1 Introduction↩︎

We consider stochastic gradient dynamics formally described by \[\label{eq} dX_t=-A\nabla F(X_t) \, dt+\sqrt{2\beta^{-1}A} \, dW_t .\tag{1}\] Here \(W\) is a cylindrical Brownian motion on a separable Hilbert space \(H\), the potential \(F \in C^2(H)\) satisfies suitable coercivity assumptions at infinity and \(A\) is a selfadjoint uniformly positive operator on \(H\). The parameter \(\beta>0\) is the inverse noise strength, and we are interested in the small-noise regime \(\beta\to\infty\).

Heuristically, the drift in 1 drives the dynamics towards the local minima of the functional \(F\), while the noise creates rare transitions between different wells. When \(F\) has several local minima, this competition gives rise to metastable behavior: on moderate time scales the process appears trapped near one minimum, whereas on exponentially long time scales transitions between different wells become visible. A central quantity encoding this phenomenon is the spectral gap \(\lambda_\beta\), namely the first nonzero eigenvalue of the generator. Its inverse \(\lambda_\beta^{-1}\) corresponds to the relaxation time of the dynamics and is therefore expected to be exponentially large in the small-noise regime. More generally, if the energy landscape has \(k_0\) local minima, one expects a cluster of \(k_0\) low-lying eigenvalues: the zero eigenvalue corresponding to equilibrium, together with \(k_0-1\) positive eigenvalues which are exponentially small in \(\beta\). These eigenvalues describe the effective transitions between the metastable wells, while the rest of the spectrum remains bounded away from zero and corresponds to faster relaxation modes within the wells.

When \(\dim(H)<\infty\), this picture is by now classical. Large deviations theory identifies the exponential scale of rare transitions [1], while sharp asymptotic estimates can be obtained through potential-theoretic and semiclassical methods; see, for instance, [2]–[7], and see also [8]–[10] for non-reversible extensions. For example, suppose that \(H=\mathbb{R}^d\), that \(A\) is a symmetric positive definite matrix, and that \(F\) has exactly two local minima \(x_-\) and \(x_+\) with \(F(x_-)<F(x_+)\), separated by a unique nondegenerate saddle point \(z\) of index one. Let \(-\alpha<0\) be the unique negative eigenvalue of the Hessian of \(F\) with respect to the modified scalar product \(\langle A^{-1}\cdot, \cdot \rangle\) and assume that \(\nabla^2F(x_\pm)>0\). Then, as \(\beta\to\infty\), the spectral gap satisfies the Eyring–Kramers law \[\label{FDEK} \lambda_\beta = \frac{\alpha}{2\pi} \sqrt{ \tfrac{\det \nabla^2F(x_+)} {|\det \nabla^2F(z)|} } e^{-\beta[ F(z)-F(x_+)]} \, \left(1+o(1) \right).\tag{2}\] Further, \(\lambda_\beta\) is separated from the rest of the spectrum by a gap of order one. The determinant ratio in 2 comes from Gaussian approximations of the invariant measure \(Z_\beta^{-1}e^{-\beta F(x)} dx\) near the minimum \(x_+\) and the saddle \(z\), while the factor \(\alpha/(2\pi)\) accounts for the unstable direction at the saddle.

The goal of this paper is to establish an analogue of this picture in infinite dimensions. In this setting, the cylindrical Brownian motion in 1 is not \(H\)-valued, but can be realized only after embedding \(H\) into a larger separable Banach space \(B\). Our basic structural assumption is that \(F\) is a sufficiently regular perturbation of the quadratic Gaussian energy. More precisely, we assume that there exists a function \(U\in C^2(B)\) such that \[U|_{H} = F-\frac{1}{2}|\cdot|_H^2 .\] As a consequence of this assumption the invariant measure, formally proportional to \(e^{-\beta F(x)} dx\), can be represented as the Gibbs perturbation \[d\mu_\beta = Z_\beta^{-1}e^{-\beta U} d\gamma_\beta ,\] where \(\gamma_\beta\) is obtained by rescaling the centered Gaussian measure on \(B\) with Cameron-Martin space \(H\). Despite this enlargement of the state space, and despite the fact that \(\mu_\beta (H) = \gamma_\beta(H)=0\) in infinite dimensions, the metastable behavior is still governed by the geometry of the energy landscape on the Cameron–Martin space \(H\); in this sense, the relevant low-energy picture is the one that would arise if everything happened on \(H\). To make this principle precise, we prove two main results.

Main results. The first one, Theorem 1, is a spectral separation theorem. Suppose that \(F\) has \(k_0\) local minima and satisfies suitable coercivity and nondegeneracy assumptions, see Assumption 1 below. Then the \(k_0\)-th eigenvalue is bounded away from zero uniformly in the small-noise limit: \[\varliminf_{\beta\to\infty}\lambda_\beta^{(k_0)}>0 .\] Equivalently, at most \(k_0\) eigenvalues vanish as \(\beta\to\infty\). Since each local minimum gives rise to a quasimode with exponentially small energy, this identifies the number of exponentially small eigenvalues with the number of local minima. The proof is based on the variational characterization of the eigenvalues through the infinite-dimensional Dirichlet form [11]–[13] \[\mathcal{E}_\beta[f] = \frac{1}{\beta} \int_B \| \sqrt A \nabla f\|_H^2 \, d\mu_\beta ,\] and on a localization procedure: away from the critical set, after a ground-state transformation, a large-deviation estimate yields a uniform lower bound, with the logarithmic Sobolev inequality for the Gaussian reference measure \(\gamma_\beta\) as a key ingredient; near minima, the analysis reduces to convex models, which yield one low-energy direction, while near saddles and higher-index critical points one instead exploits an unstable direction to rule out additional low-energy modes.

Our second main result, Theorem 2, gives the sharp Eyring–Kramers asymptotics in the double-well case, i.e. assuming that \(F\) has exactly two local minima \(x_-\) and \(x_+\). For example, if \(F(x_-)<F(x_+)\) and there is a unique nondegenerate saddle point \(z\) of index one, Theorem 2 states that 2 still holds, with \(-\alpha\) the unique negative eigenvalue of the selfadjoint operator associated with \(\nabla^2 F(z)\) in the metric induced by the modified scalar product \(\langle A^{-1} \cdot, \cdot\rangle_H\). We emphasize that the determinants appearing in the prefactor of \(\eqref{FDEK}\) are well-defined as Fredholm determinants of trace class perturbations of the identity. Indeed, under our assumptions, \[\nabla^2 F(x) = \text{Id} + \nabla^2 U(x) , \; \; \; \text{ for every } x\in H,\] and \(\nabla^2 U(x)\) is automatically trace class by a general result of Goodman. Theorem 2 also describes the modifications that occur to the prefactor in the case \(F(x_-)= F(x_+)\) and in the case of several relevant saddle points. The proof is based on the choice of a suitable approximate eigenfunction, a precise estimate of its Dirichlet energy and an application of the spectral separation given by Theorem 1 together with a Kato–Temple type inequality.

