bound is obtained in a similar way, using Assumption 2.1 (U4).
The gradient of V (z) and the Fréchet derivative of V [u] are related by where e k is defined in (2.9) or (2.10), respectively. Thus, by (2.25) and Parseval's identity, Lemma 4.6 (Lower bound on ∇ V 2 2 ). For any ρ > 0 there exists M 1 (ρ) such that (4.26) Proof: We expand the square in (4.25) and evaluate the terms separately. Using Assumptions 2.1 (U5) und (U6) and integration by parts, we have for any γ > 0 (4.28) For any ρ > 0, we can find a γ such that the term in brackets is bounded below by 2ρ(1 + k 2 ), uniformly in k. This proves the result.

Normal forms
We will rely on normal forms when analysing the system for L near a critical value. In this situation we will always assume that the local potential U is in C 5 , so that we can write its Taylor expansion as (4.29) Then for small u, the potential energy admits the expansion shows that the potential energy in Fourier variables can be decomposed as where the V n (z) are given by the convolutions where λ k = ν k − 1. It follows from (4.16), applied for p = 5, that the remainder satisfies Proposition 4.8. Let L be such that λ k is bounded away from 0 for all k = ±1. Then there exists a map g : and the remainder satisfies Proof: In the course of the proof, we will need Sobolev norms with indices q, r, t, satisfying the relations 0 < q, r < t < s < 1/2 , t < 2s − 1/2 , r < 2t − 1/2 and q < 3t − 1 .
This is always possible for 5/12 < s < 1/2. In this proof, we will denote by C 0 any constant appearing when applying Lemma 4.3. Its value may change from one line to the next one.
1. Let g (2) : R Z → R Z be homogeneous of degree 2, and satisfy g k (z). Then, expanding and grouping terms of equal order we get with remainders that can be written as for some θ 1 , θ 2 ∈ [0, 1]. We want to choose g (2) (z) in such a way that the terms of order 3 in V (z + g (2) (z)) cancel. This can be achieved by taking The choice of these coefficients is not unique, but we can make it unique by imposing the symmetry Indeed, these are all the terms contributing to the monomial z k z l z −k−l in the first sum in (4.37). Then simple combinatorics show that all b k,l belong to the interval [1/6, 6]. This choice has the further advantage that on the set whenever |k|, |l| = 1. The term of order 4 of V (z + g (2) (z)) is given by Note that the convolution structure is preserved. In order to show that the sums indeed converge, we first note that since t < 2s − 1/2, we have The second sum in (4.42) can be bounded as follows: This shows that V 4 (z) indeed exists, and satisfies Next we estimate the remainders. The remainder r 1 (z) can be bounded as follows: A similar computation yields We have thus obtained can be deduced from (4.41). By construction, A(z) is self-adjoint with respect to the scalar product weighted by the λ k , and thus has real eigenvalues. Its 1 -operator norm satisfies (4.51) Hence the spectral radius ρ(z) of A(z) has order z H s . Now if ρ(z) < 1 and we denote the eigenvalues of A(z) by a k (z), (4.52) It follows that |det(1l + A(z))| e Tr(A(z)) , and one easily shows that Tr(A(z)) = O(a 3 |z 0 |) O(a 3 z H s ). A matching lower bound can be obtained in a similar way. This proves that the Jacobian of the transformation where the coefficients b k 1 ,k 2 ,k 3 are invariant under permutations of k 1 , k 2 and k 3 . In addition, we require invariance under sign change and This guarantees in particular that holds on the set {z k = z −k }. The coefficients b k 1 ,k 2 ,k 3 can now be chosen in such a way that k z kg contains only one term, proportional to z 2 −1 z 2 1 , which cannot be eliminated becausẽ g for some constant C 4 . Along the lines of the above calculations, one checks that R 1 (z) = O( z 5 H s ), and that the Jacobian of the transformation This proves (4.35) and (4.36). 4. It remains to compute the coefficient C 4 of the resonant term. To do this, it is sufficient to compute the terms containing z ±1 ofg (2) 0 (z) andg (2) ±2 (z), which are the only ones contributing to the resonant term. One finds This result has important consequences for the behaviour of the potential near bifurcation points. In the case of Neumann b.c., λ 0 = −1 and λ 2 = (4π 2 /L 2 ) − 1. Thus the coefficient C 4 of the term z 2 1 z 2 −1 is given by (4.60) In particular, at the bifurcation point we have The expression (4.33) for the normal form shows that if C 4 (π) > 0, the system undergoes a supercritical pitchfork bifurcation at L = π. This means that the origin is an isolated stationary point if L < π, while for L > π two new stationary points appear at a distance of order √ L − π from the origin. They correspond to the functions we denoted u * 1,± . As a consequence, the period T (E) defined in (2.18) must grow for small positive E, to be compatible with the existence of nonconstant stationary solutions for L > π. An analysis of the Hessian matrices of V at u * 1,± shows that they have one negative eigenvalue for L slightly larger than π. This must remain true for all L > π because we know that the stationary solutions u * 1,± remain isolated when L grows.
In the case of periodic b.c., λ 0 = −1 and λ 2 = (16π 2 /L 2 ) − 1. Thus the coefficient C 4 of the term z 2 1 z 2 −1 is given by (4.62) The value C 4 (2π) at the bifurcation point is equal to the value (4.61) of C 4 (π) for Neumann b.c. Thus the condition on the bifurcation being supercritical is exactly the same as before.
The difference is that instead of being equal, z 1 and z −1 are only complex conjugate, and thus the centre manifold at the bifurcation point is two-dimensional. The invariance of the potential under translations u(x) → u(x + ϕ) for any ϕ ∈ R implies that V (z) is invariant under z k → e i kϕ2π/L z k . This and the expression (4.33) for the normal form show that for L > 2π, there is a closed curve of stationary solutions at distance of order √ L − 2π from the origin. It corresponds to the family of solutions we denoted u * 1,ϕ . An analysis of the Hessian of V at any u * 1,ϕ shows that it has one negative and one vanishing eigenvalue (due to translation symmetry).
Finally note that a similar normal-form analysis can be made for the other bifurcations, at subsequent multiples of π or 2π. We do not detail this analysis, since only saddles with one negative eigenvalue are important for metastable transition times.

