This work is concerned with time-periodic perturbations of the stochastic differential equation
describing the overdamped motion of a Brownian particle in a double-well potential V 0 : R → R, which is bounded below and grows at least quadratically at infinity. Let us start by recalling some well-known properties of the unperturbed system (1.1). It's unique invariant measure has density Z −1 e −2V 0 (x)/σ 2 with respect to Lebesgue measure, where Z is the normalisation. Furthermore, the dynamics is reversible with respect to this measure. Denote the local minima of V 0 by x * ± , and its local maximum by x * 0 , with x * − < x * 0 < x * + . Let τ + = inf{t > 0 : x t = x * + } be the first-hitting time of x * + . Then one has the explicit expression
for the expectation of τ + when starting at any x < x * + . This result is obtained by solving an ordinary differential equation (ODE) satisfied by the function x → E x τ + , owing to Dynkin's formula. In particular, the Laplace method shows that when starting in x * − , this expectation satisfies the so-called Eyring-Kramers law [23, 29] 
Furthermore, in [17] , Day has shown that the law of τ + is asymptotically exponential, in the sense that lim
holds for all s > 0. While the expected transition time from x * − to x * + is exponentially long, the actual successful transition, also known as the reactive or transition path, takes much less time. In [16] , Cérou, Guyader, Lelièvre and Malrieu have shown that for any fixed a < x 0 < x * 0 < b in (x * − , x * + ), one has the convergence in law
where T (x 0 , b) is an explicit deterministic quantity independent of σ, and Z is a standard Gumbel variable, that is, P{Z t} = exp{− e −t } holds for all t ∈ R. Therefore, the duration of a transition is of order log(σ −1 ). See also Bakhtin's works [1, 2] for insights on the relation of this result to extreme-value theory.
Several of these results have been generalised to multidimensional diffusions of the form
where now V 0 : R d → R. These are still reversible with respect to the invariant measure Z −1 e −2V 0 /σ 2 . A weaker form of the Eyring-Kramers law (that is, without a sharp control of the prefactor of the exponential in (1.2)), known as Arrhenius law, follows from the theory of large deviations developed for diffusions by Freidlin and Wentzell [24] . In [14, 15] , Bovier, Eckhoff, Gayrard and Klein used potential theory to prove a generalisation of (1.2) to the multidimensional gradient case, as well as the asymptotically exponential character (1.3) of the law of transition times. Similar results have been obtained by Helffer, Klein and Nier in [26] using methods from semiclassical analysis. See also [31, 32, 34, 37] for generalisations to diffusions on manifolds with or without boundary. The potential-theoretic approach has also been successfully applied to obtain Eyring-Kramers laws for stochastic PDEs [10, 3, 7] . See also [4, 6] and references therein, as well as [13] for a comprehensive account of the potential-theoretic approach. The situation is much less understood for non-gradient diffusions, whose invariant measure is not explicitly known in general, and which are not reversible. While the theory of large deviations in [24] allows to derive Arrhenius laws for these systems as well, determining precise asymptotics on transition times of Eyring-Kramers type is much harder than in the reversible case. Some partial results in this direction have nevertheless been obtained. In [12] , Bouchet and Reygner proposed an Eyring-Kramers law for non-reversible diffusions in a bistable situation, based on formal asymptotic computations. In [30] , Landim, Mariani and Seo obtained a generalisation of the potential-theoretic approach of [14, 15] to non-reversible systems. This allowed them in particular to justify the formal result of Bouchet and Reygner for a particular class of systems whose invariant measure is known explicitly. See also the work [33] by Le Peutrec and Michel for semiclassical results on non-reversible diffusions with known invariant measure. In a different direction, a reactive path theory for multidimensional, non-reversible diffusions was developed by Lu and Nolen in [35] , based on ideas by E and Vanden-Eijnden [22] .
In this work, we are concerned with extensions of (1.2) to systems of another type, namely to periodically perturbed versions of (1.1) of the form
where {W x t } t and {W y t } t are independent standard Wiener processes. The parameter ̺ has to be strictly positive for technical reasons (we need the diffusion to be elliptic), but our results do not depend on ̺ to leading order. This system is a particular case of systems studied by the author and Barbara Gentz in [11] . The main result in that work gives a rather sharp description of the density of τ 0 , the first-passage time at the saddle x * 0 (y) of x → V 0 (x, y) (or, more precisely, at the deterministic periodic solution tracking the saddle). A slightly less precise, but more transparent way of formulating this result is that
where • θ(y) is a convenient and explicit parametrisation of the periodic orbit tracking x * 0 (y); • λ + is the Lyapunov exponent of this orbit; • Z follows again a standard Gumbel law; • and Y σ is asymptotically geometric, meaning that it has positive integer values and satisfies lim n→∞ P{Y σ = n + 1|Y σ > n} = p(σ) , for a constant p(σ) that is exponentially small in σ 2 . (In fact, we have slighly simplified the precise result, which is given in [5, Theorem 4.2] .) The most striking feature of (1.6) is that the law of θ(y τ 0 ) is shifted by an amount log(σ −1 ) as σ decreases, and thus does not admit a limit as σ → 0. This is the phenomenon of cycling discovered by Day [18, 19, 20, 21] . In fact, this shift by log(σ −1 ) is also present in (1.4) . As for Y σ , its interpretation is as follows: under a non-degeneracy assumption, the system has a "window of opportunity" during each period to make a transition, which is defined by the minimisers of its large-deviation rate function. The integer variable Y σ simply gives the period during which the actual transition takes place.
The expectation E[τ 0 ] can be deduced from (1.6) , and is close to the inverse of the parameter p(σ) (see [8] ). Since transitions from the saddle to the local minima x * ± (y) take a time of order log(σ −1 ) (see [5, Theorem 6.2] ), the expectation of the first-hitting time τ + of x * + (y) has the same sharp asymptotics as E[τ 0 ]. In [11] , we did not attempt to obtain sharp asymptotics for p(σ), but only showed that it is close, in the sense of logarithmic equivalence, to e −I/σ 2 where I is the Freidlin-Wentzell quasipotential, which can be expressed as the solution of a variational principle.
The aim of the present work is to obtain sharp asymptotics of Eyring-Kramers type for E[τ + ], which is equivalent to getting precise asymptotics for p(σ). Our main result, Theorem 2.4, states that for any starting point on x * − (y),
where R 1 (ε, σ) is some (complicated) error term. The result applies to values of ε which are large with respect to the integrand in this expression, but still have to be exponentially small in σ 2 owing to technical reasons. Note that (1.7) is indeed a generalisation of the static Eyring-Kramers law (1.2). The remainder of this paper is organised as follows. In Section 2, we define precisely the considered equations, and state all main results. These are proved in Sections 3 to 7, see Section 2.7 for a more precise outline of the structure of the proofs. Finally, the appendix contains some of the more technical proofs.
The system studied in this work depends on two small parameters ε and σ. We write X Y to indicate that X cY for a constant c independent of ε and σ, as long as ε and σ are small enough. The notation X ≍ Y indicates that one has both X Y and Y X, while Landau's notation X = O(Y ) means that |X| Y . The canonical basis of R 2 is denoted (e x , e y ). If a, b ∈ R, a ∧ b denotes the minimum of a and b, and a ∨ b denotes the maximum of a and b. Finally, we write 1 D (x) for the indicator function of a set or event D.
This work is supported by the ANR project PERISTOCH, ANR-19-CE40-0023. To a large extent, it was written during the Spring 2020 lockdown due to the Covid-19 epidemics, which probably goes a long way toward explaining why it contains a lot of detailed computations. The author wishes to thank all members of the probability community who helped keeping the community's spirits intact by organising and giving many interesting online seminars.