Applications. Our framework is designed to provide a unified treatment of a broad class of reversible infinite-dimensional stochastic gradient systems. The original motivation, however, comes from two concrete stochastic partial differential equations: the stochastic Allen–Cahn and stochastic Cahn–Hilliard equations on bounded intervals \((0,\ell)\). They are given, respectively, by \[\label{SPDE:AC} dX_t = \bigl(\Delta X_t - X_t^3 + X_t\bigr)\,dt + \sqrt{2\beta^{-1}}\,dW^{L^2}_t\tag{3}\] and \[\label{SPDE:CH} dX_t = -\Delta\bigl(\Delta X_t - X_t^3 + X_t\bigr)\,dt + \sqrt{2\beta^{-1} (-\Delta) }\,dW^{L^2}_t .\tag{4}\] Here \(W^{L^2}_t\) denotes a cylindrical Brownian motion on \(L^2((0,\ell))\), whose formal time derivative is space-time white noise; in the Cahn–Hilliard case the zero-average constraint is imposed. These are prototypical phase-field models for relaxation in a double-well energy landscape. In the terminology of dynamic critical phenomena, they correspond respectively to Model A and Model B dynamics: Allen–Cahn describes a non-conserved order parameter, whereas Cahn–Hilliard describes a conserved order-parameter dynamics and is classically associated with phase separation. We refer, for instance, to [14], [15] for background on these models in the deterministic and stochastic settings. Related small-noise large-deviation problems for stochastic reaction-diffusion equations go back, in particular, to [16]; see also [17]. From the classical gradient-flow perspective, these equations correspond to evolving the same energy \[F(x) = \int_0^\ell \left( \frac{1}{2} |x'(s)|^2 + \frac{1}{4}\bigl(x(s)^2-1\bigr)^2 \right)\,ds\] with respect to different geometries: the Allen–Cahn equation is the \(L^2\)-gradient dynamics, whereas the Cahn–Hilliard equation is the \(H^{-1}\)-gradient dynamics on the zero-average subspace. A key point of our approach is that this geometric distinction is expressed relative to the Cameron–Martin space of the Gaussian reference measure, rather than relative to the usual \(L^2\)-based description of the equations. Indeed, for Allen–Cahn we take \(H=H^1((0,\ell))\), and the classical \(L^2\)-geometry is recovered through the modified metric generated by \(A^{-1}=(-\Delta+1)^{-1}\) on \(H\). With this choice, the term \(\sqrt A\,dW_t\) in 1 , where \(W_t\) is cylindrical on \(H\), has the same law as \(dW^{L^2}_t\). Analogously, for Cahn–Hilliard we take \[H=\dot{H}^1((0,\ell)) := \left\{ x\in H^1((0,\ell)):\int_0^\ell x(s)\,ds=0 \right\}.\] With this choice, the mobility is \(A=(-\Delta)^2\), so that the metric induced by \(A^{-1}=(-\Delta)^{-2}\) is two Sobolev orders weaker than the Cameron–Martin metric. Thus, the usual \(H^{-1}\)-gradient structure over \(L^2\) is represented in our framework as an \(H^{-2}\)-type geometry relative to \(H=\dot{H}^1((0,\ell))\). This shift of viewpoint reflects the guiding principle that the relevant low-energy geometry is encoded on \(H\), and it places both equations in the common abstract form 1 , where \(W_t\) is cylindrical on \(H\). Similar changes of Hilbertian structure appear in infinite-dimensional convergence-to-equilibrium problems; see, for instance, [18].

We apply our abstract results to both equations 3 , 4 , see respectively Corollary 1 and Corollary 2, each providing a spectral separation result and Eyring-Kramers formula for the spectral gap. Corollary 1 (Allen–Cahn) generalizes a result obtained in our earlier work [19], which assumed sufficiently small \(\ell\), leading to a particularly simple saddle structure.

We remark that for the stochastic Allen–Cahn equation, Eyring–Kramers laws for metastable transition times have been studied before, for instance through finite-dimensional approximation and potential-theoretic methods; see e.g. [20]–[22]. The more challenging Cahn–Hilliard case, by contrast, appears to be substantially less developed. Related metastability results for mass-conserving spatially extended diffusions were obtained in [23], but that work concerns a discretized mass-conserving Allen–Cahn-type dynamics rather than the continuum stochastic Cahn–Hilliard equation considered here. To the best of our knowledge, Corollary 2 gives the first rigorous Eyring–Kramers formula for the stochastic Cahn–Hilliard equation 4 .

We remark that the length \(\ell\) of the interval is kept fixed. The joint regime in which both \(\beta\to \infty\) and \(\ell \to \infty\) leads to a sharp-interface problem, which is beyond the scope of the present work; see [24] for related asymptotics of the \(\phi^4_1\) measure in this regime. Although we focus on Allen–Cahn and Cahn–Hilliard, the abstract framework may also apply to other one-dimensional stochastic gradient systems, such as the sine-Gordon model [25], or to models with non-local interaction terms.

Outline of the paper. In Section 2 we introduce the precise setting, define the Gibbs measures and Dirichlet forms, and state the main results. In Section 3 we prove Theorem 1 on the spectral separation. Section 4 is devoted to the proof of Theorem 2 giving the Eyring–Kramers formula in the double-well case. Finally, in Section 5 we verify the assumptions in the main examples, namely the stochastic Allen–Cahn and stochastic Cahn–Hilliard equations, proving Corollary 1 and Corollary 2.

2 Setting and main results↩︎

We fix a separable Hilbert space \(H\) and a functional \(F:H \to\mathbb{R}\). It is standard  [26], [27] that \(H\) admits abstract Wiener realizations: one may embed \(H\) continuously and densely into a separable Banach space \(B\) carrying a centered nondegenerate Gaussian measure \(\gamma\) whose Cameron–Martin space is \(H\).