The truncated potential
Let V (d) be the restriction of the potential V to the subspace of Fourier modes z k such that |k| d. For given d, let us write z = (v, w), where v is the vector of Fourier components with |k| d and w contains the vector of remaining components. Then (4.63) Proposition 4.9. There exists d 0 < ∞ such that for d d 0 , the potentials V (d) and V have the same number of nondegenerate critical points, and with the same number of negative eigenvalues.
Proof: A critical point (v * , w * ) of V has to satisfy the conditions Lemma 4.6 implies that all critical points of V have an H 1 -norm bounded by some constant M . Let us prove that for v H 1 M . Indeed it follows from (4.19) that where u(x) = | | d v e i b πx/L . Since v H 1 M and U is at least continuously differentiable, the Fourier components of U (u(x)) decay like k −1 at least, which implies (4.66).
Let (v * , w * ) be a critical point of V and consider the function (4.68) Then F (0, w * ) = 0 and ∂ ξ F (0, w * ) = ∂ vv V (v * , w * ). Thus if (v * , w * ) is nondegenerate, the implicit function theorem implies that in a neighbourhood of w = w * , there exists a continuously differentiable function h with h(w * ) = 0 and such that all solutions of F (ξ, w) = 0 in a neighbourhood of (0, w * ) are given by ξ = h(w). In particular, choosing d large enough, we can assume that h is defined for w = 0 , and we get , which is unique in the neighbourhood of (v * , w * ). Conversely, let v * be a stationary point of V (d) . The same implicit-function-theorem argument shows that if v * is nondegenerate, then there exists a continuously differentiable function h, with h(0) = 0, such that all solutions of . Now let us consider the function (4.70) Then g(0) = ∂ w V (v * , 0) has an 2 -norm of order d −1/2 by (4.66). Furthermore, (4.71) The first matrix on the right-hand side has eigenvalues of order d 2 , while the second one is small as a consequence of (4.66). Thus ∂ w g is invertible near w = 0 for sufficiently large d, and the local inversion theorem shows that g(w) has an isolated zero at a point w * near w = 0. This yields the existence of a unique stationary point (v * = h(w * ), w * ) of V in the vicinity of (v * , 0).

A priori estimates
This section has two major aims: • Show that the first-hitting time of a given set B admits a second moment, bounded uniformly in the dimension d; • Derive a priori bounds on the equilibrium potential h A,B (x) = P x {τ A < τ B }.
We start in Section 5.1 by recalling some general bounds involving sup and Hölder norms of solutions of the SPDE (2.1). In order to estimate moments of first-hitting times, the space being unbounded, we repeatedly need the Markov property to restart the process when it hits certain sets. This is most efficiently done using Laplace transforms, and we prove some useful inequalities in Section 5.2. Section 5.3 recalls some large-deviation results. Sections 5.4 and 5.5 contain the main estimates on moments, respectively, for the infinite-dimensional system and for its Galerkin approximation. Finally, Section 5.6 contains the estimates of the equilibrium potential.

A priori bounds on solutions of the SPDE
The solution of the heat equation ∂ t u = ∆u with initial condition u 0 ∈ L 2 (T 1 ) can be written u t = e ∆t u 0 , where e ∆t stands for convolution with the heat kernel Here the e k are the eigenfunctions of the Laplacian, defined in (4.2).
Lemma 5.1 (Smoothing effect of the heat semigroup). For any s 0, there is a finite constant C(s) such that for all u 0 ∈ L 2 (T 1 ) The result follows by computing the H s -norm, and using the fact that (2xt) s e −2xt is bounded by a constant, depending only on s.
Note that by Lemma 4.1, this implies where the constant C depends only on α and s. Consider now the stochastic convolution where W (t) is a cylindrical Wiener process on L 2 (T 1 ). It is known that almost surely, for all t > 0 and all s < 1/2 and α < 1/2 (see e.g. [Hai09, p. 50]). We will need to control the sup and Hölder norms of the rescaled process √ 2εW ∆ (t).
Proposition 5.2 (Large-deviation estimate for W ∆ ). For any α ∈ [0, 1/2) and T > 0, there exists a constant κ > 0 such that for all H, η > 0, there exists an ε 0 > 0 such that for all ε < ε 0 , Proof: Let H denote the Cameron-Martin space of the cylindrical Wiener process, defined by Schilder's theorem for Gaussian fields shows that the family { √ 2ε W } ε>0 satisfies a largedeviation principle with good rate function From the large-deviation principle for parabolic SPDEs established in [FJL82,CM97], it follows in particular, that the family { √ 2εW ∆ } ε>0 satisfies a large-deviation principle with good rate function otherwise . (5.12) Now observe that if ψ = Z[ϕ], for any T 1 ∈ [0, T ] and any s ∈ [0, 1) one has by Lemma 5.1 Since s < 1, the integral is finite (and increasing in T 1 ). Together with Lemma 4.1, this proves that for all T 1 ∈ [0, T ], 0 α < 1/2 and s satisfying α + 1/2 < s < 1, where C 1 is increasing in T 1 . By a standard application of the large-deviation principle (see e.g. [FJL82,CM97]) (5.15) The bound (5.14) implies that the right-hand side is bounded above by −H 2 /C 1 (T, α), which concludes the proof.
We now turn to properties of mild solutions of the full nonlinear SPDE, given by 1. There exists γ > 0 such that for any u 0 and any t 0,  Observe that in the case ε = 0, we can find a constant M uniform in t such that u t L ∞ M (1 + u 0 L ∞ ) for all t 0. Hence Proposition 5.2 shows that for all H 1 > 0, for some κ(T ) > 0 and ε small enough. Combining (5.16) and Proposition 5.3, we obtain the following estimate on the Hölder norm of u T at a given time T > 0.