We will consider a version of (1.5) in which time has been scaled by a factor ε, given by
where {W x t } t 0 and {W y t } t 0 are independent Wiener processes on a filtered probability space (Ω, F , P, {F t } t 0 ), ε, σ and ̺ are strictly positive parameters, and the drift term b satisfies the following assumptions:
• b : R 2 → R is of class C 4 and is periodic, of period 1, in its second argument.
• For any y ∈ [0, 1], the map x → b(x, y) vanishes at exactly 3 points x * − (y) < x * 0 (y) < x * + (y), and the derivative ∂ x b(x, y) is nonzero for these 3 values of x. • There are constants M, L > 0 such that xb(x, y) −M x 2 whenever |x| L. The above conditions guarantee existence of a pathwise unique strong solution (x t , y t ) t 0 for any initial condition (x 0 , y 0 ). We denote by P x,y {·} the law of the process starting in (x, y), and by E x,y [·] expectations with respect to P x,y {·}.
We define the potential
The assumptions on b imply that for any y, x → V 0 (x, y) has local minima at x * ± (y), a local maximum at x * 0 (y), and grows at least quadratically for large |x|. We say that V 0 is a double-well potential ( Figure 1 ). We denote the well depths by
, y) , and measure the curvatures at stationary points by
The assumptions on b imply that all these quantities are finite and bounded away from zero, uniformly in y. We further write ∆(y) = h + (y) − h − (y) for the difference of the two potential well depths.
We recall some well-known properties of the static system
in which y is kept constant. Its direct and adjoint infinitesimal generators are the differential operators
In particular, the kernel of L x is spanned by constant functions, while the kernel of L † x is spanned by the density π 0 (x|y) of the invariant measure of (2.2), which is given by Figure 2 ). We denote the eigenvalues of L x and L † x by
and the corresponding L 2 -normalised eigenfunctions by φ n (·|y) and π n (·|y). These are related by π n (x|y) = π 0 (x|y)φ n (x|y) .
There is a spectral gap of order 1, separating λ 1 (y) from λ 2 (y) and all subsequent eigenvalues, which is why an important role will be played by λ 1 (y) and the associated eigenfunctions. The eigenvalue satisfies λ 1 (y) = r + (y) + r − (y) 1 + O(σ 2 ) ,
Sketch of the static eigenfunctions π 0 (x|y) and φ 1 (x|y) for the potential of Figure 1 .
where
The corresponding eigenfunction can be approximated in terms of the committor
which satisfies L x h 0 = 0 with boundary conditions h 0 (x * − (y)|y) = 1 and h 0 (x * + (y)|y) = 0. Solving this equation, one obtains that for all
The first eigenfunction of L x is related to h 0 (x|y) by
where∆(y) is defined by
The function x → φ 1 (x|y) is almost constant except near x * 0 (y), with a value close to e∆ (y)/σ 2 for x < x * 0 (y) and close to − e −∆(y)/σ 2 for x > x * 0 (y) (Figure 2 ). We give a precise statement of (2.6), including bounds on derivatives of φ 1 , in Section 4.1.
The spectral-gap property implies that for small σ, the dynamics of the static system (2.2) is well-approximated by a two-state Markovian jump process with rates r ± (y). It is thus natural to expect that the dynamics of the fast-slow system (2.1) is well-approximated by a time-dependent two-state process, in which y plays the role of time ( Figure 3 ). Its law (p − (y), p + (y)) satisfies the system
Time-dependent two-state markovian jump process.
Let
and let δ(y) be the 1-periodic solution of
Then it is straightforward to check that the solution of System (2.8) with initial condition (p + (y 0 ), p − (y 0 )) satisfying p + (y 0 ) + p − (y 0 ) = 1 is given by
Note that δ(y) admits the explicit integral representation
Two regimes are of particular interest:
• In the fast forcing regime ε ≫ max y∈[0,1] λ 1 (y), the dynamics is averaged, and δ(y) satisfies
In this case, δ(y) and p ± (y) are asymptotically almost constant. • In the super-adiabatic regime ε ≪ min y∈[0,1] λ 1 (y), integration by parts shows that
Thus δ(y) tracks A(y), which is close to the sign of∆(y), meaning that with high probability, the jump process is found in the currently deepest potential well. It is also possible to compute explicitly the expectation of the transition time τ jump + from the − state to the + state. We give the simple proof of the following result in Appendix A.
A similar expression holds for the transition time τ jump − from the + state to the −state.
The same distinction between regimes as above can be made here:
• If ε ≫ max y∈[0,1] r − (y), then the expected jump time does not depend on y 0 to leading order, and is given by the average
• If ε ≪ min y∈[0,1] r − (y), then the expected jump time is much shorter than the oscillation period, and thus given by the instantaneous value
We now return to the fast-slow SDE (2.1). In order to be able to apply the potential-theoretic approach of [30] , it is necessary to control the invariant measure of the system. The main result of this section is the following theorem, which will be proved in Section 4.
. For sufficiently small σ and ε, the invariant measure of the system (2.1) has the density π(x, y) = π 0 (x|y) 1 + α 1 (y)φ 1 (x|y) + Φ ⊥ (x, y) ,
Here∆(y) is given by (2.7), and δ 1 (y) is the unique periodic solution of the linear second-order equation
(2.14)
Furthermore, the error term Φ ⊥ (x, y) in (2.12) is orthogonal to the span of φ 0 and φ 1 , and satisfies
where ·, · denotes the standard inner product for L 2 (R, dx).
As we will see in Section 4, the periodic solution of (2.13) is in fact close to the periodic solution of the first-order equation
which is similar to (2.10). The function δ 1 (y) also has a similar interpretation as δ(y) in (2.11). Indeed, one has (by a similar argument as in the proof of Corollary 5.2)
By analogy with (2.11), p − (y) can be interpreted as the "instantaneous" probability to be in the left-hand potential well at equilibrium.
Given a function f : [0, 1] → R, we introduce the notation
We will mainly be concerned with the fast-forcing regime ε ≫ λ 1 . Then δ 1 (y) is actually nearly constant, in the sense that
One should note that the main limitation of Theorem 2.2 lies in the error term proportional to λ 1 (y) in (2.14), which causes the error term in √ λ 1 in (2.18). This is due to technical difficulties in controlling Φ ⊥ , and will limit the applicability of our results to the regime
In fact, there is already a substantial amount of work involved in getting an error term proportional to (ε/σ 2 ) 3 √ λ 1 , rather than (ε/σ 2 ) 2 √ λ 1 . This improvement is due to the fact that we are able to prove that Φ ⊥ (x, y) = Φ * ⊥ (x, y) + Φ 1 ⊥ (x, y) , where Φ * ⊥ is explicit, and has a contribution of order λ 1 (y) to q 3 (y), while Φ 1 ⊥ satisfies a bound of the form (2.15), but with a larger power of ε. See Corollary 4.13 for details.
In order to formulate our main result, we introduce two functions
where ρ > 0 is a parameter of order 1 that will be taken sufficiently small. We then define two set
see Figure 4 . Our first main result gives a general expression for the expected first-hitting time of B, when starting with a specific distribution on ∂A, the so-called equilibrium measure.
Definition of the sets A and B, in relation with the extrema x * ± (y) and x * 0 (y) of the map x → V 0 (x, y).
There exists a probability measure ν AB , supported on ∂A, such that
where R 0 (ε, σ) is an error term satisfying
As such, this result has two main limitations. First, it is not immediately apparent for which values of ε and σ the remainder R 0 (ε, σ) is actually small. And second, we do not know the equilibrium measure ν AB .
We will address both issues in the fast-forcing regime ε ≫ λ 1 . In fact, the discussion of the two-state jump process in Section 2.3 suggests that if ε < λ 1 , the expected first-hitting time depends strongly on the starting point, whereas it is almost constant if ε ≫ λ 1 .