Our standing assumption on \(F\) is that it admits an abstract Wiener realization for which it can be written as a regular perturbation of the quadratic Cameron–Martin energy. More precisely, we assume that there exist an abstract Wiener space \((B,H,\gamma)\) and a function \(U\in C^2(B)\) such that \[\label{Eq:Def95F} F = U|_H + \frac{1}{2} \|\cdot\|_H^2 .\tag{5}\] We fix one such realization \((B,H,\gamma)\) once and for all; the corresponding function \(U\) is then uniquely determined by 5 . Here and throughout, differentiability on Banach spaces is understood in the Fréchet sense.

We denote by \(\|\cdot\|_B\) the norm on \(B\), while \(\|\cdot\|_H\) and \(\langle\cdot,\cdot\rangle_H\) denote respectively the norm and inner product on \(H\). We write \[\langle \ell,x\rangle := \ell(x), \qquad \ell\in B^*,\;x\in B,\] for the dual pairing between \(B^*\) and \(B\). For \(f\in C^1(B)\) and \(x\in B\), we write \(Df(x)\in B^*\) for its Fréchet differential at \(x\). The Cameron–Martin gradient \(\nabla f(x)\) is the unique element of \(H\) satisfying \[\langle \nabla f(x),h\rangle_H = Df(x)[h], \qquad h\in H .\] The map \(x\mapsto \nabla f(x)\) is continuous whenever \(f\in C^1(B)\). If \(f\in C^2(B)\) and \(x\in B\), let \(D^2f(x)\) be the second Fréchet differential of \(f\) at \(x\), viewed as a bounded symmetric bilinear form on \(B\times B\). The Cameron–Martin Hessian \(\nabla^2 f(x)\) is then the bounded selfadjoint operator on \(H\) associated with the restricted bilinear form \[(h,k)\longmapsto D^2f(x)[h,k], \qquad h,k\in H,\] that is, \[\langle \nabla^2 f(x)h,k\rangle_H = D^2f(x)[h,k], \qquad h,k\in H .\] By Goodman’s theorem, see Theorem 4.6 in [28], this operator is trace class on \(H\) and the map \(x\mapsto \operatorname{tr}\nabla^2 f(x)\) is continuous. Hence the Cameron–Martin Laplacian \[\Delta f(x):=\operatorname{tr}\nabla^2 f(x)\] is well defined for every \(x\in B\), and \(x\mapsto \Delta f(x)\) is continuous whenever \(f\in C^2(B)\).

It follows from the assumption 5 that \(F\in C^2(H)\) and that the gradient and Hessian of \(F\) are given by \[\nabla F(x)=x+\nabla U(x), \qquad \nabla^2F(x)=\mathrm{Id}+\nabla^2U(x), \quad x\in H ,\] where \(\nabla U\) and \(\nabla^2 U\) are understood in the Cameron–Martin sense introduced above. We shall denote by \[\mathcal{S}:=\left\{x\in H:\nabla F(x)=0\right\}\] the set of critical points of \(F\). In finite dimensions, one can take \(B=H\) and a natural object associated with \(F\) is the Witten, or effective, potential \[\label{Eq:Def95Witten} \frac{\beta}{4}\|\nabla F(x)\|_H^2-\frac{1}{2}\Delta F(x), \qquad x\in H,\tag{6}\] depending on the parameter \(\beta>0\). In the present infinite-dimensional setting this expression is only formal. Indeed, although \(\nabla^2U(x)\) is trace class by Goodman’s theorem, the identity term in \(\nabla^2F=\mathrm{Id}+\nabla^2U\) has infinite trace. Thus \(\Delta F\) is not defined as a finite quantity. A natural renormalized expression, see Proposition 3 below, is obtained by removing the divergent quadratic contribution coming from the quadratic part \(F-U|_{H}=\frac{1}{2}\|\cdot\|_H^2\). Formally, this amounts to subtracting \(\frac{\beta}{4}\|\nabla(F-U)\|_H^2 -\frac{1}{2}\Delta(F-U)\) from the expression 6 . Accordingly, we introduce for each \(\beta>0\) the effective potential \(V_\beta \in C(B)\) defined by \[\label{Eq:Def95V95beta} V_\beta(x) := \frac{\beta}{4} \|\nabla U(x)\|_H^2+ \frac{\beta}{2} \langle DU(x),x\rangle - \frac{1}{2}\, \Delta U(x) , \qquad x\in B .\tag{7}\] We now collect the assumptions on the realized potential \(F\) needed for the spectral separation result.

For \(\beta>0\), let \(\gamma_\beta\) denote the push-forward of \(\gamma\) under the scaling map \(x\mapsto \beta^{-1/2} x\). Under Assumption 1, the function \(U\) is bounded from below, hence \(Z_\beta = \int_B e^{- \beta U } d \gamma_\beta < \infty\). We then define the probability measure \[\begin{align} \label{Eq:def95mu} \mathrm{d}\mu_{\beta}:=Z_\beta^{-1}e^{-\beta U}\mathrm{d}\gamma_\beta . \end{align}\tag{10}\] We shall use in the sequel the space \(\operatorname{Lip}_b(B)\) of bounded, globally Lipschitz continuous functions on \(B\). By the infinite-dimensional Rademacher theorem of Enchev and Stroock [29], every \(f\in \operatorname{Lip}_b(B)\) admits a Cameron–Martin gradient \(\nabla f(x)\) for \(\gamma_\beta\)-almost every \(x\in B\), for each fixed \(\beta>0\). Since \(\mu_\beta\) is equivalent to \(\gamma_\beta\), the same holds \(\mu_\beta\)-almost everywhere. Moreover, denoting by \(C_{\operatorname{Lip(H)}}(f)\) the Lipschitz constant of \(f\) along \(H\)-directions, it holds \(\|\nabla f(x) \|_H \le C_{\operatorname{Lip(H)}}(f)\) at \(x\in B\) where \(\nabla f(x)\) exists.