Laplace transforms
Let (E, · ) be a Banach space, and let (x t ) t 0 be an E-valued Markov process with continuous sample paths. All subsets of E considered below are assumed to be measurable with respect to the Borel σ-algebra on E.
Recall that the Laplace transform of an almost surely finite positive random variable τ is given by for any λ ∈ C . There exists a c ∈ [0, ∞] such that the Laplace transform is analytic in λ for Re λ < c.
To control first-hitting times of bounded sets B ⊂ E, we will introduce an auxiliary set C with bounded complement, B ∩ C = ∅, such that the process is unlikely to hit C before B. On the rare occasions the process does hit C before B, we will use the strong Markov property to restart the process on the boundary ∂C. The following proposition recalls how the restart procedure is encoded in Laplace transforms.
Proposition 5.5 (Effect of restart on Laplace transform). Let B, C ⊂ E be disjoint sets, and let x ∈ B ∪ C. Then (5.26) In the same way, or by differentiating (5.25) with respect to λ and evaluating in λ = 0, the moments of first-hitting times can be expressed. Assuming their existence, for the first two moments we find for any choice of disjoint sets B and C, and any x ∈ (B ∪ C). Below we will use the notations It follows that for any three pairwise disjoint sets A, B and C, and a similar relation holds for the second moment.
Lemma 5.6 (Moment estimate based on the Markov property). Let B ⊂ E be such that for some T > 0. Then for any n ∈ N , (5.32) Proof: The Markov property implies that for any m ∈ N and any x ∈ B c ,

Integration by parts shows that
The result is thus a consequence of the inequality and satisfies . (5.37) A sharper bound on the Laplace transform can be obtained by a direct integration by parts, but this does not automatically lead to better bounds on the moments.
Next, we will iterate the estimate (5.30) in order to get a better bound on the moments of first hitting times.
Corollary 5.8 (Three-set argument). Let A, B, C ⊂ E be such that A, B and C are pairwise disjoint, and assume P A τ C < τ B < 1, E A τ k B < ∞ and E ∂C τ k B < ∞ for k = 1, 2. Then (5.39) Proof: We introduce the shorthands for k = 1, 2. Note that X k , Y k < ∞ and p < 1 according to our assumptions. Applying (5.30), once to the triple (A, B, C) and once to the triple (∂C, B, A), yields where we have bounded P ∂C {τ A < τ B } by 1. In addition, we used that hitting B requires first exiting from A which is necessarily realized by passing through ∂A. This implies which proves (5.38). Starting from (5.28), we find Together with (5.42), this gives (1 − p)(X 2 + pY 2 ) + 2(X 2 1 + (1 + p)X 1 Y 1 + p 2 Y 2 1 ) , (5.44) and the result follows after some algebra, using Jensen's inequality. Note that we have overestimated some terms in order to obtain a more compact expression.

Large deviations
As shown in [FJL82,Fre88], the family {u t } ε>0 of mild solutions of the SPDE with initial condition u 0 ∈ E = C 0 (T 1 ) satisfies a large-deviation principle in E, equipped with the sup norm, with rate function The right-hand side being independent of T , the result follows.
By a direct application of the large-deviation principle to the set of paths starting in u 0 and reaching A in a time less or equal to T , we obtain the following estimate.

Bounds on moments of τ B in infinite dimension
We will now apply the results of the previous sections to the mild solution u t of the SPDE, with E = C 0 (T 1 ) equipped with the sup norm. We fix α ∈ (0, 1/2) and introduce two families of sets Note that A 2 ⊂ A 1 , and that A 2 is a compact subset of E, while A 1 is not compact as a subset of E.
Let B ⊂ C 0 (T 1 ) be a non-empty, bounded open set in the · L ∞ -topology. We let be the cost, in terms of the rate function, to reach the set B from either one of the local minima. Our aim is to estimate the first two moments of τ B , using the three-set argument Corollary 5.8 for A = A 2 (R 2 ) \ B and C = A 1 (R 0 ) c , with appropriately chosen R 0 and R 2 . We thus proceed to estimating the quantities appearing in the right-hand side of (5.38) and (5.39). We will use repeatedly the fact that Proposition 5.2 and (5.18) yield the estimate valid for all ε < ε 0 (T, η, H).
Proof: We start by fixing an initial condition u 0 ∈ A 2 (R 2 ). By the large-deviation principle, we have Following a classical procedure similar to the one in the proof of [FJL82,Theorem 9.1], we construct a path ϕ * u 0 connecting u 0 to a point u * ∈ B such that I(ϕ * u 0 ) 2H 0 +η. This can be done by following the deterministic flow from u 0 to the neighbourhood of a stationary solution of the deterministic PDE at zero cost, then connecting to that stationary solution at finite cost. Any two stationary solutions and u * can also be connected at finite cost, and ϕ * u 0 is obtained by concatenation. It follows that there exists ε 0 (η, u 0 ) > 0 such that The set A 2 (R 2 ) being compact, we can find, for any δ > 0, a finite cover of A 2 (R 2 ) with N (δ) balls of the form D n = {u ∈ C 0 (T 1 ) : u − u n L ∞ < δ}. Hence where I * (δ) = max n I(ϕ * un ) < 2H 0 + η and ε 1 (η, δ) = min n ε 0 (η, u n ) > 0. Consider now two solutions u (2) 0 ∈ D n . By a Gronwall-type argument similar to the one given in [FJL82,Theorem 5.10] and the bound (5.21), for any κ 1 > 0 there exist K(κ 1 ) > 0 such that The set B being open, it contains a ball {u ∈ C 0 (T 1 ) : u − u * L ∞ < ρ} with ρ > 0. Thus choosing δ = e −KT ρ, combining (5.60) and (5.61), we get for all sufficiently small ε. Now choosing, e.g., κ 1 = I(ϕ * u 1 ) and δ = e −K(κ 1 )T ρ guarantees that the term e −κ 1 /2ε is negligible.
Proof: For any T 1 > 1 and R 1 > 0, we have The third term on the right-hand side is bounded by 1 − 1 2 e −(H 0 +η)/ε by Proposition 5.12. Proposition 5.4 shows that (5.65) while (5.53) shows that there exists κ 3 (T 1 ) > 0 such that (5.66) We have estimated the probability in (5.64) by three terms. By first choosing R 1 large enough so that the exponent in (5.66) is smaller than −(H 0 +2η)/ε, and then R 2 sufficiently large for the exponent in (5.65) to be smaller than −(H 0 + 2η)/ε as well, we see that the third summand in (5.64) is of leading order. This shows that the probability in (5.64) is smaller than 1 − 1 4 e −(H 0 +η)/ε for sufficiently small ε, and therefore, the result follows from Lemma 5.6 with T 0 = 4T 1 .
Proposition 5.14. For any R 2 > 0, there exists a constant R 0 > R 2 such that for sufficiently small ε.
Proof: For any T > 0 and n ∈ N , we have We introduce the quantities Using the Markov property, they can all be expressed in terms of p 1 , q 1 and r 1 . Namely, and one easily shows by induction that Splitting again according to whether u nT belongs to A 2 (R 2 ) or not, we get In a similar way, we have It follows by induction that p n p n 1 + n 2 q 1 (1 − p 1 ) . (5.74) Putting together the different estimates, we obtain P A 2 (R 2 ) τ A 1 (R 0 ) c τ B p n + r n p n 1 + r 1 + nq 1 1 + n(1 − p 1 ) .
(5.75) It remains to estimate p 1 , q 1 and r 1 . Proposition 5.12 shows that where H 1 = H 0 + η. We can estimate q 1 by and both terms can be bounded as in the proof of Proposition 5.13. An appropriate choice of R 1 , R 2 ensures that q 1 e −H 1 /ε . Finally, by (5.19) we also have for sufficiently large R 0 . The choice n = (4 log 2) e H 1 /ε (5.79) yields log(p n 1 ) − 1 2 n e −H 1 /ε −2 log(2) so that p n 1 1/4, while the other terms in (5.75) are exponentially small. This concludes the proof.
Combining Propositions 5.11, 5.13 and 5.14, we finally get the main result of this section.
Corollary 5.15 (Main estimate on the moments of τ B ). Let B ⊂ C 0 (T 1 ) be a non-empty, bounded open set in the · L ∞ -topology. Then for all R 0 , η > 0, there exist constants ε 0 > 0 and T 0 < ∞ such that for all ε < ε 0 .
Proof: Making R 0 larger if necessary, we choose R 2 and R 0 > R 2 large enough for the three previous results to hold, and such that B A 1 (R 0 ). We apply the three-set argument n!T n 0 (5.82) by Proposition 5.11. Finally, P A {τ C < τ B } 1/2 by Proposition 5.14. This shows the result for initial conditions in A 2 (R 2 ) \ B. Now we can easily extend these bounds to all initial conditions in A 1 (R 0 ) by using Proposition 5.11 and restarting the process when it first hits A 2 (R 2 ) \ B.
Note that in the proof of Proposition 3.2, we apply this result when B is a neighbourhood of u * + . In that case, H 0 (B) is equal to the potential difference between the transition state and the local minimum u * − .