Recalling the expressions (2.4) for λ 1 (y) and (2.9) for A(y), we obtain
Furthermore, if ε ≫ λ 1 , (2.17) and (2.18) imply that δ 1 (y) is close to
It follows that the leading term in (2.20) is given by
which agrees with (1.7) (recall that we have scaled time by a factor ε). In order to quantify error terms, we introduce minimal barrier heights 
In the symmetric case h min − = h min + , we have H = H − = 0, and the error term takes the simpler form
Disregarding powers of σ with respect to exponential terms, we see that Theorem 2.4 is applicable when
In the asymmetric case h min − = h min + , the error term is larger, and results in stronger conditions on ε. One can however check (see Section 7) that there exists a non-empty interval of values of ε for which Theorem 2.4 is still meaningful as long as
that is, as long as the asymmetry between the potential wells is not too large.
Theorem 2.4 provides a generalisation of the static Eyring-Kramers law (1.2) to slowly oscillating double-well potentials, when the forcing frequency ε lies in an interval given by (2.27) if the oscillation is symmetric, in the sense that h min − = h min + . We now provide some comments on what we expect to happen outside this domain of validity. This will also serve as a "reality check" of our main results.
If ε λ 1 , Theorem 2.3 is still valid, but perhaps not as useful. The main limitation of the result in that case is that E (x,y) [τ B ] is no longer expected to be almost constant, so that the equilibrium measure ν AB matters. In fact, we do have an explicit expression for ν AB , which is given (cf. (3.10)) by
where all notations are defined in Section 3. In particular, n is the unit normal vector to ∂A, and dλ is the Lebesgue measure on ∂A, so that n dλ = e x − a ′ (y)e y dy .
Using the estimates on the capacity cap(A, B) given in Theorem 6.1, the expression (3.11) of the diffusion matrix D, the estimate on the adjoint committor h * AB obtained in Proposition 5.1, and Theorem 2.2 on the invariant measure π, we obtain that to leading order,
Substituting this in the result (2.20) of Theorem 2.3, and using the fact that λ 1 (y)[1 + A(y)] = 2r − (y), we obtain that to leading order,
• In the fast-forcing regime ε ≫ λ 1 , the expectation E (a(y 0 ),y 0 ) [τ B ] being nearly constant, we recover indeed (2.25). • In the superadiabatic regime ε ≪ min y λ 1 (y), the discussion in Section 2.3 on the two-state jump process suggests that we have
which is indeed consistent with (2.29). • In the intermediate regime min y λ 1 (y) ε max y λ 1 (y), the situation is more complicated owing to the phenomenon of stochastic resonance (see for instance the discussion in [9, Section 4.1.2]). What we expect then is the following. If the process starts at a point (a(y 0 ), y 0 ) ∈ ∂A such that ε < r − (y 0 ), the mean hitting time of B will still satisfy (2.30). Otherwise, the transition to B will occur near the smallest y > y 0 such that r − (y) = ε, and thus the expectation of τ B is dominated by y − y 0 . This picture is also consistent with large-deviation results obtained in [25] .
The other regime not covered by our results is when ε λ 1 1/4 . As noted above, this is mainly due to technical difficulties in controlling the part of the invariant measure π which is orthogonal to the span of the first two eigenfunctions π 0 and π 1 of L †
x . In fact, it seems quite plausible that the expression (2.25) for the mean transition time still holds as long as ε ≪ σ 2 (for larger ε, the slow-fast structure of the equation for π changes). To establish such a result, however, new ideas are needed to achieve a better control of the invariant measure.
As already mentioned, the main ingredient of our proof is the potential-theoretic approach to metastability, which was developed in [14, 15] for reversible diffusions, and extended in [30] to general diffusions. We give a quick overview of this approach in Section 3. Its key result relates the expected first-hitting time of a set B, when starting with in the equilibrium measure ν AB on the boundary of another set A, with the invariant measure of the diffusion and the so-called capacity cap(A, B). See Proposition 3.4 below.
The main difficulty in our case is to determine the invariant measure π of the system. While this measure is explicitly known for reversible systems, this is no longer the case here. As shown in [30] , π is related to the solution of a Hamilton-Jacobi equation, see Lemma 3.1. However, obtaining an approximate solution of this equation with good enough control of error terms turns out to be difficult. Therefore we adopt another approach, which consists in expanding π on a basis of eigenfunctions of the generator of the system with frozen y, and analysing the resulting system of ODEs. This is done in Section 4, which contains in particular the proof of Theorem 2.2.
In Section 5, we investigate the adjoint system that enters the expression for the mean firsthitting time in Proposition 3.4. In particular, we obtain approximate expressions for the committors P{τ A < τ B } of the original and adjoint system in Proposition 5.1, using a perturbation theory argument around the committors of the frozen systems. The necessary estimate for Proposition 3.4 is then obtained in Corollary 5.2.
The other quantity that needs to be determined for the potential-theoretic approach to work is the capacity cap(A, B). This is comparatively easy once the invariant measure is known, since the capacity obeys variational principles (the Dirichlet and Thomson principle) that give upper and lower bounds once one makes a sufficiently good guess of test functions to feed into them. It turns out that the system with frozen y provides such sufficiently good guesses, the only difficulty being to account for the fact that these guesses are not strictly divergence-free. The main result is Theorem 6.1, which provides upper and lower bounds on the capacity. Section 7 contains the last steps of the proof of Theorems 2.3 and 2.4. While Theorem 2.3 follows directly from the obtained bounds on the invariant measure, committors and capacity, Theorem 2.4 requires a little more work, which consists in simplifying the expressions for the dominant term and error terms, and getting rid of the equilibrium measure ν AB .
In order to increase readability, we have relegated some of the more technical proofs to the appendix. Appendix A contains the proof of Proposition 2.1 on the two-state jump process, Appendix B contains the proofs of the potential-theoretic results in Section 3, Apppendix C contains the estimates on static eigenfunctions required for determining the invariant measure, and Appendix D gathers a few auxiliary results involving Laplace asymptotics.
In this section, we give a short overview of the potential-theoretic results contained in [30, Section 4] , slightly adapted to our situation. All proofs are given in Appendix B.
The infinitesimal generator of the system (2.1) is given by
A key idea in [30] is to decompose L into a symmetric and an antisymmetric part. This allows to define an adjoint stochastic process, and both the direct and adjoint process play a role in the expressions for mean first-passage times.
where V solves the Hamilton-Jacobi equation
Lemma 3.2. The infinitesimal generator (3.1) can be written as
satisfies the vanishing divergence condition
We decompose L into a symmetric and an antisymmetric part by writing L = L s + L a , where
We write T = R/Z for the circle, and endow L 2 (R × T) with the inner product
where dπ = π(x, y) dx dy. Then one checks that L s is self-adjoint with respect to this inner product, while (3.5) and the divergence theorem imply
showing that L a is anti-self-adjoint (skew-symmetric), that is L † a = −L a . By definition, the adjoint process has the generator
The corresponding SDE is given by
We denote by P *
x,y {·} the law of the adjoint process starting in (x, y), and by E *
x,y [·] the corresponding expectations.
The capacities of the direct and adjoint process are defined via the Dirichlet form associated with L s , that is
where n is the inward-pointing unit normal vector to ∂A, and dλ is the arclength on ∂A. An analogous relation, with h AB replaced by h * AB , holds for cap * (A, B). Furthermore,
The AB-equilibrium measure ν AB is the probability measure supported on ∂A defined by
Then we have the following fundamental relation.
