# Geometric Characterization of the Eyring-Kramers Formula

Benny Avelin  Benny Avelin, Department of Mathematics, Uppsala University,
S-751 06 Uppsala, Sweden<EMAIL_ADDRESS>,  Vesa Julin  Vesa Julin,
Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box
35, 40014 Jyväskylä, Finland<EMAIL_ADDRESS>and  Lauri Viitasaari  Lauri
Viitasaari, Department of Mathematics, Uppsala University, S-751 06 Uppsala,
Sweden<EMAIL_ADDRESS>
(Date: August 27, 2024)

###### Abstract.

In this paper we consider the mean transition time of an over-damped Brownian
particle between local minima of a smooth potential. When the minima and
saddles are non-degenerate this is in the low noise regime exactly
characterized by the so called Eyring-Kramers law and gives the mean
transition time as a quantity depending on the curvature of the minima and the
saddle. In this paper we find an extension of the Eyring-Kramers law giving an
upper bound on the mean transition time when both the minima/saddles are
degenerate (flat) while at the same time covering multiple saddles at the same
height. Our main contribution is a new sharp characterization of the capacity
of two local minimas as a ratio of two geometric quantities, i.e., the
smallest separating surface and the geodesic distance.

###### Contents

  1. 1 Introduction
    1. 1.1 Assumptions and statement of the main results
  2. 2 Preliminaries
    1. 2.1 Potential theory
    2. 2.2 Classical pointwise estimates
  3. 3 Technical lemmas
    1. 3.1 Rough estimates for potentials
    2. 3.2 Laplace asymptotics for log-concave functions
  4. 4 Proofs of Theorem 1 and Theorem 2

##  1\. Introduction

In this paper we investigate the so called metastable exit times for the
stochastic differential equation

$\displaystyle dX_{t}=-\nabla F(X_{t})dt+\frac{\sqrt{\varepsilon}}{2}dB_{t},$
(1.1)

where $F$ is a smooth potential with many local minimas and $\varepsilon$ is a
small number.

The main question of metastability is to determine how long time does the
process 1.1 take from going from one local minima to another one. We call
these the metastable exit times. This question has a rich history and in the
double well case with non-degenerate minimas and a saddle point this is
characterized by a formula called Eyring-Kramers law [11, 17] which can be
stated as follows: Assume that $x$ and $y$ are quadratic local minimas of $F$,
separated by a unique saddle $z$ which is such that the Hessian has a single
negative eigenvalue $\lambda_{1}(z)$. Then the expected transition time from
$x$ to $y$ satisfies

$\displaystyle\mathbb{E}^{x}[\tau]\simeq\frac{2\pi}{|\lambda_{1}(z)|}\sqrt{\frac{\det(\nabla^{2}F(z))}{\det(\nabla^{2}F(x))}}e^{(F(z)-F(x))/\varepsilon},$
(1.2)

where $\simeq$ denotes that the comparison constant tends to $1$ as
$\varepsilon\to 0$.

The validity of the above formula has been studied, from a qualitative
perspective, quite extensively, starting from the work of Freidlin and
Wentzell. For more information, see the book [12]. Roughly 15 years ago,
Bovier et. al. produced a series of papers [6, 7, 8, 9] (see also [5]) which
provided the first proof of 1.2 in the general setting of Morse functions.
Specifically, they showed that the comparison function is like
$1+\mathcal{O}(\varepsilon^{1/2}|\log\varepsilon|^{3/2})$. In these papers,
they utilized the connection to classical potential theory in order to reduce
the problem of estimating metastable exit times to the problem of estimating
certain capacities sharply. This approach was later used in [4] to generalize
1.2 to general polynomial type of degeneracies.

In this paper we are interested in estimating the metastable exit times in the
case of general type of degenerate critical points. This requires new
techniques and effective notation from geometric function theory which we will
describe below. Our motivation comes from the field of non-convex optimization
where we cannot expect the minimas/saddles to be quadratic or even to have
polynomial growth in any direction. In particular, such situations are well
known in the context of neural networks, where the minimas and saddles may be
completely flat in some directions [15]. Furthermore, it seems that they are
preferrable, see [19] for a discussion, see also [3] for an explicit example.