Uniform bounds on moments of τ B in finite dimension
In this section, we derive bounds on the moments of first-hitting times, similar to those in Corollary 5.15, for the finite-dimensional process, uniformly in the dimension.
For E = C 0 (T 1 ) and d ∈ N , we denote by E d the finite-dimensional space be an open ball of radius r in the · L ∞ -norm. We assume that the centre of B is some w ∈ E d . As in Corollary 5.15, we define A = A 2 (R 2 ) \ B. Then, there exist constants ε 0 > 0 and H 1 , T 1 < ∞ such that for any ε ∈ (0, ε 0 ), there is a d 0 (ε) ∈ N such that for all d d 0 (ε).
Proof: We fix constants δ, T > 0, and let Ω d be the event Theorem 3.1 shows that for given γ < 1/2, there exists an almost surely finite random variable Z such that P(Ω c d ) P Z > δd γ . (5.86) Given D ⊂ C 0 (T 1 ), we define the sets Then for any initial condition u 0 ∈ E d , we have the two inequalities , Let R 0 be as in the proof of Corollary 5.15, and define the sets We assume δ to be small enough for B to be non-empty. Note that C and C are the complements of open balls in the · L ∞ -norm while B is the open ball of radius r − δ around the center w of B.
Applying the three-set argument (5.38) to the triple (A d,+ \ B d , B d , C d,− ), where the sets are disjoint for sufficiently large R 0 , we get (5.90) Using the facts that A d ⊂ A d,+ , (B ) d,+ = B d and (C ) d,+ ⊂ C d,− , we now reduce the estimation of each of the terms on the right-hand side of (5.90) to probabilities that can be controlled, via (5.88), in terms of the infinite-dimensional process.
(5.91) By (5.88) we have for any u 0 ∈ E d \ (A 2 (R 2 )) d,+ (5.92) As we have seen in (5.56), the first term on the right-hand side can be bounded by 1/2. As for the second term, (5.86) shows that it is smaller than 1/4 for d d 0 (ε) large enough. Hence the right-hand side of (5.91) can be bounded by 4T /3.

The term E
(B ) d,+ ∪(C ) d,+ can be estimated in a similar way, by comparing with P A d,+ {τ B ∪C > T } and proceeding as in the proof of Proposition 5.13, cf. (5.64).
Finally, we have the bounds (5.93) Proposition 5.12 and (5.78) show that the sum of the first two terms on the right-hand side can be bounded by 1 − 1 2 e −H 1 /ε . The third term can be bounded by 1 4 e −H 1 /ε , provided d is larger than some (possibly large) d 0 (ε).
This completes the bound on the first moment, and the second moment can be estimated in the same way.