For ϕ, ψ two vector fields on (A ∪ B) c , we define the bilinear form
and we denote D(ϕ, ϕ) by D(ϕ). For γ ∈ R, we write F γ AB for the closure with respect to the norm D(·) of the set of flows ϕ which are divergence-free, i.e.
We further denote by H α,β AB the set of functions f ∈ L 2 (dπ) which have constant values α in A, and β in B. For such an f , we use the notations
Note in particular that
Lemma 3.5 is all we need to prove the Dirichlet and Thomson principles.
where the infimum is reached for f =f := 1 2 (h AB +h * AB ) and ϕ =φ :
holds for any ϕ satisfying (3.12) with γ = 0.
Proposition 3.7 (Thomson principle). We have
where the supremum is reached for
holds for any ϕ satisfying (3.12) with γ = 1.
This section is devoted to the proof of Theorem 2.2. Lemma 3.1 shows that the invariant density π(x, y) can be obtained by solving the Hamilton-Jacobi equation (3.2) . However, it turns out to be difficult to obtain a good control of error terms when trying to do so. We thus use another approach instead, which consists in expanding π(x, y) on the basis of eigenfunctions of L † x , and analysing the resulting system of infinitely many coupled ODEs. In order to do so, we will need a number of bounds involving these eigenfunction, which we will derive in Section 4.1. The actual proof of Theorem 2.2 will then be given in Section 4.2.
The aim of this section is to obtain estimates on the eigenfunction φ 1 , and on the inner products
Note that taking derivatives of the orthonormality relations π m , φ n = δ nm yields
In particular, since φ 0 is constant, f 0m (y) = 0 and g 0m (y) = 0 for all m ∈ N. Using standard Laplace asymptotics (cf. Appendix D), it is rather easy to obtain estimates on integrals against π 0 up to multiplicative errors of the form 1 + O(σ 2 ). In particular, the normalisation of π 0 (x|y)
Similarly, the normalisation of the committor (2.5) satisfies
and bounds of the same type can be obtained for f 1i and g 1i for i ∈ {0, 1}. We will, however, need much sharper estimates with exponentially small errors of order λ 1 (y), which requires more work.
We start by providing sharp estimates on the first eigenfunction φ 1 of L x and its derivatives. Let τ ± = τ x * ± (y) be the first-hitting times of x * ± (y) for the static SDE (2.2), and let τ = τ − ∧τ + . The Feynman-Kac formula allows us to write
where we use the shorthands φ ± (y) = φ 1 (x * ± (y), y), while h 0 (x|y) = P x {τ − < τ + } is the committor and
Recall that h 0 (x|y) is given by (2.5) for x ∈ (x * − (y), x * + (y)). Furthermore, h 0 is constant equal to 1 for x < x * − (y), and constant equal to 0 for x > x * + (y). It will be convenient to define∆(y) by the relations
(4.5)
Indeed, standard Laplace asymptotics (see Lemma D.1) show that this definition is compatible to leading order with (2.7). We further introduce
.
Note carefully that B(y) ∈ (0, 1] may be exponentially small, and that we have the relations
Combining (4.3) and (4.2) with the expression (2.4) of λ 1 (y), we obtain the very useful relation
Finally, to lighten notations, we set
and we will sometimes omit the argument y.
The following results establish some properties of h 0 , h 1 and φ 1 . Their proofs are postponed to Appendix C.1. 
Similar bounds hold forh 1 (x|y), with h 0 replaced by 1 − h 0 .
Combining the last two propositions with (4.4), we obtain the following representations of φ 1 and its derivatives:
It is then straightforward to obtain the following expressions for inner products involving derivatives of the first two eigenfunctions. . We have
A consequence of these estimates is that we have, for instance,
where R 1 is a remainder, dominated by the term in ∂ y h 0 in (4.10b). One checks that it satisfies
In other words, ∂ y φ 1 lies almost in the space spanned by φ 0 and φ 1 .
A useful observation is that the expression (4.10a) for φ 1 implies
where we have used the definition (4.5) of∆. This is often better than the bound π 0 , |φ 1 | 1 provided by the Cauchy-Schwarz inequality. ♦
When analysing the system of ODEs giving the invariant density, we will also need a number of bounds involving other eigenfunctions than φ 0 and φ 1 . All proofs of these bounds are postponed to Appendix C.2. We start with some simple ℓ 2 estimates. Proposition 4.6. There exists a constant M 0 , uniform in σ and n 1, such that
14)
The following result shows that h 0 is almost orthogonal to the span of π 0 and π 1 .
Proposition 4.7. We have
We can also get exponentially small bounds for a number of sums involving f nm and g nm . :
Finally, the following result provides exponentially small bounds on similar sums, but with all terms divided by λ n . These bounds are not consequences of the previous ones, since the terms of these sums do not have the same sign, so that their smallness is due to cancellations between terms. 
Since for each y, the eigenfunctions π n (·|y) form a complete orthonormal basis of L 2 (R, π 0 dx), we can decompose the density π of the invariant measure as π(x, y) = n 0 α n (y)π n (x|y) = π 0 (x|y) n 0 α n (y)φ n (x|y) . We write the adjoint generator as
The stationarity condition L † π = 0 becomes n 1 λ n (y)α n (y)π n (x|y) = ε n 0 L † y α n (y)π n (x|y) ,
where the right-hand side can be evaluated using L † y (α n π n ) = −α ′ n + ̺ 2 σ 2 2 α ′′ n π n + −α n + ̺ 2 σ 2 α ′ n ∂ y π n + ̺ 2 σ 2 2 α n ∂ yy π n .
We now project (4.16) on each eigenfunction φ n . Since ∂ y π n , φ 0 = ∂ y π n , φ 0 = 0, and similarly for the second derivative, the projection on φ 0 yields
Using periodicity in y and the fact that π is normalised, one easily gets α 0 (y) = 1 .
The projections on the remaining φ n result in the following statement, whose proof is a simple computation. 
It is instructive to consider first the case ̺ 2 = 0. Then β n (y) = α n (y), and α n (y) satisfies the linear inhomogeneous system
Note that for n 2, α n (y) is a fast variable, which, by the general theory of singularly perturbed ordinary differential equations, is expected to remain ε-close to a value α * n (y) such that the righthand side of the system vanishes.
The case n = 1, however, is special since λ 1 (y) is exponentially small. This makes the system hard to study in the form (4.18), because α 1 (y) can become exponentially large. The solution is to observe that, disregarding for a moment the terms α n with n 2 we have by Lemma D.1
This suggests setting
, (4.20) so that p − (y) ≃ 1 2 (1 − δ 1 (y)), and therefore δ 1 (y) remains of order 1. Then a computation shows that
where
The unique periodic solution of this equation is given by
where we have setλ 1 (y) = λ 1 (y) + ε σ 2 p 1 (y) and
In particular, for ε ≫Λ 1 (1, 0) = λ 1 , δ 1 (y) is almost constant, that is, we have
To analyse the dynamics of the remaining coefficients α n (y) with n 2, we introduce a vector α * ⊥ (y) with components
and examine in particular the behaviour of α 1 ⊥ (y) = α ⊥ (y) − α * ⊥ (y). The bound (4.26) is a direct consequence of the bound (4.13) on the sum of f 2 ni . In order to establish (4.27), we first note that the α 1 n satisfy the equation
We will show that the set
is invariant under the flow of (4.28) for sufficiently large C 0 . Assume α 1 ⊥ belongs to ∂H, and pick n such that α 1 n = ±(ε 2 C 0 )/(σ 4 λ n B). The Cauchy-Schwarz inequality yields
for a constant C 1 , where we have used (4.14) to bound the first sum, and (4.26) and the definition of H to bound the second one. The derivative of α * n (y) can be bounded using the relations
and the Hellmann-Feynman theorem (cf. (C.16)), which shows that σ 2 λ ′ n (y) has order 1. The result is that
for a constant C 2 . Plugging these bounds into (4.28) shows that for C 0 large enough, the sign of ε(α 1 n ) ′ is the opposite of the sign of α 1 n . This shows the invariance of H, and therefore the bound (4.27). 
uniformly in y ∈ [0, 1].