In addition to the realized potential \(F\), which determines the Gibbs measures \(\mu_\beta\) as above, we fix throughout a possibly unbounded selfadjoint operator \(A\) on \(H\), defined on a dense linear space \(\operatorname{dom}(A) \subset H\). We assume that \(A\) is uniformly positive and denote by \(a>0\) its lower bound. We also assume the following compatibility condition between \(A\) and the abstract Wiener realization. For \(\ell\in B^*\), we denote by \(\ell_H\in H\) its Cameron–Martin representative, characterized by \[\ell(ih)=\langle \ell_H,h\rangle_H, \qquad h\in H ,\] where \(i: H \to B\) is the Cameron–Martin embedding and set \(B_H^* := \{ \ell_H: \ell\in B^* \} \subset H\). We shall assume throughout that the set of \(A\)-admissible Cameron-Martin directions \[\mathcal{K}_A := \left\{ h \in B_H^* \cap \operatorname{dom}(A): Ah \in B_H^* \right\} \text{ is dense in } H.\] We denote by \(\mathcal{F}_A C_b^\infty(B)\) the set of cylindrical functions adapted to \(A\), namely of functions \(f: B \to\mathbb{R}\) such that \[f(x)=\varphi\bigl(\ell_1(x),\ldots,\ell_n(x)\bigr)\] for some \(n\in\mathbb{N}\), \(\varphi\in C_b^\infty(\mathbb{R}^n)\) and \(\ell_1,\ldots,\ell_n\in B^*\) satisfying \((\ell_j)_H \in \mathcal{K}_A\). Because \(\mathcal{K}_A\) is dense in \(H\), the linear functionals \(\ell\in B^*\) with \(\ell_H\in\mathcal{K}_A\) generate the full Gaussian first chaos. By the Wiener chaos decomposition, the corresponding smooth cylindrical functions are therefore dense in \(L^2(\gamma_\beta)\). Since \(\mu_\beta\) is equivalent to \(\gamma_\beta\) and has bounded density with respect to \(\gamma_\beta\), a standard truncation argument shows that \(\mathcal{F}_A C_b^\infty(B)\) is also dense in \(L^2(\mu_\beta)\), for every \(\beta>0\). For a function \(f\in \mathcal{F}_A C_b^\infty(B)\) the Cameron–Martin gradient is given by \[\nabla f(x) = \sum_{j=1}^n \partial_j\varphi\bigl(\ell_1(x),\ldots,\ell_n(x)\bigr)\,(\ell_j)_H , \qquad x\in B.\] Since \((\ell_j)_H\in K_A\), the vector \(\nabla f(x)\) belongs to \(\operatorname{dom}(A)\). The coefficients in the above formula being bounded, it follows that \(x\mapsto \|\sqrt A \,\nabla f(x)\|_H\) is bounded on \(B\).

In view of the preceding discussion we may define on \(\mathcal{F}_A C_b^\infty(B)\) the bilinear form \[\mathcal{E}_\beta[f,g] := \frac{1}{\beta} \int_B \langle \sqrt A \, \nabla f , \sqrt A \nabla g\rangle_H \, d\mu_\beta ,\] and set for short \(\mathcal{E}_\beta[f]:=\mathcal{E}_\beta[f,f]\). For \(f\in \mathcal{F}_A C_b^\infty(B)\) and \(x\in B\) we also define \[\label{Eq:Def95L95Operator} \mathcal{L}_\beta f(x) := -\frac{1}{\beta} \operatorname{tr} \left(A\nabla^2 f(x)\right) +\langle A\nabla f(x),x\rangle +\langle A\nabla f(x),\nabla U(x)\rangle_H .\tag{11}\] Each term on the right-hand side is well defined, and \(\mathcal{L}_\beta f\in L^2(\mu_\beta)\) for all sufficiently large \(\beta\). Indeed, \(A\nabla f(x)\in B_H^*\). Hence, through the identification of \(B_H^*\) with \(B^*\), the dual pairing \(\langle A\nabla f(x),x\rangle\) is well defined for every \(x\in B\). Since this term is a finite linear combination of \(B^*\)-linear functionals with bounded coefficients, it belongs to \(L^2(\mu_\beta)\). Moreover, the growth assumption 8 on \(DU\), together with the lower boundedness of \(U\) and Fernique’s theorem applied to the rescaled Gaussian measure \(\gamma_\beta\), implies that \(\langle A\nabla f,\nabla U\rangle_H\in L^2(B,\mu_\beta)\) for all sufficiently large \(\beta\). Finally, since \(\nabla^2f(x)\) has finite rank and range contained in \(\operatorname{dom}(A)\), the operator \(A\nabla^2f(x)\) is finite rank and the trace term is bounded.

An integration-by-parts formula gives, for all \(f,g\in\mathcal{F}_A C_b^\infty(B)\), \[\mathcal{E}_\beta[f,g] = \int_B f\,\mathcal{L}_\beta g\,d\mu_\beta .\] In particular, \(\mathcal{L}_\beta\) is symmetric and non-negative on \(\mathcal{F}_A C_b^\infty(B)\). Since \(\mathcal{F}_A C_b^\infty(B)\) is dense in \(L^2(\mu_\beta)\), the nonnegative symmetric operator \(\mathcal{L}_\beta\) admits a Friedrichs extension. Equivalently, the quadratic form \(\mathcal{E}_\beta\) is closable. We keep the notation \(\mathcal{E}_\beta\) for its closure and still denote by \(\mathcal{L}_\beta\) the associated nonnegative selfadjoint operator.

The min–max values associated with the closed form \(\mathcal{E}_\beta\) are defined by \[\label{Eq:min95max95definition95eigenvalue} \lambda_\beta^{(j)} := \inf_{\substack{V\subset \operatorname{dom}(\mathcal{E}_\beta)\\ \dim V=j+1}} \sup_{f\in V\setminus\{0\}} \frac{\mathcal{E}_\beta[f]}{\|f\|_{L^2(\mu_\beta)}^2}, \qquad j\ge 0 .\tag{12}\] Equivalently, since \(\mathcal{F}_A C_b^\infty(B)\) is a form core for \(\mathcal{E}_\beta\), the same values are obtained if the infimum is restricted to subspaces \(V\subset\mathcal{F}_A C_b^\infty(B)\). Since \(\mathcal{E}_\beta[f]\ge0\) and \(\mathcal{E}_\beta[1]=0\), we have \(\lambda_\beta^{(0)}=0\).

In order to obtain a sharp asymptotic on the low–lying eigenvalues, we make the additional Assumption 2.