Bounds on the equilibrium potential in finite dimension
The aim of this subsection is to obtain bounds on the equilibrium potential when A and B are small open balls, in the L ∞ -norm, around the local minima u * ± of the potential V , and as before A d = A ∩ E d and B d = B ∩ E d . We denote the centre of A by u * 1 and the centre of B by u * 2 , where either u * 1 = u * − and u * 2 = u * + or vice versa. We now derive a bound on h For 0 < δ < r, we define Ω d = Ω d (δ) as in (5.85). Then we have, in a way similar to (5.88), , We choose δ small enough that H(u 0 , A + ) η/2. The large-deviation principle shows that lim sup ε→0 2ε log P u 0 τ A + T = −2H(u 0 , A + ) −η . (5.98) Thus there exists ε 0 (η) > 0, independent of T , such that for ε < ε 0 and d d 0 .
To estimate the second term in (5.97), we assume δ < r/2 and let B − = { u−u * 2 L ∞ < r − 2δ}. Assume T is large enough that the deterministic solution starting in u 0 reaches B − in time T . Then the large-deviation principle in [Fre88] shows that the stochastic sample path starting in u 0 is unlikely to leave a tube of size δ in the L ∞ -norm around the deterministic solution before time T , which implies that there exists κ > 0 such that for ε small enough. This implies the result, with H 0 = κδ 2 ∧ η/4. As for the uniformity in u 0 , note that after a first finite time T 1 we may assume that the process has reached a compact subset, cf. the proof of Proposition 5.13. Restarting from this compact subset a standard compactness argument yields the uniformity of ε 0 , δ 0 and H 0 in u 0 .
Next we derive a more precise bound, which is useful in situations where we know V [u 0 ] explicitly. Proof: Fix a u 0 with V [u 0 ] M . We decompose the equilibrium potential in the same way as in (5.96) and (5.97). It follows from (5.98) and Lemma 5.9 that there exists ε 0 (η) > 0, independent of T , such that holds for all ε < ε 0 . We choose δ in the definition of A + small enough that V (u 0 , A + ) V (u 0 , A) − η/2, and finally d d 0 (η, ε) where d 0 is large enough that P(Ω c d 0 ) e −1/ηε . This shows that for ε < ε 0 and d d 0 .
In order to estimate P u 0 {τ B − > T }, we let D(κ) be the set of u ∈ E such that V (u, A + ) > 0 and ∇V [u] L 2 > κ. Then we can decompose where F (κ) = D(κ) c ∩ B c − . Now the same argument as above shows that Note that lim κ→0 V (u 0 , F (κ)) = V (u 0 , F (0)). Let us show that V (u 0 , F (0)) = V (u 0 , A + ) provided δ is small enough. We proceed in two steps: The fact that A and B have disjoint closure implies that A + ∩ B − = ∅ for sufficiently small δ. The fact that V (u, A + ) > 0 in D(0) shows that A + ∩ D(0) = ∅. It follows that A + ∩ F (0) c = ∅, and thus A + ⊂ F (0), which implies V (u 0 , F (0)) V (u 0 , A + ). 2. Assume by contradiction that V (u 0 , F (0)) < V (u 0 , A + ). Then there must exist a path ϕ, connecting u 0 to a point u ∈ F (0), on which the potential remains strictly smaller than V (u 0 , A + ) + V [u 0 ]. If we can show that V (u, A + ) = 0, then this implies that we can connect u 0 to A + , via u, by a path on which the potential remains strictly smaller than V (u 0 , A + ) + V [u 0 ], contradicting the definition of V (u 0 , A). It thus remains to show that V (u, A + ) = 0. Note that Since u ∈ F (0) = D(0) c ∩ B c − , we have u = u * 2 and either V (u, A + ) = 0, or ∇V [u] = 0. However, the assumptions imply that u * 2 is the only stationary point in the set {u : V (u, A + ) > 0}, so that necessarily V (u, A + ) = 0.
We have thus proved that V (u 0 , F (0)) = V (u 0 , A + ), and it follows that there exists a δ 0 (η) such that for δ < δ 0 (η) for all κ < κ 0 . It remains to estimate the first term on the right-hand side of (5.105). Let ϕ be a continuous path starting in u 0 and remaining in D(κ) up to time T . Then its rate function satisfies where we have used the fact that the second integral is proportional to ∇V 2 L 2 , cf. (4.25). The large-deviation principle implies that Since V [u 0 ] M , we can find for any κ > 0, a T = T (κ, M ) such that the right-hand side is smaller than −V (u 0 , A). Finally note that {u 0 : V [u 0 ] M } is contained in a closed ball in the C 1/2 -norm, so that a standard compactness argument allows to choose ε 0 and d 0 uniformly in u 0 from this set. This concludes the proof. 6 Estimating capacities 6.1 Neumann b.c.
We consider the potential energy for functions u(x) containing at most 2d + 1 nonvanishing Fourier modes and satisfying Neumann boundary conditions, that is where y 0 = z 0 and y k = √ 2z k = √ 2z −k for k 1. The expression V of the potential in Fourier variables follows from (4.31) and (4.32), with the sums restricted to −d k d.
Note that