: This follows directly from the decomposition (4.25). Indeed, the contribution of the α * m can be bounded via Proposition 4.9, and yields the first term on the right-hand side. The contribution of the α 1 m can be bounded via the Cauchy-Schwarz inequality, using Proposition 4.8 and (4.27).
One consequence of this result that will be useful when estimating capacities is the following. Integrating the ODE (4.21) satisfied by δ 1 (y) over one period, and using (4.22) and (4.29), we obtain 
hold for all y ∈ [0, 1]. Furthermore, the bounds
hold for all x ∈ R and all y ∈ [0, 1].
The first two L 2 -bounds follow directly from the fact that
while the L 2 -bound on the derivative is a consequence of Lemma C.5. As for the L ∞ -bounds (4.32), they follow from the fact that L x is conjugated to a Schrödinger operator (cf. (C.12) in Appendix C.2), whose eigenfunctions ψ n are bounded by a constant of order 1, so that
The ℓ 1 -norm of α * ⊥ can be bounded using the previous proposition.
Part of the importance of Φ * ⊥ lies in the following estimate, which shows that functions bounded by h 0 (1 − h 0 ) are almost orthogonal to Φ * ⊥ , and thus allows to improve a certain number of error bounds when estimating the capacity. Its proof is close in spirit to the proof of Proposition 4.6, so we also give it in Appendix C.2. 
Consider now the case ̺ 2 > 0. We again carry out the change of variables (4.20) , and in addition set
The resulting system for (δ 1 , γ 1 ) is given by
and the contribution of the other variables is contained in the term
It follows from Proposition 4.4 that
It is straightforward to check that the system (4.33) is equivalent to the second-order equation In order to analyse the behaviour of the remaining coefficients α n (y) and β n (y) with n 2, we introduce, analogously to (4.24), α * n (y) = ε σ 2 1 λ n (y) c n0 (y) + c n1 (y)α 1 (y) + d n1 (y)β 1 (y) . 
The bound (4.34a) is again a direct consequence of (4.13), noting that
To show (4.34b) and (4.34c), we will use the fact that the pairs (α 1 n , β 1 n ) satisfy the system
We will argue that the unique periodic solution of this equation has to be entirely contained in the set
provided C 0 is a sufficiently large constant of order 1. Indeed, similar estimates as in the proof of Proposition 4.11 show that whenever (α 1 n , β 1 n , y) lies in H, one has
For sufficiently large C 0 , this vector field has the following properties on the boundary ∂H ( Figure 5 ):
• on the upper boundary of H, it points to the left, and changes from pointing outward H to pointing inward as α 1 n increases; • on the right boundary of H, it points downward, and changes from pointing outward H to pointing inward as β 1 n increases; • the situation is reversed on the lower and left boundaries of H. Combined with the fact that the equation is linear, these properties imply that a solution leaving H cannot enter it again. Therefore, the unique periodic solution has to lie within H.
As α * ⊥ and α 1 ⊥ satisfy the same bounds as for ̺ 2 = 0, it is straightforward to check that Corollary 4.12, Corollary 4.13 and Proposition 4.14 still hold in the present case.
Recall from Lemma 3.1 that the invariant density can be written as π(x, y) = Z −1 e −2V (x,y)/σ 2 . Since we also have π(x, y) = π 0 (x|y)Φ(x, y) , Φ(x, y) = 1 + n 1 α n (y)φ n (x|y) ,
solving for V gives the expression
One can get a better idea of the difference between V and V 0 by writing
Note that by (4.9a) we have B(y) 2 Φ 0 (x, y) = 1 − A(y)δ 1 (y) + (2h 0 (x|y) − 1)(A(y) − δ 1 (y)) 1 + O(λ 1 ℓ) . 
and the adjoint SDE has the form (3.7) with
where we have defined the adjoint potential by
Using the expression (4.10a) for φ 1 , and approximating Φ(x, y) by Φ 0 (x, y), one can deduce from (5.2) and (5.5) that
In other words, the potential well depths are symmetrised for V , and inverted for the adjoint potential V * 0 with respect to the initial potential V 0 (see also Figure 1 ). We will need some a priori estimates on the committors h AB and h * AB . We expect h AB to be close to the static committorh 0 given for a(y) < x < b(y) bỹ
Note thath 0 only slightly differs from h 0 owing to the different boundary conditions. The difference is however exponentially small in σ 2 , with an exponent that can be made large by taking ρ in (2.19) small. Similarly, h * AB should be close tõ
and similarly for π 0 , (g * ) 2 1/2 . P : Since L xh0 = 0, the function g satisfies the equation
with Dirichlet boundary conditions. Consider first the case ̺ 2 = 0, in which L y = ∂ y , and define the Lyapunov function V (g) = 1 2 π 0 , g 2 .
Changing y into −y, we obtain ε∂ y V = π 0 g, ε∂ y g + 1 2 ε ∂ y π 0 , g 2 = π 0 g, L x g + ε π 0 g, ∂ yh0 + ε
where c 1 > 0 is a constant of order 1 related to the spectral gap of L x with Dirichlet boundary conditions onD = (a(y), b(y)). Using the bounds (C.1) and (C.2) on ∂ yh0 obtained in the proof of Proposition 4.1, we get
.
for some constantc 1 ,c 2 > 0. The result then follows by applying Gronwall's inequality to W = √ V . It thus remains to deal with the case ̺ 2 > 0. To this end, we introduce k(x, y) = √ ε σ g(x, y) + ̺ 2 σ 2 2 ∂ y g(x, y) .
Then the pair (g, k) satisfies the system of hyperbolic type
This system can be made isotropic via a shearing transformation
where for any γ ∈ R, we set, in terms of eigenvaluesλ n and eigenfunctionsπ n andφ n of L x with Dirichlet boundary conditions onD,
This results in the system √ ε̺σ
The result then follows in a similar way as above, by working with the Lyapunov functions
and showing that for a periodic solution of the system, both V + and V − have to remain small.
We decompose
and estimate the contribution of each term separately. A similar argument as in (4.19) shows that
The other terms can be bounded via the Cauchy-Schwarz inequality. Namely, we obtain
where we have used Proposition 4.7 (which applies toh 0 as well), Corollary 4.13 and Proposition 5.1.
Recall from (2.16) the notation
The purpose of this section is to establish the following estimates on the capacity cap(A, B). 
The proof of this result is naturally divided into two parts. We will prove the upper bound in Section 6.1, and the lower bound in Section 6.2.
The upper bound on the capacity will follow from the defective-flow Dirichlet principle (3.14). The expressions for the minimisers given in Proposition 3.6 suggest taking as test functions 
We introduce a probability measure µ onD = (a(y), b(y)) with density (see (5.6))
where κ can be made large by taking ρ in (2.19) large. Applying the definition (3.11) of D to Ψh 0 , we obtain
In order to estimate the first inner product, we use the decomposition π = π 0 (Φ 0 + Φ * ⊥ + Φ 1 ⊥ ), expand, and consider the resulting terms separately. For the first term, using (4.10a) to compute µ, φ 1 , we find
. The second term can be directly bounded via Proposition 4.14 by
It remains to estimate the contribution of π, (∂ yh0 ) 2 . We split it into two parts, which satisfy
(where we used the sharper estimate (C.3) to bound ∂ yh0 ), and
To complete the proof of the upper bound on the capacity, it remains to control the error due to the fact that ϕ is not exactly divergence-free. Note that in view of (3.5), we have
3)
The contributions of the two brackets to the error term can be estimated separately. They are small becauseh 0 andh * 0 are both approximately harmonic with respect to L and L * . Proposition 6.3. We have the bound
P : We will consider the contribution of the first bracket in (6.3). The expression (5.4) for c x shows that the derivatives with respect to x cancel exactly, while the expression (3.4) for c y shows that the remaining part is equal to
The first error term is thus given by
Bounding h AB by 1, and using
we find that the contribution of ∂ yh0 satisfies the claimed bound. The contribution of ∂ yyh0 is bounded similarly. The proves the result for the first bracket in (6.3), and the proof for the second bracket is similar.