The main goal is to estimate the dependency of the metastable exit times with
respect to the geometry of the potential $F$. In the proof of 1.2 in [8] this
is reduced to estimating the ratio of the $L^{1}$ norm of the hitting
probability and the capacity. Thus in order to estimate the metastable exit
times, one needs to produce

  1. (1)

Estimates of the integral of the hitting probability, i.e. the integral of
capacitary potentials with respect to the Gibbs measure.

  2. (2)

Estimates of the capacity itself, i.e. estimates of the energy of the
capacitary potentials.

The interesting point is that the influence of $F$ on 1 and 2 is in a sense
dual. Specifically, the shape of minimas of $F$ influence 1 while the shape of
saddles between minimas influence 2. As is well known, the main difficulty is
to estimate 2, which is an interesting topic of its own.

Our main contribution is a sharp capacity estimate for a very general class of
degenerate saddle points. In order to achieve this, we phrase the problem in
the language of geometric function theory, where the capacity estimates are
central topic [13, 20]. We introduce two geometric quantities which allow us
to estimate the capacity in a sharp and natural way. As a byproduct, we see
that in the case of several saddle points at the same height, the topology
dictates how the local capacities add up. Specifically, we consider two cases,
which we call the parallel and the serial case, and it turns out that the
formulas for the total capacity have natural counterparts in electrical
networks of capacitors, see Theorem 1. Even in the context of non-degenerate
saddles, our formulas provide a generalization of the result of [8] where the
authors consider only the parallel case. As we mentioned, we allow the saddle
points to be degenerate but we have to assume that saddles are non-branching,
see 1.6.

###  1.1. Assumptions and statement of the main results

In order to state our main results we first need to introduce our assumptions
on the potential $F$. We also need to introduce notation from geometric
function theory which might seem rather heavy at first, but it turns out to be
robust enough for us to treat the potentials with possible degenerate critical
points.

Let us first introduce some general terminology. We say that a critical point
$z$ of a function $f\in C^{1}(\mathbb{R}^{n})$ is a local minimum (maximum) of
$f$ if $f(x)\geq f(z)$ ($f(x)\leq f(z)$) in a neighborhood of $z$. If $f$ is
not locally constant at a critical point $z$, then $z$ is a saddle point if it
is not a local minimum / maximum. For technical reasons we also allow saddle
points to include points $z$ where $f$ is locally constant. We say that a
local minimum at $z$ is _proper_ if there exists a $\hat{\delta}>0$ such that
for every $0<\delta<\hat{\delta}$ there exists a $\rho$ such that

$\displaystyle f(x)\geq f(z)+\delta\quad\text{for all }\,x\in\partial
B_{\rho}(z),$

where $B_{\rho}(z)$ denotes open ball with radius $\rho$ centered at $z$. When
the center is at the origin we use the short notation $B_{\rho}$.

Let us then proceed to our assumptions on the potential $F$. Throughout the
paper we assume that $F\in C^{2}(\mathbb{R}^{n})$ and satisfies the following
quadratic growth condition

$F(x)\geq\frac{|x|^{2}}{C_{0}}-C_{0}$ (1.3)

for a constant $C_{0}\geq 1$. We assume that every local minimum point $z$ of
$F$ is proper, as described above, and that there is a convex function
$G_{z}:\mathbb{R}^{n}\to\mathbb{R}$ which has a proper minimum at $0$ with
$G(0)=0$ such that

$\big{|}F(x+z)-F(z)-G_{z}(x)\big{|}\leq\omega\big{(}G_{z}(x)\big{)},$ (1.4)

where $\omega:[0,\infty)\to[0,\infty)$ is a continuous and increasing function
with

$\lim_{s\to 0}\frac{\omega(s)}{s}=0.$ (1.5)

We denote by $\delta_{0}$ the largest number for which
$\omega(\delta)\leq\frac{\delta}{8}$ for all $\delta\leq 4\delta_{0}$. We
define a neighborhood of the local minimum point $z$ and $\delta<\delta_{0}$
as