L < π
We consider first the case where L π − c for some constant c > 0. We know that in this case, V has only three stationary points, all lying on the y 0 -axis. One of them is the origin O, where the Hessian of V has eigenvalues Thus O is a saddle with one-dimensional unstable manifold, which in this case is contained in the y 0 -axis. Let W s (O) denote the d-dimensional stable manifold of the origin. Proof: Let y ⊥ = (y 1 , . . . , y d ). The centre-stable manifold theorem for differential equations in Banach spaces [Gal93, Theorem 1.1] shows that W s (O) can be locally described by a graph of the form y 0 = g(y ⊥ ). More precisely, the nonlinear part of ∇ V (y) being of order y 2 H s for s > 1/4, there exist constants ρ, M > 0 such that holds for all s > 1/4, we have in particular whenever y ⊥ H 1 ρ. Thus using (6.9) to bound g(y ⊥ ) 2 , we obtain the existence of constants m 1 , ρ 1 > 0 such that (6.12) We have used the fact that y 2 H 1 = |y 0 | 2 + y ⊥ 2 H 1 and estimated |y 0 | 2 on the stable manifold by applying (6.9) once more. A similar computation shows that −∇ V (y) · ∇( y 2 H 1 ) < 0 ∀y ∈ W s (O) : y H 1 ρ 1 , (6.13) that is, the vector field −∇ V (y) points inward the ball of radius ρ 1 on the stable manifold. By definition of the stable manifold, V has to decrease on W s (O) along orbits of the gradient flowẏ = −∇ V (y). We thus conclude from (6.12) and (6.13) that V (y) m 1 ρ 2 1 ∀y ∈ W s (O) : y H 1 ρ 1 . (6.14) Next, recall that by Lemma 4.5, there exist constants α, β > 0 such that for all y ∈ R d+1 . Define γ > 0 by −α + βγ 2 = 1. Then for all y ∈ W s (O) such that ρ 1 y H 1 γ, we have Together with (6.12) and (6.15) for y H 1 > γ, this proves (6.8), with the choice m 0 = min{m 1 , (m 1 ρ 2 1 /γ 2 ), (1/γ 2 )}.
Proof: Choosing the radius r of the balls A and B small enough, we can ensure that A and B lie at a L ∞ -distance of order 1 from the stable manifold W s (O). By the variational principle (6.5), it is sufficient to construct a particular function h + ∈ H A,B for which the claimed upper bound holds. We define h + separately in different sets D, S defined below, and the remaining part of R d+1 . Let with c k = c 0 (1 + log(1 + k)). We will choose c 0 sufficiently large below. We set Note that for any s < 1/2 one has The contribution of h + on D to the Dirichlet form is given by Using the expression (6.21) of the potential, one readily gets e −λ k y 2 k /2ε dy k 1 + O ε 1/2 |log ε| 3/2 , (6.24) which implies that Φ D (h + ) its bounded above by the right-hand side of (6.17), provided c 0 is chosen large enough. We now continue h + outside the set D. Let S be a layer of thickness of order ε|log ε| in · 2 -norm around the stable manifold W s (O). We set h + = 1 in the connected component of R d+1 \ S containing A, h + = 0 in the connected component of R d+1 \ S containing B, and interpolate h + in an arbitrary way inside S, requiring only ∇h + (y) 2 2 M/(ε|log ε|) for some constant M . Then the contribution Φ R d+1 \S (h + ) to the capacity is zero, and it remains to estimate Φ S\D (h + ). By Lemma 6.1, we have where κ > 0 depends only on m 0 and L. Recalling the choice c k = c 0 (1 + log(1 + k)), we find uniformly in d, provided κc 0 |log ε| > 1. Thus we can ensure that Φ S\D (h + ) is negligible by making c 0 large enough.
Proof: We write as before y = (y 0 , y ⊥ ), where y ⊥ = (y 1 , . . . , y d ). Let where the constantsĉ k are of the formĉ k =ĉ 0 (1 + log(1 + k)). Note that as in the previous proof, this implies y ⊥ H s = O( ε|log ε| ) for y ⊥ ∈ D ⊥ and all s < 1. Given ρ > 0, we set D = [−ρ, ρ] × D ⊥ . (6.29) Let h * = h A,B denote the equilibrium potential defined by cap A (B) = Φ (A∪B) c (h A,B ), cf. (6.5). Then the capacity can be bounded below as follows: Solving a one-dimensional Euler-Lagrange problem, we obtain that the infimum is realised by the function f such that Substituting in (6.30) and carrying out the integral over y 0 , we obtain (6.32) By (6.9) (which also applies to the infinite-dimensional system) and the fact that y ⊥ H s = O( ε|log ε|), any point (ρ, y ⊥ ) lies on the same side of the stable manifold W s (O) as u * + . This implies that H((ρ, y ⊥ ), B) = 0 while H((ρ, y ⊥ ), A) η, where η is uniform in y ⊥ ∈ D ⊥ . We can thus apply Proposition 5.17 to obtain the existence of H 0 > 0 such that h * (ρ, y ⊥ ) = P (ρ,y ⊥ ) τ A < τ B 4 e −H 0 /ε , (6.33) provided ε is small enough and d is larger than some d 0 (ε). For similar reasons, we also have Substituting in (6.32), we obtain Consider now, for fixed y ⊥ ∈ D ⊥ , the function y 0 → g(y 0 ) = V (y 0 , y ⊥ ). It satisfies, for all 1/4 < s < 1/2, The assumption on U being a double-well potential, the definitions of A, B and the implicitfunction theorem imply that g admits a unique maximum at y * 0 = O ε|log ε| , and we have Thus by applying standard Laplace asymptotics, we obtain Substituting in (6.35) yields and the result follows from the same estimate as in (6.26), takingĉ 0 1.