To obtain a lower bound on the capacity, we will apply the defective Thomson principle (3.15). Since 
where we have used the expression (5.3) for B 2 Φ 0 , the expression (4.30) for λ 1 [A − δ 1 ] , Corollary 4.13 to estimate the contribution of Φ ⊥ , as well as (2.4).
. P : Substituting the expression (6.4) of the test flow in the definition (3.11) of D and using (4.6), we obtain
where µ is the probability density µ introduced in (6.1). The leading contribution comes from the term (cf. (5.3) )
Therefore, when integrating against λ 1 (y)B(y) 2 , (6.2) and (4.30) imply that the contribution of the term in (A(y) − δ 1 (y)) satisfies the claimed bound. Furthermore, it follows from Proposition 4.14 that
Combining this with the bound
which follows from (4.32), yields the result.
Since the test flow ϕ is not exactly divergence-free, to complete the proof of the lower bound it remains to control the error term in (3.15).
The definition (6.4) of the test flow implies
We decompose the inner product as
and estimate the resulting terms separately. For the first term, we note that
As in (6.2) above, the integral is exponentially close to 1 2 , which results in a negligible term when integrating against λ 1 (y)B(y) 2 , owing to (4.30) .
Regarding the second term, we observe that (5.7) . The contribution of this term is thus of the order of (ε/σ 2 ) λ 1 .
The third term in (6.7) satisfies (cf. (6.5))
which results in a contribution of order (εℓ/σ 2 ) λ 1 . The fourth term in (6.7) can be bounded using (4.31a) by
Using (4.6) and integrating against λ 1 B 2 , we obtain indeed a quantity bounded by the second term on the right-hand side of (6.6), though with a slightly better power of σ. For the last term in (6.7), we use the quick-and-dirty bound
implied by (4.31b), which accounts for the second summand in (6.6).
Theorem 2.3 follows immediately from Proposition 3.4, Corollary 5.2 and the estimate of the capacity given in Theorem 6.1. We thus proceed with the proof of Theorem 2.4. To this end, we start by evaluating more precisely the expression (2.20) of the expected transition time integrated with respect to the equilibrium measure.
Using (2.17) and (2.18), we obtain
Using the expressions (2.22) for λ 1 and λ 1 A , we obtain
In a similar way, we get
Substituting in (2.20) yields
and R 0 is defined in (2.21) . Discarding error terms already accounted for in the previous estimate, and using (7.1) and (7.2), we obtain
In order to simplify the expression of R 1 , we start by noting than Jensen's inequality implies
Then we define
A short computation shows that
This expression is indeed equivalent to (2.26), since we have R e 2H/σ 2 , R − e 2H − /σ 2 for the constants H and H − introduced in (2.24). For R 1 to be small, ε has to satisfy the condition
Since λ 1 has order e −2(h min
have been defined in (2.23), by treating separately the cases h min + > h min − and h min + < h min − , one readily obtains that this condition can be satisfied for a non-empty interval of values of ε if and only if Condition (2.28) is met.
It remains to show that we can replace the expectation when starting in the equilibrium measure ν AB by the expectation when starting in a single point on ∂A. This will follow if we can show that E z [τ B ] depends little on the starting point z ∈ ∂A. We will do this by adapting an argument used in the proof of [10, Proposition 3.6].
We first fix a point z ∈ ∂A, and show that Ez[τ B ] is close to E z [τ B ] for allz in a ball of small radius of order 1 centred in z. Let Ω be an event of probability close to 1, on which − + r − (y)/ε F 6. Absorbing variant of the two-state markovian jump process.
where the sets B − ⊂ B ⊂ B + have boundaries close to each other. Using the Cauchy-Schwarz inequality, we obtain
Using a standard large-deviation estimate on Pz{τ B > T } for a fixed T > 0, one easily obtains a bound of the form
with T 1 (η) < ∞ for all η > 0 (see for instance [10, Sections 5.2 to 5.4] , which applies to a much harder infinite-dimensional setting). It thus suffices to show that P(Ω c ) e −κ/σ 2 for some κ > 0 and to apply (7.3) with η < κ/2 to show that Ez[τ B ] and E z [τ B ] are exponentially close to each other. We do this by choosing
Indeed, in [36] it is shown that this event has a probability exponentially close to 1 for appropriate values of c, m > 0 (see also [39] for a more streamlined version of the proof of [36] in a more general setting). It remains to show that E z [τ B ] changes little when z moves along the boundary ∂A. To do this, we fix, say,z = (y, a(y)) with 0 < y < 1 and z = (1, a(1)) on ∂A. Let τ 1 (z) = inf{t > 1 : y t = 1} be the first time at which the sample path starting inz hits the line {y = 1}. Using for instance [11, Proposition 6.3] , one easily obtains that with probability exponentially close to 1, τ 1 (z) is bounded by a constant of order 1, and z τ 1 − z is smaller than an arbitrary constant of order 1. From that we deduce as above that
which concludes the proof of Theorem 2.4.
A The two-state jump process P P 2.1. Consider a modified two-state process in which the + state has been made absorbing ( Figure 6 ). Its first-hitting time τ + , starting from the − state at a fixed time y 0 , agrees with the corresponding first-hitting time of the original process. The occupation probability p − (y) of the − state satisfies εp ′ − (y) = −r − (y)p − (y) with initial value p − (y 0 ) = 1, and is thus given by p − (y) = P y 0 τ + > y = e −R − (y,y 0 )/ε . The expectation of τ + is thus given by
Noting that by periodicity,
we obtain
Summing the geometric series yields the claimed result.
In this section, we provide quick proofs of the potential-theoretic results stated in Section 3. Except for a small addition in the case of test flows which are not divergence-free, all theses proofs are contained in [30] . We provide them here for convenience, as we use slightly different notations and scalings.
The adjoint in L 2 (R × (R/Z)) of L is given by
The condition L † e −2V /σ 2 = 0 is equivalent to (3.2). P L 3.2. Relation (3.3) follows from a short computation. Using the explicit form (3.4) of c, one obtains
which vanishes by (3.2). To prove the first equality in (3.8), we use integration by parts, that is,
By the divergence theorem and the boundary conditions for h AB ,
Since L h AB vanishes on (A ∪ B) c , the second term on the right-hand side of (B.1) is equal to 2ε/σ 2 times
The same skew-symmetry argument as in (3.6) implies that the last integral vanishes. Since the first term on the right-hand side of (B.1) is proportional to the capacity, the first equality in (3.8) follows. To prove the second equality, we use the fact that owing to the vanishing divergence condition (3.5), we have
Since h AB = 1 on ∂A, the integral of h AB (c · n)π dλ indeed vanishes. To prove the first equality in (3.9), we use a similar computation as in (B.1) to obtain
The first term on the right-hand side is proportional to the capacity, yielding the claimed result. The second equality in (3.9) then follows from (3.6), while the last equality is obtained by exchanging the roles of h AB and h * AB in the above computation.