$O_{z,\delta}:=\\{x\in\mathbb{R}^{n}:G_{z}(x)<\delta\\}+\\{z\\}.$

For the saddles, we assume that for every saddle point $z$ of $F$ there are
convex functions $g_{z}:\mathbb{R}\to\mathbb{R}$ and
$G_{z}:\mathbb{R}^{n-1}\to\mathbb{R}$ which have proper minimum at $0$ with
$g_{z}(0)=G_{z}(0)=0$, and that there exists an isometry
$T_{z}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ such that, denoting
$x=(x_{1},x^{\prime})\in\mathbb{R}\times\mathbb{R}^{n-1}$, it holds

$\big{|}(F\circ
T_{z})(x)-F(z)+g_{z}(x_{1})-G_{z}(x^{\prime}))\big{|}\leq\omega(g_{z}(x_{1}))+\omega(G_{z}(x^{\prime})),$
(1.6)

where $\omega:[0,\infty)\to[0,\infty)$ is as in 1.5. The assumption 1.6 allows
the saddle point to be degenerate, but we do not allow them to have many
branches, i.e., the sets $\\{F<F(z)\\}\cap B_{\rho}(z)$ cannot have more than
two components. Note that the convex functions $g_{z},G_{z}$ and the isometry
$T_{z}$ depend on $z$, while the function $\omega$ is the same for all saddle
points. We define a neighborhood of the saddle point $z$ and
$\delta<\delta_{0}$ as

$O_{z,\delta}:=T_{z}^{-1}\left(\\{x_{1}\in\mathbb{R}:g_{z}(x_{1})<\delta\\}\times\\{x^{\prime}\in\mathbb{R}^{n-1}:G_{z}(x^{\prime})<\delta\\}\right),$
(1.7)

where $T_{z}$ is the isometry in 1.4. Note that, since the saddle may be flat,
we should talk about sets rather than points. However, we adopt the convention
that we always choose a representative point from each saddle (set) and thus
we may label the saddles by points $z_{1},z_{2},\dots$. Moreover, we assume
that there is a $\delta_{1}\leq\delta_{0}$ such that for $\delta<\delta_{1}$
we have that if $z_{1}$ and $z_{2}$ are two different saddle points, then
their neighborhoods $O_{z_{1},3\delta}$ and $O_{z_{2},3\delta}$ defined in 1.7
are disjoint. We assume the same for local minimas (or more precisely, the
representative points of sets of local minimas).

Let us then introduce the notation related to geometric function theory, see
[13, 20]. Let us fix two disjoint sets $A$ and $B$ in a domain $\Omega$ (open
and connected set). We say that a smooth path $\gamma:[0,1]\to\mathbb{R}^{n}$
_connects $A$ and $B$ in the domain $\Omega$_ if

$\displaystyle\gamma(0)\in A,\quad\gamma(1)\in
B\quad\text{and}\quad\gamma([0,1])\subset\Omega.$

In this case we denote $\gamma\in\mathcal{C}(A,B;\Omega)$. We follow the
standard notation from geometric function theory and define a dual object to
this by saying that a smooth hypersurface $S\subset\mathbb{R}^{n}$ (possibly
with boundary) _separates $A$ from $B$ in $\Omega$_ if every path
$\gamma\in\mathcal{C}(A,B;\Omega)$ intersects $S$. In this case we denote
$S\in\mathcal{S}(A,B;\Omega)$. We define the geodesic distance between $A$ and
$B$ in $\Omega$ as

$d_{\varepsilon}(A,B;\Omega):=\inf\left(\int_{\gamma}|\gamma^{\prime}|e^{\frac{F(\gamma)}{\varepsilon}}\,dt:\gamma\in\mathcal{C}(A,B;\Omega)\right)$
(1.8)

and its dual by

$V_{\varepsilon}(A,B;\Omega):=\inf\left(\int_{S}e^{-\frac{F(x)}{\varepsilon}}\,d\mathcal{H}^{n-1}(x):S\in\mathcal{S}(A,B;\Omega)\right).$
(1.9)

Here $\mathcal{H}^{k}$ denotes the $k$-dimensional Hausdorff measure. Finally,
we define the communication height between the sets $A$ and $B$ as

$F(A;B):=\inf_{\gamma\in\mathcal{C}(A,B;\mathbb{R}^{n})}\sup_{t\in[0,1]}\,F(\gamma(t)).$

Let us then assume that $x_{a}$ and $x_{b}$ are local minimum points and
denote the height of the saddle between $x_{a}$ and $x_{b}$ as