L near π
We now turn to the case |L − π| c, with c small. Then the eigenvalue λ 1 associated with the first Fourier mode satisfies |λ 1 | η, where we can assume η to be small by making c small. Recall from Proposition 4.8 that if the local potential U is of class C 5 , there exists a change of variables y = z + g(z), with g(z) H t = O( z 2 H s ) for all 5/12 < s < 1/2 and t < 2s − 1/2, such that with C 4 > 0. Note that the factor 1/2 in front of C 4 results from the change from complex to real Fourier series. In order to localise this change of variables, it will be convenient to introduce a C ∞ cut-off function θ : R d+1 → [0, 1] satisfying θ(z) = 1 for z H s 1 , 0 for z H s 2 . (6.41) Given ρ > 0, we consider the potential which is equal to V (z) for z H s 2ρ, and to the normal form (6.40) for z H s ρ. It what follows, we will always assume that ρ > |λ 1 |.
The expression (6.40) of the normal form shows that for sufficiently small ρ, • if λ 1 0, the origin O is the only stationary point of V ρ in the ball z H 1 < ρ, and λ 0 = −1 is the only negative eigenvalue of the Hessian of V ρ at O; • if λ 1 < 0, the origin O is a stationary point with two negative eigenvalues, and there are two additional stationary points P ± with coordinates z ± 1 = ± 2|λ 1 |/C 4 + O(λ 1 ) , z ± k = O(λ 2 1 /λ k ) for k = 0, 2 . . . , d . (6.43) The symmetry (6.4) implies that z + k = (−1) k z − k and V (P + ) = V (P − ). The eigenvalues of the Hessian of the potential at P ± are the same, owing to the symmetry, and of the form which shows that P + and P − are saddles with a one-dimensional unstable manifold, and a d-dimensional stable manifold. The unstable manifolds necessarily converge to the two local minima of the potential.
The basins of attraction of the two minima of the potential are separated by a ddimensional manifold that we will denote W s . For λ 1 > 0, W s = W s (O) is the stable manifold of the origin. For λ 1 < 0, we have W s = W s (O) ∪ W s (P − ) ∪ W s (P + ). See for instance [Jol89] for a picture of the situation.
Lemma 6.4 (Growth of the potential along W s ). Let z ⊥ = (z 2 , . . . , z d ). There exist constants ρ > 0, m 0 > 0 and η > 0 such that for |λ 1 | < η and all z ∈ W s , one has Proof: The manifold W s can be locally described by a graph z 0 = ψ(z 1 , z ⊥ ), where |ψ(z 1 , z ⊥ )| M (z 4 1 + z ⊥ Proposition 6.5 (Upper bound on the capacity). There exist constants ε 0 , η, c + > 0 such that for ε < ε 0 and d 1, Consider first the case λ 1 0. Let D be a box defined by (6.19), where we take δ k as in (6.18) for k = 1, while δ 1 is the positive solution of u 1 (δ 1 ) = c 1 ε|log ε|, which satisfies Note that if z ∈ D, then z H s = O(δ 1 ) for all s < 1. This ensures that the potential V ρ is given by the normal form (6.40). The rest of the proof then proceeds exactly as in Proposition 6.2. We have slightly overestimated the logarithmic part of the error terms to get more compact expressions. For −c ε|log ε| λ 1 < 0, the proof is the same, with δ 1 of order (ε|log ε|) 1/4 . Note that in this case, the potential at the saddles P ± has order ε|log ε|, so that e − Vρ(P ± )/ε is still close to 1 for small c.
Finally, for −η λ 1 < −c ε|log ε|, we evaluate separately the capacities on each half-space {z 1 < 0} and {z 1 > 0}. Each Dirichlet form is dominated by the integral over a box around P + , respectively P − , where the extension of the box in the z 1 -direction is of order ε|log ε|/µ 1 . The main point is to notice that Remark 6.6. As shown in [BG10, Section 5.4], the integrals of e −u 1 (y 1 )/ε and e −u ± 2 (y 1 )/ε can be expressed in terms of Bessel functions, yielding the functions Ψ ± given in (2.30) and (2.31).
Proof: For −c ε|log ε| λ 1 η, the proof is exactly the same as the proof of Proposition 6.3, except thatδ 1 is defined in a similar way as δ 1 in (6.54), and thus the error terms are larger. For −η λ 1 < c ε|log ε|, the definition of the set D has to be slightly modified. Since the same modification is needed for all L − π of order 1, we postpone that part of the proof to the next subsection.

L > π
We finally consider the case L π + c. Recall from Section 2.2 the following properties of the deterministic system: 1. The infinite-dimensional system has exactly two saddles of index 1, given by functions u * ± (x) of class C 2 (at least). The fact that u * ± (x) ∈ C 2 implies that their Fourier components decrease like k −2 . 2. The Hessian of V corresponds to the second Fréchet derivative of V at u * ± , given by the map The eigenvalues µ k of the Hessian are solutions of the Sturm-Liouville problem v (x) = −U (u * ± (x))v(x). They satisfy µ 0 < 0 < µ 1 < . . . and −γ 1 + γ 2 k 2 µ k γ 3 k 2 ∀k , (6.59) for some constants γ 1 , γ 2 , γ 3 > 0 (this follows from expressions for the asymptotics of the eigenvalues of Sturm-Liouville equations, see for instance [VS00]). In Fourier variables, we have ∇ 2 V (u * ± ) = Λ + Q(u * ± ) , (6.60) where Λ is a diagonal matrix with entries λ k , and the matrix Q represents the second summand in the integral (6.58). Thus if v has Fourier coefficients z, we have (6.62) 3. As shown in Section 4.4, similar statements hold true for the finite-dimensional potential for sufficiently large d. As above we denote the two saddles by P ± , and the eigenvalues of the Hessian by µ k = µ k (d). Let S ± be the orthogonal change-of-basis matrices such that S ± ∇ 2 V (u * ± )S T ± = diag(µ 0 , . . . , µ d ) . (6.63) We denote again by W s the basin boundary, which is formed by the closure of the stable manifolds of P + and P − . Lemma 6.9 (Growth of the potential along W s ). There exists a constant m 0 > 0 such that for all y ∈ W s , (6.74) Proof: We have for any s > 1/4. Since the stable manifold can be described locally by an equation of the form z 0 = g(z ⊥ ), we obtain, as in the proof of Lemma 6.1, the existence of constants m 1 , ρ 1 > 0 such that By Lemma 6.8, this implies A similar bound holds in the neighbourhood of P − . Now choose a γ > 0 such that −α + βγ 2 /4 1, where α and β are the constants appearing in (6.15), and such that γ 3 P + H 1 . We want to consider the case of y ∈ W s satisfying √ β 0 ρ 1 y − P + H 1 ∧ y − P − H 1 γ. Without loss of generality we may assume y − P + H 1 y − P − H 1 . As in the proof of Lemma 6.1, we use the fact that the vector field −∇ V (y) is pointing inward. Thus, (6.78) Together with (6.15) for y−P + H 1 ∧ y−P − H 1 > γ, this proves (6.74) for all y ∈ W s . Proposition 6.10 (Upper bound on the capacity). There exist r 0 , ε 0 > 0 and d 0 < ∞ such that for r < r 0 , ε < ε 0 and d d 0 , µ k e − V (P ± )/ε 1 + c + ε 1/2 |log ε| 3/2 , (6.79) where the constant c + is independent of ε and d.