By the divergence theorem, we have
where we have used the facts that (L s − L a )h * AB vanishes on (A ∪ B) c , while ∇h * AB = 0 on A. Furthermore, since h * AB vanishes on B, we have
Since w B solves the Poisson problem (B.2), we have L s w B = −1 − c · ∇w B , so that substitution in (B.3) yields
By the skew-symmetry property (3.6) and the boundary conditions, the contribution of the last two summands in the integral on the right-hand side vanishes.
We start by noting that
Integrating by parts with respect to ∇f , we obtain
The second term on the right-hand side cancels the c-dependent term in (B.4). Furthermore, we have
The first term on the right-hand side is equal to −γ by (3.12) , while the second one vanishes since ϕ is divergence-free.
Remark B.1. If ϕ is only approximately divergence-free, the above proof yields
This can be used to obtain bounds from flows that are not exactly divergence-free. ♦ P P 3.6. Pick f ∈ H 1,0 AB and ϕ ∈ F 0 AB . By (3.13) with α = 1 and γ = 0 and the Cauchy-Schwarz inequality, we have
showing that cap(A, B)
Since clearlyf ∈ H 1,0 AB , it remains to show thatφ ∈ F 0 AB . Noting that
where we have used the fact that ∇ · (e −2V /σ 2 c) = 0 in the second line. Furthermore,
which vanishes by Lemma 3.3. The bound (3.14) is obtained by using (B.5) instead of (3.13). P P 3.7. Pick f ∈ H 0,0 AB and ϕ ∈ F 1 AB . By (3.13) with α = 0 and γ = 1 and the Cauchy-Schwarz inequality, we have 
Using the fact that ∇ · (e −2V /σ 2 c) = 0, we obtain
showing thatφ has flux −1 as required. The bound (3.15) is again obtained by using (B.5) instead of (3.13).
C.1 Bounds on h 0 , h 1 and φ 1
In this section, to lighten notations, we will often drop the dependence of the functions on y. P P 4.1. First note that ∂ y h 0 (x) vanishes for x ∈ (x * − , x * + ). We thus assume henceforth that x ∈ (x * − , x * + ). Taking the derivative with respect to y of the expression (2.5) for the committor, we obtain
where
By standard Laplace asymptotics (see Appendix D), we obtain
Similarly, using the fact that
Substituting in (C.1) yields
which implies (4.7a). The bound (4.7b) follows in an analogous way from the fact that
The bound (4.8) on η(y) is a consequence of the fact that (C.2) yields
combined with (4.3) and (4.6).
Remark C.1. Actually, Lemma D.3 provides the sharper bound
and similarly for J − I(x). ♦
In order to prove Proposition 4.2, we will use the fact that h 1 and its derivatives satisfy certain Poisson boundary value problems. For this purpose, we will repeatedly use the following lemma.
Lemma C.2. Let D = (x * − , x * + ), and let ϕ satisfy the Poisson problem
Assume that there exists a constant c such that |ψ(x)| ch 0 (x) for all x ∈ D. Then
holds for all x ∈ D. Analogous bounds hold with h 0 (x) replaced by 1 − h 0 (x) throughout.
P : Consider first the simpler Poisson problem L x ϕ = ψ, with zero boundary conditions as in (C.4). Using the second expression for L x in (2.3), it is easy to check that its solution is given by
Using the assumption on ψ and the bounds (C.2) on h 0 , one obtains that for x x * 0 ,
A similar conclusion is obtained for x x * 0 using the equivalent expression
Corresponding bounds on ∂ x (L −1 x ψ) are obtained in a similar way, using the derivative with respect to x of (C.5). It thus remains to extend the bounds to (L x + λ 1 ) −1 ψ. This follows readily from the Neumann-type series 
Taking the derivative with respect to y of the equation for h 1 , we obtain that ∂ y h 1 satisfies (C.4) with
. The bounds on h 0 , ∂ y h 0 and (C.6) imply that ψ(x) has order ℓλ 1 h 0 (x)/σ 2 , so that Lemma C.2 yields
. The bound on ∂ yy h 1 is obtained in an analogous way, by taking one more derivative with respect to y.
We introduce the variables
The orthogonality condition
where we have used (4.5) and Proposition 4.2 to obtain the last equality. The function v(y) is then determined via the normalisation condition
owing to Propositions 4.1 and 4.2. Substituting in (C.9), using the expressions (4.5) of π 0 , h 0 and π 0 , 1 − h 0 and solving for v(y) yields v(y) = 1 + O(λ 1 (y)ℓ) thanks in particular to the bound (4.8) on η(y). Expressing φ ± (y) in terms of u(y) and v(y) yields (4.9a). The other relations then follow essentially by taking derivatives with respect to y of the above expressions. Differentiating (C.7), we obtain
(C.10)
Differentiating (C.8) yields
while the derivative of (C. 9) gives
Substituting in (C.10) yields (4.9b). In the same spirit, one obtains
and plugging this into the derivative of (C.10) yields (4.9c).
The expression for f 10 = −σ 2 π 0 , ∂ y φ 1 follows from the expression (4.10b) for ∂ y φ 1 , the definition (4.5) of∆(y), and the fact that σ 2 π 0 , |∂ y h 0 | has order η(y) by Proposition 4.1. A similar argument applies to f 11 = −σ 2 π 0 , φ 1 ∂ y φ 1 .
The expression for g 10 is obtained by evaluating separately the two summands on the right-hand side of (4.1). Proceeding as for f 10 , we obtain
In order to determine ∂ y π 0 , ∂ y φ 1 , we note that on one hand,
while on the other hand, (4.5) implies
and implies the stated expression for g 10 . The computation of g 11 is similar.
For further reference, we list here a few more expressions of particular inner products, which can be derived in the same way as in the above proof:
To obtain estimates involving other eigenfunctions than φ 1 , it will sometimes be useful to take advantage of the fact that L x is conjugated to the Schrödinger operator
where U 0 is the three-well potential
In particular,L x has the same eigenvalues −λ n as L x , and its eigenfunctions ψ n satisfy
In particular, we have the relations
between derivatives of eigenfunctions, where we have used
Note that by the Feynman-Hellmann theorem, we have λ ′ n (y) = − ψ n , ∂ yLx ψ n = 1 2σ 2 ψ n , ∂ y U 0 ψ n , (C. 16) while first-order perturbation theory shows that if λ n = λ m , then
This entails in particular the following useful estimate.
For any function f ∈ L 2 (π 0 [1 + W 2 + σ 2 ∂ y W ] 2 dx),
∂ yy π n , f 2 π 0 , [1 + W 2 + σ 2 ∂ y W ] 2 f 2 .
: By (C.14), we have σ 4 ∂ y π n , f 2 2 √ π 0 , W f ψ n 2 + 2σ 4 √ π 0 , ∂ y ψ n f 2 .
Summing over n yields two terms, the first one being equal to
As for the second sum, the Cauchy-Schwarz inequality and (C.17) show that it is bounded by
The last sum is bounded, because λ n grows like n 2 , while ψ n , ∂ y U 0 ψ m is bounded uniformly in n and m. This proves the first inequality, and the second one is proved in a similar way, taking one more derivative with respect to y.
Remark C.4. In may happen that two eigenvalue λ n (y) and λ m (y) cross for particular value of y.
In that case, the bound (C.17) becomes useless, but an equivalent result can be obtained by locally modifying the basis of eigenfunctions. For simplicity, we will not give details of this procedure here, but refer the reader to [27, 28] . ♦
In the same spirit, the following lemma allows to estimate derivatives of functions expanded in the eigenbasis.
Then
λ n α 2 n .