$\displaystyle F(x_{a};x_{b}):=F(\\{x_{a}\\};\\{x_{b}\\}).$

Notice that $F(x_{a}),F(x_{b})\leq F(x_{a};x_{b})$. For $s\in\mathbb{R}$,
denote

$\displaystyle U_{s}:=\\{x\in\mathbb{R}^{n}:F(x)<F(x_{a};x_{b})+s\\}.$

We note that the points $x_{a}$ and $x_{b}$ lie in different components of the
set $U_{-\delta/3}$ while they are in the same component of the set
$U_{\delta/3}$. We will always denote the components of $U_{-\delta/3}$
containing the points $x_{a}$ and $x_{b}$ by $U_{a}$ and $U_{b}$,
respectively. It is important to notice that if $z$ is a saddle point and
$F(z)<F(x_{a};x_{b})+\delta/3$, then the neighborhood $O_{z,\delta}$ defined
in 1.7 intersects the set $U_{-\delta/3}$. We will sometimes call the
components of the set $U_{-\delta/3}$ _islands_ and the neighborhoods
$O_{z,\delta}$ _bridges_ since we may connect islands with bridges, see Fig.
1. (The terminology is obviously taken from Seven Bridges of Königsberg). We
say that the set of saddle points $Z_{x_{a},x_{b}}=\\{z_{1},\dots,z_{N}\\}$
_charge capacity_ if it is the smallest set with the property that every
$\gamma\in\mathcal{C}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{a});U_{\delta/3})$
intersects the bridge $O_{z_{i},\delta}$, defined in 1.7, for some $z_{i}\in
Z_{x_{a},x_{b}}$. In particular, it holds that $Z_{x_{a},x_{b}}\subset
U_{\delta/3}$.

$U_{b}$$U_{a}$$z$$x_{b}$$B_{\varepsilon}(x_{b})$$x_{a}$$B_{\varepsilon}(x_{a})$$O_{z,\delta}$
Figure 1. The neighborhood $O_{z,\delta}$ of the saddle point $z$ connects the
sets $U_{a}$ and $U_{b}$.

We will focus on two different topological situations, where the saddle points
in $Z_{x_{a},x_{b}}$ are either _parallel_ or in _series_. We say that the
points in $Z_{x_{a},x_{b}}$ are parallel if for every $z_{i}\in
Z_{x_{a},x_{b}}$ there is a path

$\displaystyle\gamma\in\mathcal{C}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b});U_{\delta/3})$

passing only through $z_{i}$. We say that the points in $Z_{x_{a},x_{b}}$ are
in series if every path
$\gamma\in\mathcal{C}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b});U_{\delta/3})$
passes through the bridge $O_{z_{i},\delta}$, defined in 1.7, for all
$z_{i}\in Z_{x_{a},x_{b}}$. In other words, if the points in
$Z_{x_{a},x_{b}}=\\{z_{1},\dots,z_{N}\\}$ are parallel, then the islands
occupied by the points $x_{a}$ and $x_{b}$ respectively are connected with $N$
bridges and we need to pass only one to get from $x_{a}$ to $x_{b}$. If they
are in series, then we have to pass all $N$ bridges in order to get from
$x_{a}$ to $x_{b}$, see Fig. 2.

$x_{a}$$x_{b}$$z_{1}$$z_{2}$$z_{3}$ $x_{a}$$x_{1}$$x_{b}$$z_{1}$$z_{2}$

Figure 2. Left picture is the parallel case and the right is the series case.

Recall that $U_{a}$ and $U_{b}$ denote the islands, i.e., the components of
$U_{-\delta/3}$, which contain the points $x_{a}$ and $x_{b}$. If the points
in $Z_{x_{a},x_{b}}=\\{z_{1},\dots,z_{N}\\}$ are parallel, then we may connect
$U_{a}$ and $U_{b}$ with one bridge, i.e., for every $z_{i}$ the set

$U_{z_{i},\delta}:=O_{z_{i},\delta}\cup U_{a}\cup U_{b}$ (1.10)

is connected, again see Fig. 1. Then all paths
$\gamma\in\mathcal{C}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b});U_{z_{i},\delta})$
pass through the bridge $O_{z_{i},\delta}$. If the points in $Z_{x_{a},x_{b}}$
are in series, then it is useful to order them
$Z_{x_{a},x_{b}}=\\{z_{1},\dots,z_{N}\\}$ as follows. Let us consider a path
$\gamma\in\mathcal{C}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b});U_{\delta/3})$
which passes through each point in $Z_{x_{a},x_{b}}$ precisely once. This
means that there are