In the following we denote by \(\overline{H}\) the completion of \(H\) with respect to the inner product \(\langle x,y\rangle_A:=\langle A^{-1}x,y\rangle\) and we write \(\mathcal{T}\) for the set of critical points \(x\in \mathcal{S}\) such that \(\nabla^2 F(x)\) has exactly a single negative eigenvalue. We obtain by Assumption 13 that the unbounded quadratic form \[\begin{align} q(y):=\langle y,\nabla^2 F(x)y\rangle_H \end{align}\] defined on the subspace \(H\subseteq \overline{H}\) (equipped with the norm \(\|\cdot \|_A\)) is complete, with a corresponding self-adjoint operator \(Q\) defined on \(\overline{H}\) satisfying \[\begin{align} \label{Eq:Q95Operator} A^{-1}Q=\nabla^2 F(x)|_{\operatorname{dom}(Q)}. \end{align}\tag{14}\] While the spectrum of \(Q\) and \(\nabla^2 F(x)\) is different in general, we note that the kernel of \(Q\) is trivial and the number of negative eigenvalues of \(Q\) agrees with the number of negative eigenvalues of \(\nabla^2 F(x)\). In particular, we have for \(x\in \mathcal{T}\) that there is, up to a sign, a unique eigenvector \(\nu_x\in H\) of \(Q\) with a negative eigenvalue \(-\alpha_x<0\) and \(\|\nu_x\|_A=1\). By Assumption 2 it is clear that \(Q\) is invertible, i.e. \(0\notin \sigma(Q)\), and hence we find a \(\delta>0\) such that \[\begin{align} \langle w, Qw\rangle_A \geq \delta \|w\|_A^2 \end{align}\] for all \(w\in \overline{H}\) with \(\langle \nu_x,w\rangle_A=0\). Defining the vector \[\begin{align} \label{Eq:def95of95eta95x} \eta_x:=A^{-1}\nu_x, \end{align}\tag{15}\] we therefore obtain, together with Assumption 13 , for all \(w\in H\) with \(\langle \eta_x,w\rangle=0\) \[\begin{align} \left\langle w, \nabla^2 F(x)w \right\rangle\geq \delta' \|w\|^2, \end{align}\] for a suitable \(\delta'>0\). In particular, the vector \(\eta_x\) separates the level set of the quadratic form \([q\leq 0]\) into two connected components, or equivalently the orthogonal space \(\langle \eta_x\rangle^{\perp}\) has a trivial intersection with \([q\leq 0]\).
The preceding local picture motivates the following standard Morse-theoretic description of the critical sublevel set at the communication height. Let \(h\) be the infimum of the maximal value attained along paths connecting \(x_\pm\), and let \(\mathcal{T}_h\) denote the set of critical points \(x\in [F\le h]\) whose Hessian has exactly one negative eigenvalue. We use the corresponding decomposition of the critical sublevel set: the set \([F\le h]\setminus \mathcal{T}_h\) has two path-connected components, denoted by \(A_-\) and \(A_+\), containing respectively \(x_-\) and \(x_+\). Equivalently, the two components meet only through the saddle region \(\mathcal{T}_h\). In particular, after removing small balls around the points of \(\mathcal{T}_h\), the two components are separated by a positive distance in the Banach-space topology: \[\label{eq:dist} \operatorname{dist}_B \bigl(A_- \setminus B_r(\mathcal{T}_h), A_+ \setminus B_r(\mathcal{T}_h)\bigr)>0 .\tag{16}\] Here \[B_r(\mathcal{T}_h):=\bigcup_{y\in\mathcal{T}_h} B_r(y), \qquad B_r(y):=\{z\in B:\|z-y\|_B<r\}.\] Moreover, if \(x\in \mathcal{T}_h\), then the sign of \(\langle \eta_x,y-x\rangle\) distinguishes the two local branches of \([F\le h]\) near \(x\). More precisely, for \(R>0\) sufficiently small, all points \(y\in [F\le h]\cap B_R(x)\) with \[\langle \eta_x,y-x\rangle<0\] belong to one of the two components, denoted by \(A_{\sigma_1(x)}\), while all points \(y\in [F\le h]\cap B_R(x)\) with \[\langle \eta_x,y-x\rangle>0\] belong to the other component, denoted by \(A_{\sigma_2(x)}\).

In order to state our second main Theorem 2, let us sort \[\begin{align} \label{Eq:enummeration95crit} \mathcal{T}_h=\{x_1,\dots ,x_{m_0},x_{m_0+1},\dots ,x_m\}, \end{align}\tag{17}\] such that \(\sigma_1(x_j)\neq \sigma_2(x_j)\) for \(j\leq m_0\) and \(\sigma_1(x_j)=\sigma_2(x_j)\) otherwise, and let \(\{-\alpha_1,\dots,-\alpha_m\}\) as well as \(\{\eta_1,\dots,\eta_{m}\}\) be the corresponding enumeration of the eigenvalues \(-\alpha_x\) and vectors \(\eta_j:=A^{-1}\nu_j\) defined in 15 . Then, using the rank one projection \(P_{\eta,\eta'}(y):=\langle \eta',y\rangle \eta\), we define the Gaussian measures \[\begin{align} \mathrm{d}\gamma_{\pm}: & =C_\pm^{-1} e^{-\frac{1}{2}\langle x,\nabla^2 U(x_\pm)x\rangle}\mathrm{d}\gamma,\\ \mathrm{d}\gamma_{j}: & =C_j^{-1} e^{- \frac{1}{2}\left\langle x,\left(\nabla^2 U(x_j)+2\alpha_j P_{\eta_j,\eta_j}\right)x\right\rangle }\mathrm{d}\gamma, \end{align}\] with the normalization constants \(C_\pm\) and \(C_j\) given as \[\begin{align} \nonumber C_\pm: & =\mathrm{det}\Big[\mathbf{1}+\nabla^2 U(x_\pm)\Big]^{-\frac{1}{2}}=\mathrm{det}\Big[\nabla^2 F(x_\pm)\Big]^{-\frac{1}{2}},\\ \label{Eq:def95C95j95const} C_j: & = \mathrm{det}\Big[\mathbf{1}+\nabla^2 U(x_j)+2\alpha_j P_{\eta_j,\eta_j}\Big]^{-\frac{1}{2}}= \left|\mathrm{det}\Big[\nabla^2 F(x_j)\Big]\right|^{-\frac{1}{2}}. \end{align}\tag{18}\] The identity in 18 follows easily from the fact that \[\begin{align} \mathbf{1}+\nabla^2 U(x_j)+2\alpha_j P_{\eta_j,\eta_j}=\nabla^2 F(x_j)R_j , \end{align}\] where \(R_j:=I-2P_{\nu_j,\eta_j}\) is an isometry with respect to the \(\|\cdot\|_A\) metric. Furthermore, we define the re-scaled versions \[\begin{align} \mathrm{d}\gamma_{\beta,\pm}: & =C_\pm^{-1} e^{-\frac{\beta}{2} \langle x,\nabla^2 U(x_\pm)x\rangle}\mathrm{d}\gamma_{\beta},\\ \mathrm{d}\gamma_{\beta,j}: & =C_j^{-1} e^{- \frac{\beta}{2} \left\langle x,\left(\nabla^2 U(x_j)+2\alpha_j P_{\eta_j}\right)x\right\rangle }\mathrm{d}\gamma_{\beta}. \end{align}\]