: By (C.13), we have
The first term on the right-hand side satisfies the claimed bound since ∂ x V 0 is bounded on D. As for the second term, the fact that ψ m is an eigenfunction ofL x implies
This yields
which also satisfies the claimed bound. P P 4.6. To prove the first two bounds, we note that owing to the completeness of the set of eigenfunctions, one has n 0
n 0 ∂ y π 0 , φ n π n , W = 4 π 0 , W 2 , which has order 1. In a similar way, we obtain n 0
Using the Cauchy-Schwarz inequality, (4.11) and Remark 4.5, one obtains that both terms have again order 1. The proof of the bounds involving g ni are similar. The last two bounds then follow directly from Lemma C.3 with n and m interchanged, taking f = φ n , since W is bounded uniformly on compact sets, while for large |x|, the decay of π 0 (x) dominates any polynomially growing term. P P 4.7. Using again the completeness of the set of eigenfunctions, we have n 0 π n , h 0 2 = n 0 π 0 , h 0 φ n π n , h 0 = π 0 , h 2 0 = π 0 , h 0 − η(y) .
At the same time, we also have 1 n=0 π n , h 0 2 = π 0 , h 0 2 + π 0 , φ 1 h 0 2 = π 0 , h 0 2 + φ − (y) π 0 , h 2 0 + φ + (y) π 0 , h 0 (1 − h 0 ) 2 1 + O(λ 1 ℓ)
= π 0 , h 0 2 1 + e 2∆(y)/σ 2 + O η(y) .
The result follows by subtracting the two sums, and using the definition (4.5) of∆(y). P P 4.8. We will spell out the proofs in the case i = 1, since the case i = 0 is similar, though slightly easier. The first sum can be estimated by noting that we conclude that the first sum indeed has order λ 1 ℓ 2 . In the same spirit,
by (C.11b), while f 10 f 01 + f 2 11 = A 2 (∆ ′ ) 2 + O(λ 1 ℓ 2 ), showing the result for the second sum. Regarding the third sum, we use the decomposition g 1m = −ℓ 1m − 2k 1m given in (4.1) and estimate separately the sums of squares of ℓ 1m and k 1m . Noting that m 0 ℓ 2 1m = σ 8 m 0 π 0 , ∂ yy φ 1 φ m π m , ∂ yy φ 1 = σ 8 π 0 , (∂ yy φ 1 ) 2 and using (C.11c), we find that this sum is indeed equal to ℓ 2 10 + ℓ 2 11 + O(λ 1 ℓ 3 ). As for the sum of k 2 1m , we note that (4.11) implies k 1m = σ 2∆′ ∂ y π m , Aφ 1 + B + σ 4 ∂ y π m , R 1 =∆ ′ Af 1m + σ 2 ∂ y π m , R 1 .
We have already bounded m 2 f 2 1m , and the sum involving the error term R 1 can be bounded using Lemma C.3 and (4.12). The proof is similar for the other sums.
In order to prove Proposition 4.9, we introduce two linear operators Π ⊥ and L −1 ⊥ defined by
The operator Π ⊥ is the projection on the complement of the span of φ 0 and φ 1 , while L −1 ⊥ is the Green function of L x restricted to this complement. Note that L −1 ⊥ = L −1 ⊥ Π ⊥ = Π ⊥ L −1 ⊥ .
Lemma C.6. Let G 0 be the Green function with Dirichlet boundary conditions, given by (C.5) for x ∈ D = (x * − , x * + ), and by
Then we have the representation
where the boundary values f ± are given by The boundary values f ± follow from the conditions π 0 , L −1 ⊥ f = π 1 , L −1 ⊥ f = 0, which are equivalent to the linear system π 0 , h 0 π 0 , 1 − h 0 π 1 , h 0
Solving this system, using (4.5) and the fact that 1 λ m f 1m f m1 = σ 4 ∂ y π 1 , L −1 ⊥ ∂ y φ 1 .
Applying Lemma C.6 and using the representation (4.11) of ∂ y φ 1 , we obtain S 1 = σ 4 (f − − f + ) ∂ y π 1 , h 0 + σ 4 ∂ y π 1 , G 0 Π ⊥ R 1 = −σ 4 (e∆ /σ 2 + e −∆/σ 2 ) ∂ y π 1 , h 0 π 1 , G 0 Π ⊥ R 1 + σ 4 ∂ y π 1 , G 0 Π ⊥ R 1 .
By the expressions (4.10a) of φ 1 , we have f 11 = σ 2 ∂ y π 1 , φ 1 = σ 2 (e∆ /σ 2 + e −∆/σ 2 ) ∂ y π 1 , h 0 [1 + O(λ 1 ℓ)] + O(λ 1 ℓ) , which yields σ 2 (e∆ /σ 2 + e −∆/σ 2 ) ∂ y π 1 , h 0 = f 11 + O(λ 1 ℓ) = −A∆ ′ + O(λ 1 ℓ) .
Using the fact that ∂ y π 1 = ∂ y π 0 φ 1 + π 0 ∂ y φ 1 and the expression (C.15) for ∂ y π 0 , we arrive at S 1 = σ 2 2π 1 W +∆ ′ A + O(λ 1 ℓ 2 ) + π 0 ∆ ′ B + R 1 , G 0 Π ⊥ R 1 π 0 , |G 0 Π ⊥ R 1 | + |π 1 |, |G 0 Π ⊥ R 1 | .
• f ′ /f and f ′′ /f are bounded uniformly in σ in I ± . Then π 0 , f = f (x * − (y)) e −∆(y)/σ 2 +f (x * + (y)) e∆ (y)/σ 2 e −∆(y)/σ 2 + e∆ (y)/σ 2 1 + O(σ 2 ) . P : Using the change of variables x = x * ± (y) + σz/( √ 2ω − (y)), one obtains
whereĨ ± = √ 2ω ± (y)(x − x * ± (y))/σ. Furthermore, the integral over R \ (I − ∪ I + ) is negligible with respect to the sum of these two integrals. The result then follows from applying (D.1) first to f = 1 to estimate Z 0 (y), and then to general f satisfying the stated assumptions.
We will also need estimates involving the integral of e 2V 0 /σ 2 against a function vanishing polynomially at x = x * 0 . To ease notation, we will assume that x * 0 = 0, and write
where W (x) = O(x 3 ). Consider the integrals
x n e 2V 0 (x)/σ 2 dx , J n (x) = e −2V 0 (x)/σ 2 δ x x n 1 e 2V 0 (x 1 )/σ 2 dx 1 ,
where n ∈ N 0 and δ has order 1.
if n is odd .
(D.2) P : The case of even n follows from a direct application of [38, Theorem 8.1] , where the fact that the error has order σ 2 is due to the leading term of W being odd. When n is odd, we use integration by parts to obtain
(D.3) If n = 1, the integral has order σ 3 by (D.2) with n = 2, while the boundary terms are negligible. For n 3, we obtain I n = O(σ 2 I n−1 ) + O(σ n+2 ), so that the result follows by induction.
In particular, we have 
x n , whose right-hand side vanishes for J n (x) = J ⋆ n (x) = O(σ 2 x n−1 ). Since J n (x) − J * n (x) is a decreasing function of x for x < 0, and (D.4) is satisfied for negative x of order 1, it holds for all x ∈ [0, δ]. A similar argument applies for x > 0 by changing x into −x. To prove (D.5), we use a similar integration-by-parts argument as in (D.3) to obtain
The integral on the right-hand side can be bounded using (D.4) with n = 2, while the term e 2V 0 (δ)/σ 2 is negligible. Finally, (D.6) follows from the integration-by-parts relation e 2V 0 (x)/σ 2 J 1 (x) = − x 2 2 e 2V 0 (x)/σ 2 − e 2V 0 (δ)/σ 2 − 1 σ 2 δ x x 2 1 ∂ x V 0 e 2V 0 (x 1 )/σ 2 dx 1 , expressing the integral on the right-hand side in terms of J 3 (x).