$0<t_{1}<\dots<t_{N}<1\quad\text{ such that }\quad\gamma(t_{i})=z_{i},$ (1.11)

which gives a natural ordering for points in $Z_{x_{a},x_{b}}$. By the
assumption 1.6, we also deduce that there are $s_{1},\dots,s_{N-1}$ such that
$t_{i}<s_{i}<t_{i+1}$ and $F(\gamma(s_{i}))<F(x_{a};x_{b})-\delta/3$. We
denote

$\gamma(s_{i})=x_{i},\,\,x_{0}=x_{a}\quad\text{ and }\quad x_{N}=x_{b}.$
(1.12)

The idea is that then every point $x_{i}$ lie in a different island, i.e.,
component of $U_{-\delta/3}$, see Fig. 2.

We are now ready to state our main results. The first result is a quantitative
lower bound on the capacity between the sets $B_{\varepsilon}(x_{a})$ and
$B_{\varepsilon}(x_{b})$, where $x_{a}$ and $x_{b}$ are two local minimum
points of $F$. For a given domain $\Omega\subset\mathbb{R}^{n}$ we define the
capacity of two disjoint sets $A,B\subset\Omega$ with respect to the domain
$\Omega$ as

$\operatorname{cap}(A,B;\Omega):=\inf\left(\varepsilon\int_{\Omega}|\nabla
u|^{2}e^{-\frac{F}{\varepsilon}}\,dx\,:\,\,u=1\,\,\text{in
}\,A,\,\,u=0\,\,\text{in }\,B\right).$

Above, the infimum is taken over functions $u\in W_{loc}^{1,2}(\Omega)$. In
the case $\Omega=\mathbb{R}^{n}$ we denote

$\displaystyle\operatorname{cap}(A,B)=\operatorname{cap}(A,B;\mathbb{R}^{n})$

for short.

Finally, for functions $f$ and $g$ which depend continuously on
$\varepsilon>0$, we adopt the notation

$\displaystyle f(\varepsilon)\simeq g(\varepsilon)$

when there exists a constant $C$ depending only on the data of the problem
such that

$\displaystyle(1-\hat{\eta}(C\varepsilon))f(\varepsilon)\leq
g(\varepsilon)\leq(1+\hat{\eta}(C\varepsilon))f(\varepsilon),$

where $\hat{\eta}$ is an increasing and continuous function
$\hat{\eta}:[0,\infty)\to[0,\infty)$ with $\lim_{s\to 0}\hat{\eta}(s)=0$. In
all our estimates, the function $\hat{\eta}$ is specified and depends only on
the function $\omega$ from 1.4 and 1.6. In order to define it, we first let
$0<\varepsilon\leq\delta_{0}/2$ be fixed and let
$\varepsilon_{1}(\varepsilon)$ be the unique solution to

$\sqrt{\omega(\varepsilon_{1})\varepsilon_{1}}=\varepsilon.$ (1.13)

From the assumption that $\omega(s)<s/2$ for $s<\delta_{0}$ we see that
$\varepsilon<\varepsilon_{1}$. Furthermore, since $\omega$ is increasing we
get that $\varepsilon_{1}\to 0$ as $\varepsilon\to 0$. Now, from the
definition of $\varepsilon_{1}$ in 1.13 we see, using $\lim_{s\to
0}\frac{\omega(s)}{s}=0$ and $\varepsilon_{1}\to 0$ as $\varepsilon\to 0$,
that

$\displaystyle\frac{\varepsilon_{1}}{\varepsilon}=\frac{\sqrt{\varepsilon_{1}}}{\sqrt{\omega(\varepsilon_{1})}}\to\infty\qquad\text{as
}\,\varepsilon\to 0.$

On the other hand, again using the same facts, we see that

$\displaystyle\frac{\omega(\varepsilon_{1})}{\varepsilon}=\frac{\sqrt{\omega(\varepsilon_{1})}}{\sqrt{\varepsilon_{1}}}\to
0\qquad\text{as }\,\varepsilon\to 0.$