3 Proof of the spectral separation↩︎

It is the content of this section to verify Theorem 1, which claims that, depending on the number of minima \(k_0\) of \(F\), the eigenvalue \(\lambda^{(k_0)}_\beta\) is bounded away from zero by a constant \(\delta>0\) uniformly in \(\beta\rightarrow \infty\). From the min-max definition of \(\lambda^{(k_0)}_\beta\), it is clear that it is enough to find elements \(\Phi_\beta^{(1)},\dots, \Phi_\beta^{(k_0)}\), such that \[\begin{align} \label{Eq:lower95bound95global} \mathcal{E}_\beta[\Psi]\geq \delta \|\Psi\|_{L^2(\mu_\beta)}^2, \end{align}\tag{21}\] for all \(\Psi\) orthogonal to \(\Phi_\beta^{(1)},\dots, \Phi_\beta^{(k_0)}\). In the following, we will first verify the lower bound in 21 separately for functions \(\Psi\) supported in a neighborhood of a critical point \(x\in \mathcal{S}\), where we distinguish between minima and other critical points, and functions having a (certain) positive distance from the critical points \(\mathcal{S}\). In our final Subsection 3.3 we verify Theorem 1 by employing a localization technique and gluing together the results from the different regions.

In order to decompose \(B\) into the different relevant regions, we note that, as a consequence of Assumption 1, we can split the critical points as \(\mathcal{S}=\mathcal{S}_0\cup \mathcal{S}_1\) with

• for all \(x\in \mathcal{S}_0\) there exist \(c_x,r_x>0\), with \[\begin{align} \label{Eq:Open95Convex} \nabla^2 F(y)\geq c_x >0 \end{align}\tag{22}\] for all \(y\in B_{r_x}(x)\cap H\),
  

• for all \(x\in \mathcal{S}_1\) there exists \(c_x,r_x>0\) and a direction \(\zeta_x\in H\), with \[\begin{align} \label{Eq:Open95Saddle} \langle \zeta_x, \nabla^2 F(y)\, \zeta_x\rangle\leq -c_x<0 \end{align}\tag{23}\] for all \(y\in B_{r_x}(x)\cap H\).

In the following Subsection 3.1 we investigate functions with a support away from the critical points \(\mathcal{S}\), and subsequently we investigate functions supported in one of the regions \(B_{r_x}(x)\) in Subsection 3.2.

3.1 Analysis away from the critical points↩︎

We first recall three standard tools in Propositions 3, 4 and 5: the ground-state transformation, the NGS bound for Schrödinger-type quadratic forms, and a large deviation upper bound for exponential integrals. The first proposition is elementary and shows how to transform the underlying base measure of a Dirichlet form, at the cost of introducing the additional potential \(V_\beta\).

The next proposition is a localized form of the Nelson–Glimm–Segal bound, or NGS bound, see [30]. It follows from the Gaussian logarithmic Sobolev inequality \[\frac{1}{\beta} \int_B \|\nabla f\|_H^2\,d\gamma_\beta \ge \frac{1}{2}\,\operatorname{Ent}_{\gamma_\beta}(f^2),\] and the entropy variational formula, see also [31].

The exponential integral on the right-hand side of ?? can be controlled by large-deviation estimates. More precisely, we shall use the following variant of Varadhan’s lemma [32], [33].

With the auxiliary results Proposition 3, Proposition 4 and Proposition 5 at hand, we are in a position to verify Theorem 6, which is the main result of this subsection.

3.2 Analysis around the critical points↩︎

In the first result of this Subsection, we investigate functions \(\Psi\) that are supported in a neighborhood of a local minimum \(x\). Notably, the lower bound ?? holds only for elements that are orthogonal to the function \(\Phi_\beta\) constructed below.

For critical points \(x\in \mathcal{S}\) that are not local minima, we obtain an unconditional lower bound in the following Proposition 8.

3.3 Gluing of the regions↩︎

4 Exact Asymptotics↩︎

We assume in this Section that both Assumption 1 and Assumption 2 hold, as well as \(\eta_x\in B^*\) for all critical points \(\mathcal{T}_h\) and w.l.o.g. that \(F(x_-)=0\). We further recall the definition of \(A_\pm\) as the connected component of \([F\leq h]\setminus \mathcal{T}_h\) containing \(x_\pm\), and introduce for \(x\in \mathcal{T}_h\) the half-spaces \[\begin{align} H_1^\epsilon(x): & =[\langle \xi,\cdot \rangle < \langle \xi,x\rangle - \epsilon],\\ H_2^\epsilon(x): & =[\langle \xi,\cdot \rangle > \langle \xi,x\rangle + \epsilon]. \end{align}\] As is discussed below 16 , locally, the points in \(H_i^0(x)\cap [F\leq h]\) are all in the same connected component, which is denoted by \(A_{\sigma_i(x)}\).

In order to verify Theorem 2, we will construct a trial state that is piece-wise defined on the regions \(B_\delta \big(A_{\pm}\setminus B_r(x)\big)\) and \(B_r(x)\) respectively for \(x\in \mathcal{T}_h\). In order for this piece-wise definition to be consistent, it will be imperative to have a precise understanding of the intersections \(B_r(x)\cap B_\delta \big(A_{\pm}\setminus B_r(x)\big)\), which is the content of the subsequent Proposition 9.

In order to avoid issues that arise due to our Banach space being possibly infinite dimensional, we will further use the subsequent auxiliary Proposition 10. We recall the enumeration in, and below, 17 of the critical points and of the corresponding elements \(\{\eta_1,\dots ,\eta_m\}\).