Proof:
The proof is similar to the proof of Proposition 6.2. We first compute the Dirichlet form over a box D + , defined in rotated coordinates z = S + (y − P + ) by |z k | c k ε|log ε|/|µ k |. Constructing h + as a function of z 0 as before yields a contribution equal to half the expression in (6.79). The other half comes from a similar contribution from a box D − centred in P − . The remaining part of the Dirichlet form can be shown to be negligible with the help of Lemmas 6.8 and 6.9. Proposition 6.11 (Lower bound on the capacity). There exist r 0 , ε 0 > 0 and d 0 (ε) < ∞ such that for r < r 0 , ε < ε 0 and d d 0 (ε), where the constant c − is independent of ε and d.
Proof: We perform the change of variables y = P + + S T + z in the Dirichlet form, which is an isometry, and thus of unit Jacobian. Let A , B denote the images of A and B under the inverse isometry.
(6.83) Substituting this into (6.82), the Dirichlet form Φ D + (h * ) can be estimated as in (6.39). Now a similar estimate holds for the Dirichlet form Φ D − (h * ) on a set D − constructed around P − . The two sets may overlap, but the contribution of the overlap to the capacity is negligible.

Periodic b.c.
We turn now to the study of capacities for periodic b.c. Since most arguments are the same as for Neumann b.c., we only give the main results and briefly comment on a few differences.
The potential energy (6.1) is invariant under translations u → u(· + ϕ). As a consequence, when expressed in Fourier variables it satisfies the symmetry where the error term is uniform in d.
For 2π − c < L 2π, the capacity can again be estimated by using the normal form. The only difference is that the centre manifold is now two-dimensional, which leads to the expression where u(r 1 , ϕ 1 ) = 1 2 λ 1 r 2 1 + C 4 r 4 1 (6.88) results from the terms in z ±1 written in polar coordinates, and R(ε, λ 1 ) is the same as in (6.50). The integral can be expressed in terms of the distribution function of a Gaussian random variable, cf. [BG10, Section 5.4]. For 2π L 2π + c, the expression for the capacity is given by (6.87) with an extra term e V /ε , where V −λ 2 1 /(16C 4 ε) is the value of the potential at the transition state. Finally in the case L 2π + c, we have to take into account the fact that instead of isolated transition states, there is a whole family of transition states {P (ϕ)} 0 ϕ<L , satisfying by symmetry P k (ϕ) = e 2 i πkϕ/L P k (0) . (6.89) The eigenvalues µ k of the Hessian at any transition state satisfy µ 0 < µ −1 = 0 < µ 1 < µ 2 , µ −2 < . . . where saddle is the "length of the saddle", due to the integration along the direction with vanishing eigenvalue µ −1 . It is given by When L is close to 2π, the normal form shows that |P 1 (0)| 2 = |λ 1 |/(2C 4 ) + O(λ 2 1 ), while the other components of P (0) are of order λ 2 1 . Also the eigenvalue µ 1 satisfies µ 1 = −2λ 1 + O(|λ 1 | 3/2 ). This shows that saddle = 2π µ 1 /(8C 4 ) + O(µ 1 ), and allows to check that the expressions (6.91) and (6.87) for the capacity are indeed compatible.
7 Uniform bounds on expected first-hitting times 7.1 Integrating the equilibrium potential against the invariant measure We define as before the sets A, B ⊂ E as the open balls The aim of this subsection is to obtain sharp upper and lower bounds on the integral Recall that the local minima u * ± of V are also local minima of the truncated potential, and that the eigenvalues of the Hessian of the potential at u * − are given by ν − k = (bkπ/L) 2 + U (u − ), where b = 1 and k ∈ N 0 for Neumann b.c., and b = 2 and k ∈ Z for periodic b.c. Recall that u ± denote the minima of the local potential U .
Proof: Let δ k = c k ε|log ε|/ν − k , where c k = c 0 (1 + log(1 + |k|)). We introduce two sets dz . (7.9) The volume of the sphere S 2d is given by 2π d /Γ(d), which by Stirling's formula is bounded by (M 1 /d) d for some constant M 1 . Thus choosing d 0 of order 1/ε or larger ensures that the integral (7.7) is negligible if we take η small enough. Finally, we can bound the integral of e − V (y)/ε over the remaining space in the same way as in the proof of Proposition 6.2, using again (4.20) to bound the potential below by a quadratic form. Choosing c 0 large enough ensures that this integral is negligible as well.
Proposition 7.2 (Lower bound on the integral). There exist constants r 0 > 0, ε 0 > 0 such that for any ε < ε 0 , there exists a d 0 = d 0 (ε) < ∞ such that for all 0 < r, ρ < r 0 , and all d d 0 , where the constant c − is independent of ε and d.
Proof: We define C d as in the previous proof. The fact that h (d) (7.11) The first term on the right-hand side satisfies the claimed lower bound, by a computation similar to the one in the proof of Proposition 6.3, cf. (6.39). Proposition 5.17 shows that the second term on the right-hand side is smaller than the first one by an exponentially small term.

Averaged bounds on expected first-hitting times
We define the sets A and B as in (7.1). According to (3.11), is a probability measure on ∂A d . The following result implies Proposition 3.3.
Proposition 7.3. There exist r 0 , ε 0 > 0 such that for 0 < r, ρ < r 0 and 0 < ε < ε 0 , there exists a d 0 = d 0 (ε) < ∞ such that for all d d 0 , • For L π, the value of the prefactor follows from Propositions 6.5 and 6.7. Using the computations of [BG10, Section 5.4] to determine the integral in (6.48) (note that our C 4 is equal to half the C 4 in that reference), we get where Ψ + is the function defined in (2.30). The exponent is still given by (7.16), while the error terms are of the form • For L π, again by Propositions 6.5 and 6.7 and [BG10, Section 5.4], where Ψ + is the function defined in (2.31). The exponent is again given by (7.19), and the error terms satisfy (7.21).
The expressions are similar for periodic b.c.
A Monotonicity of the period Consider the Hamiltonian system defined by the Hamiltonian (2.16). Let T (E) be the period of its periodic solution with energy E, given by (2.18). The following lemma provides a sufficient condition for T being increasing in E. = U (f E (ϕ)) 2 − 2U (f E (ϕ))U (f E (ϕ)) cos ϕ √ 2E U (f E (ϕ)) 3 . (A.5) Since cos ϕ and −U (f E (ϕ) have the same sign, the assumption (A.1) implies that the integral (A.4) is strictly increasing in E.