Thus

$\lim_{\varepsilon\to
0}\frac{\varepsilon_{1}}{\varepsilon}=\infty\quad\text{and}\quad\lim_{\varepsilon\to
0}\frac{\omega(\varepsilon_{1})}{\varepsilon}=0.$ (1.14)

In the following we will denote

$\displaystyle\eta(x)$ $\displaystyle=e^{-1/x}x^{n}$ (1.15)
$\displaystyle\hat{\eta}(C\varepsilon)$
$\displaystyle=\eta\left(C\frac{\varepsilon_{1}(\varepsilon)}{\varepsilon}\right).$
(1.16)

Finally, in our main theorems and our lemmas/propositions beyond Section 3
there is a ball $B_{R}$ which contains all the level sets of interest. The
existence of such a ball is given by the quadratic growth condition 1.3. The
constants in the estimates in our main theorems and in Section 3 are unless
otherwise stated, depending on $n,\|\nabla F\|_{B_{R}},\delta,R,C_{0}$,
specifically, this applies to the constants in $\hat{\eta}$, and as such,
gives precise meaning to $a\simeq b$.

######  Theorem 1.

Assume that $F$ satisfies the structural assumptions above. Let $x_{a}$ and
$x_{b}$ be two local minimum points of $F$ and let
$Z_{x_{a},x_{b}}=\\{z_{1},\dots,z_{N}\\}$ be the set of saddle points which
charge capacity as defined above, and let $0<\delta\leq\delta_{1}$ be fixed.
There exists an $0<\varepsilon_{0}\leq\delta$ such that if
$0<\varepsilon\leq\varepsilon_{0}$ the following holds:

If the points in $Z_{x_{a},x_{b}}=\\{z_{1},\dots,z_{N}\\}$ are parallel, then,
using the notation $U_{z_{i},\delta}$ from 1.10, it holds

$\operatorname{cap}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b}))\simeq\sum_{i=1}^{N}\operatorname{cap}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b});U_{z_{i},\delta}).$
(1.17)

Moreover for all $i=1,\dots,N$ we have the estimate

$\displaystyle\operatorname{cap}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b});U_{z_{i},\delta})\simeq\varepsilon\frac{V_{\varepsilon}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b});U_{z_{i},\delta})}{d_{\varepsilon}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b});U_{z_{i},\delta})}e^{\frac{F(z_{i})}{\varepsilon}},$

where
$d_{\varepsilon}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b});U_{z_{i},\delta})$
and
$V_{\varepsilon}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b});U_{z_{i},\delta})$
are defined in 1.8 and 1.9.

If the points in $Z_{x_{a},x_{b}}$ are in series, then, using the ordering
$z_{1},\dots,z_{N}$ from 1.11 for the points in $Z_{x_{a},x_{b}}$ and the
points $x_{0},x_{1},\dots,x_{N}$ defined in 1.12, it holds

$\frac{1}{\operatorname{cap}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b}))}\simeq\sum_{i=1}^{N}\frac{1}{\operatorname{cap}(B_{\varepsilon}(x_{i-1}),B_{\varepsilon}(x_{i}))},$
(1.18)

where we have the estimate

$\displaystyle\operatorname{cap}(B_{\varepsilon}(x_{i-1}),B_{\varepsilon}(x_{i}))\simeq\varepsilon\frac{V_{\varepsilon}(B_{\varepsilon}(x_{i-1}),B_{\varepsilon}(x_{i}))}{d_{\varepsilon}(B_{\varepsilon}(x_{i-1}),B_{\varepsilon}(x_{i}))}e^{\frac{F(z_{i})}{\varepsilon}}$

for all $i=1,\dots,N$.

Let us make a few remarks on the statement of the above theorem. First, in the
case of a single saddle $Z_{x_{a},x_{b}}=\\{z\\}$ the above capacity estimate
reduces to

$\displaystyle\operatorname{cap}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b}))\simeq\varepsilon\frac{V_{\varepsilon}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b}))}{d_{\varepsilon}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b}))}e^{\frac{F(z)}{\varepsilon}},$

where $d_{\varepsilon}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b}))$ is the
geodesic distance between $B_{\varepsilon}(x_{a})$ and
$B_{\varepsilon}(x_{a}),$ and
$V_{\varepsilon}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b}))$ is the area
of the ’smallest cross section’. This is in accordance with the classical
result on parallel plate capacitors, where the capacity depends linearly on
the area and is inversely proportional to their distance.