We are now in a position to construct good approximate eigenstates. Let \(r,\delta\) and \(\epsilon\) be as in Proposition 9, where we can assume by 16 w.l.o.g. that \(r>0\) is small enough such that \(x_\pm\in A_\pm\setminus B_r(\mathcal{T}_h)\) and \(\delta>0\) is small enough such that the sets \[\begin{align} \mathcal{A}_\pm := B_\delta\! \left(A_\pm\setminus B_r(\mathcal{T}_h)\right) \end{align}\] are disjoint (such a \(\delta>0\) exists by 16 ) and that \(r>0\) is small enough such that also the sets \(B_r(x_j)\) are disjoint for \(j\in \{1,\dots,m\}\). We furthermore note that the open set \[\begin{align} \mathcal{O}_1:=\mathcal{A}_-\cup \mathcal{A}_+\cup B_r(\mathcal{T}_h) \end{align}\] contains the compact set \([F\leq h]\), and hence there exists an open set \(\mathcal{O}_2\) such that \[\begin{align} [F\leq h]\subseteq \mathcal{O}_2\subseteq \overline{\mathcal{O}_2}^B\subseteq \mathcal{O}_1. \end{align}\] In particular, \(\inf_{H\setminus \mathcal{O}_2}F=\min_{H\setminus \mathcal{O}_2}F>h+2\kappa\), for \(\kappa\) small enough, and we choose a projection \(\pi\) as in Proposition 10 with \(E:=h+\kappa\) and \[\begin{align} \rho:=\min\{\delta,r,\kappa\}. \end{align}\] Since the embedding \(\zeta:\mathrm{ran}\pi\longrightarrow B\) with \(\zeta(y):=y\) is continuous, we have that \(\zeta^{-1}(\mathcal{O}_1)\) and \(\zeta^{-1}(\mathcal{O}_2)\) are open and \[\begin{align} \zeta^{-1}(\mathcal{O}_2)\subseteq \overline{\zeta^{-1}(\mathcal{O}_2)}^B\subseteq \zeta^{-1}(\mathcal{O}_1), \end{align}\] and hence we find a \(C^\infty\) function \(\varphi\) on the finite dimensional subspace \(\mathrm{ran}\pi\) such that \(\varphi|_{\zeta^{-1}(\mathcal{O}_2)}=1\) and \(\mathrm{supp}\varphi\subseteq \zeta^{-1}(\mathcal{O}_1)\). In particular \[\begin{align} \mathrm{supp}(\varphi\circ \pi) & \subseteq \pi^{-1}\mathcal{O}_1. \end{align}\] Finally, let \(\mathcal{T}_h=\{x_1,\dots ,x_m\}\) and let us introduce for \(x_j\in \mathcal{T}_h\) the function \(f_j:\mathbb{R}\longrightarrow [-1,1]\) as \(f_j:=1\) in case \(\sigma_1(x_j)=\sigma_2(x_j)=+\), \(f_j:=-1\) in case \(\sigma_1(x_j)=\sigma_2(x_j)=-\), and for \(\sigma_1(x_j)=-\) and \(\sigma_2(x_j)=+\) we define \[\begin{align} f_{\beta,j}(t):= \begin{cases} z(\epsilon,\beta)^{-1}\int_0^t e^{- \beta \alpha_j s^2/2-\frac{\chi(\epsilon^{-1} s)}{\epsilon^2 - s^2}} \mathrm{d}s, \text{ in case }|t|<\epsilon\\ -1, \text{ in case }t\leq -\epsilon,\\ 1, \text{ in case }t\geq \epsilon, \end{cases} \end{align}\] where \(z(\epsilon,\beta):=\int_0^\epsilon e^{- \beta \alpha_j s^2/2-\frac{\chi(\epsilon^{-1} s)}{\epsilon^2 - s^2}} \mathrm{d}s\), \(\chi\) is a smooth function with \(\chi(t)=0\) in a neighborhood of \(t=0\) and \(\chi(t)=1\) for \(|t|>1/2\), and \(-\alpha_j\) is the negative eigenvalue corresponding to \(\eta_j\). Analogously, we define \(f_j\) in case \(\sigma_1(x_j)=+\) and \(\sigma_2(x_j)=-\). We note that for a suitable constant \(C\) \[\begin{align} \label{eq:small95z95asy} \left|z(\epsilon,\beta)-\sqrt{\frac{\pi}{2\beta \alpha_j}}\right|\leq C/\beta. \end{align}\tag{29}\]

In the subsequent Theorem 12 we show that the trial state \[\begin{align} \label{Eq:def95main95trial95state} \Psi_\beta(x):=\varphi(\pi x)\times \begin{cases} 1, \text{ for }\pi x\in \mathcal{A}_+\\ -1, \text{ for }\pi x\in \mathcal{A}_-\\ f_{\beta,j}(\langle \eta_j,x-x_j \rangle), \text{ for }\pi x\in B_r(x_j)\text{ with }j\in \{1,\dots,m\}\\ 0, \text{ for }\pi x\notin \mathcal{O}_1 \end{cases} \end{align}\tag{30}\] is a well-defined element in the domain of the operator \(\mathcal{L}_\beta\) introduced in 11 and we provide precise control on its energy \[\begin{align} \mathcal{E}_\beta[\Psi_\beta]= \frac{1}{\beta}\|A^{1/2}\nabla \Psi_\beta\|_{L^2(\mu_\beta)}^2=\langle \Psi_\beta,\mathcal{L}_\beta\Psi_\beta\rangle, \end{align}\] as well as on its second moment \(\|\mathcal{L}_\beta \Psi_\beta\|_{L^2(\mu_\beta)}^2\). To this end we shall use the following sharp Laplace asymptotics.

With Theorem 12 at hand we can verify Theorem 2.

5 Examples↩︎

5.1 Allen-Cahn↩︎

We consider on the Banach space \(B:=C([0,\ell])\) the Gaussian measure \(\gamma\) with corresponding Cameron-Martin space \(H:=H^1((0,\ell))\subseteq B\), i.e. according to the natural embedding \(B^*\subseteq H^*=H\) the measure \(\gamma\) is characterized by \[\begin{align} \int_B e^{ib(x)} d\gamma = e^{-\|b\|_{H^1((0,\ell))}^2/2}. \end{align}\] On the Hilbert space \(H^1((0,\ell))\) we then study the functional \[\begin{align} \label{Eq:F95Introduction} F(x):=\frac{1}{2}\int_0^\ell x'(s)^2\mathrm{d}s+\int_0^\ell u(x(s))\mathrm{d}s, \end{align}\tag{40}\] with the function \(u:\mathbb{R}\longrightarrow \mathbb{R}\) defined as \[\begin{align} u(z):=\frac{1}{4}z^4-\frac{1}{2}z^2+\frac{1}{4}. \end{align}\] Notably, we can write \(F(x)=\frac{1}{2}\|x\|_{H^1((0,\ell))}^2+U(x)\) with \[\begin{align} U(x):=\int_0^\ell u_*(x(s))\mathrm{d}s:=\int_0^\ell \left(u(x(s))-\frac{1}{2}x(s)^2\right)\mathrm{d}s. \end{align}\] Let us define the operator \(K\) acting on \(L^2((0,\ell))\) as \(K:=-\left(\frac{\mathrm{d}}{\mathrm{d}t}\right)^2+1\) with free boundary conditions, and let \(A\) be the self-adjoint operator acting on \(H^1((0,\ell))\) given by the restriction of \(K\) to \(H^3((0,\ell))\cap \mathrm{dom}(K)\). We then consider the quadratic form corresponding to the stochastic Allen-Cahn equation \[\begin{align} \mathcal{E}^{(\mathrm{AC})}_{\beta}[f] : = \frac{1}{\beta} \int_B \|\;\sqrt A \nabla f\|_{H^1((0,\ell))}^2 d\mu^{(\mathrm{AC})}_\beta , \end{align}\] where the measure \(\mu^{(\mathrm{AC})}_\beta\) is defined according to 10 . The critical points \(x\in \mathcal{S}\) are given by the solutions \[\begin{align} 0=\nabla F(x)=x+K^{-1}(x^3-2x), \end{align}\] i.e. \(x\) is in the domain of \(K\) and we have \(0=K\nabla F(x)=-x''+x^3-x\), or equivalently \(x\) satisfies the stationary Allen-Cahn equation \[\begin{align} \label{Eq:stationary95AC} x''=x^3-x. \end{align}\tag{41}\] The solutions to 41 have been analyzed in [21] depending on the length \(\ell\). For \(n\pi<\ell\leq (n+1)\pi\), with \(n\in \mathbb{N}\), there are exactly three constant solutions \(x_-:=-1\), \(x_0:=0\) and \(x_+:=1\), as well as \(2n\) non-constant solutions \(\{x_1,\dots ,x_{2n}\}\). Introducing according to 14 the operator \(Q_x:=-\left(\frac{\mathrm{d}}{\mathrm{d}t}\right)^2+3x^2-1\) acting on \(L^2((0,\ell))\), we note \[\begin{align} Q_{x_\pm}=-\left(\frac{\mathrm{d}}{\mathrm{d}t}\right)^2+2 \end{align}\] is strictly positive, and \(Q_{x_0}=-\left(\frac{\mathrm{d}}{\mathrm{d}t}\right)^2-1\) has exactly \(n+1\) negative eigenvalues for \(n\pi<\ell\leq (n+1)\pi\) and a trivial kernel as long as \(\ell \notin \pi \mathbb{Z}\). Observe that the lowest eigenvalue equals \(-1\). Furthermore, according to [21], the non-constants solutions \(\{x_1,\dots ,x_{2n}\}\) can be arranged such that \(x_{2k-1}=-x_{2k}\), and the operators \(Q_{x_{2k-1}}=Q_{x_{2k}}\) have exactly \(k\)-many negative eigenvalues and a trivial kernel. Since the operator \(Q_x\) is a Schrödinger Operator defined on a compact domain, it is clear that \(Q_x\) is invertible as long as the kernel is trivial. In this case \(\nabla^2F(x)\) is invertible too with inverse given by \[\begin{align} \label{Eq:Inverse95nabla95square95F} Q^{-1}A=(-\Delta +3x^2-1)^{-1}(-\Delta+1)=1-Q^{-1}(3x^2-2). \end{align}\tag{42}\] Indeed, \(Q^{-1}(3x^2-2)\) is a bounded operator, as the multiplication by the smooth function \(3x^2-2\) is a bounded operator on \(H^1((0,\ell))\) and so is \(Q^{-1}\). Viewing the dual space \(B^*\) as a subset of \(H^1((0,\ell))\), we can further embed the space \(L^2((0,\ell))\) in \(B^*\) via the map \(\iota(y):=K^{-1}(y)\), i.e. \(\mathrm{dom}(K)\subseteq B^*\). Since \(\mathrm{dom}(A)\subseteq \mathrm{dom}(K)\), \[\begin{align} A^{-1}(B^*)\cap B^*=A^{-1}(B^*)\supseteq A^{-1}(\mathrm{dom}(A))=\mathrm{dom}(A^{2}) \end{align}\] is dense in \(H^1((0,\ell))\) and \(\eta_x:=A^{-1}\nu_x\) defined in 15 is an element of \(\mathrm{dom}(A)\subseteq B^*\).

Denoting with \(\lambda^{(\mathrm{AC})}_{\beta,j}\) the minmax values of \(\mathcal{E}^{(\mathrm{AC})}_{\beta}\), we obtain as a consequence of Theorem 1 and Theorem 2 the following corollary.

5.2 Cahn-Hilliard↩︎

Here the Banach space \(\widetilde{B}\subseteq C([0, \ell])\) consists of functions with zero average and \(\widetilde{\gamma}\) is the centered Gaussian measure with corresponding Cameron-Martin space \[\begin{align} \dot{H}^1((0,\ell)):=\left\{y\in H^1((0,\ell)):\int y(s)\mathrm{d}s=0\right\}, \end{align}\] equipped with the norm \(\|y\|_{\dot{H}^1((0,\ell))}:=\int (y'(s))^2\mathrm{d}s\). Furthermore, let us introduce \(\widetilde{K}x:=-x''\) defined on the Hilbert space \(\{y\in L^2((0,\ell)):\int y(s)\mathrm{d}s=0\}\) with free boundary conditions, and denote with \(\widetilde{A}\) the restriction of the operator \((\widetilde{K})^2\) to the domain \(H^5((0,\ell))\cap \mathrm{dom}(\widetilde{K}^2)\). We then consider the Allen-Cahn functional in 40 restricted to \(\widetilde{B}\) \[\begin{align} \widetilde{F}(x)=\frac{1}{2}\|x\|^2_{\dot{H}^1((0,\ell))}+\int_0^\ell \left(\frac{1}{4}x(s)^4-\frac{1}{2}x(s)^2+\frac{1}{4}\right)\mathrm{d}s, \end{align}\] and denote the corresponding measure and quadratic form by \(\mu_\beta^{(\mathrm{CH})}\) and \[\begin{align} \mathcal{E}^{(\mathrm{CH})}_{\beta}[f] : = \frac{1}{\beta} \int_{\widetilde{B}} \|\;\sqrt{\widetilde{A}} \nabla f\|_{\dot{H}^1((0,\ell))}^2 d\mu^{(\mathrm{CH})}_\beta. \end{align}\] Furthermore, let \(\lambda^{(\mathrm{CH})}_{\beta,j}\) be the minmax values of \(\mathcal{E}^{(\mathrm{CH})}_{\beta}\). Before we state the asymptotic behavior of \(\lambda^{(\mathrm{CH})}_{\beta,j}\), we analyze the critical points of \(\widetilde{F}\) in the subsequent Proposition 13.

With Proposition 13 at hand, we are in a position to verify Corollary 2.

Acknowledgements↩︎

MB gratefully acknowledges funding from the UZH Postdoc Grant FK-25-103. GDG is grateful to Martin Grothaus for inspiring discussions and to the Bernoulli Center in Lausanne for its warm hospitality during several visits in the course of this work.

References↩︎