The statement 1.17, when the saddle points are parallel, means that each
saddle point $z_{1},\dots,z_{N}$ charges capacity and the total capacity is
their sum. Again the situation is the same as in the case of parallel plate
capacitors with capacity $C_{1},\dots,C_{N}$, where the total capacity is the
sum

$\displaystyle C=C_{1}+\dots+C_{N}.$

On the other hand, if the plate capacitors are in series their total capacity
satisfies

$\displaystyle\frac{1}{C}=\frac{1}{C_{1}}+\dots+\frac{1}{C_{N}}$

which is precisely the statement in 1.18.

Using the assumption 1.6 we calculate in Proposition 4.1 and in Proposition
4.2 more explicit, but less geometric, formulas for the single saddle case in
a domain $\Omega$. Namely, we have

$\displaystyle
d_{\varepsilon}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b});\Omega)\simeq
e^{\frac{F(z)}{\varepsilon}}\int_{\mathbb{R}}e^{-\frac{g_{z}(x_{1})}{\varepsilon}}\,dx_{1}$

and

$\displaystyle
V_{\varepsilon}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b});\Omega)\simeq
e^{-\frac{F(z)}{\varepsilon}}\int_{\mathbb{R}^{n-1}}e^{-\frac{G_{z}(x^{\prime})}{\varepsilon}}\,dx^{\prime},$

and thus we recover the result in [4] . In particular, if the saddle point is
non-degenerate, i.e., $g_{z}$ and $G_{z}$ are second order polynomials and the
negative eigenvalue of $\nabla^{2}F(z)$ is $-\lambda_{1}$, we may estimate

$\displaystyle
d_{\varepsilon}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b}))\simeq\sqrt{\frac{2\pi\varepsilon}{\lambda_{1}}}\,e^{\frac{F(z)}{\varepsilon}}$

and

$\displaystyle
V_{\varepsilon}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b}))\simeq(2\pi\varepsilon)^{\frac{n-1}{2}}\frac{\sqrt{\lambda_{1}}\,e^{-\frac{F(z)}{\varepsilon}}}{\sqrt{\det(\nabla^{2}F(z))}}.$

In particular, we recover the classical formula 1.2.

Our second main theorem is an estimate on the so called metastable exit times.
However, in order to state it we need some further definitions. Assume that
the local minimas of $F$ are labelled $x_{i}$ and ordered such that
$F(x_{i})\leq F(x_{j})$ if $i\leq j$. We will group the minimas at the same
level using the sets $G_{k}$, $k=1,\ldots,K$, i.e. $x_{i},x_{j}\in G_{k}$ if
$F(x_{i})=F(x_{j})$, and $x\in G_{i}$ and $y\in G_{j}$, then $F(x)<F(y)$ for
$i<j$. We also write $F(G_{i}):=F(x)$ with $x\in G_{i}$. Furthermore, we will
denote $S_{k}=\bigcup_{i=1}^{k}G_{i}$ for $k=1,\ldots,K$. We will also
consider $G_{k}^{\varepsilon}=\bigcup_{x\in G_{k}}B_{\varepsilon}(x)$ and
$S_{k}^{\varepsilon}=\bigcup_{i=1}^{k}G_{i}^{\varepsilon}$.

In addition to the previous structural assumptions we assume further that for
$\delta_{2}\leq\delta_{1}$ small enough, it holds

$F(G_{k+1})-F(G_{k})\geq\delta_{2}$ (1.19)

for all $k=1,\ldots,K$. In our second theorem we give an upper bound on the
exit time for the process defined by 1.1 to go from local minimum point in
$G_{k+1}^{\varepsilon}$ to a lower one in $S_{k}$.

######  Theorem 2.

Assume that $F$ satisfies the structural assumptions above, and let $\Omega$
be a domain that contains $S_{k+1}^{\epsilon}$. There exists an
$0<\varepsilon_{0}\leq\delta_{2}$ such that if
$0<\varepsilon<\varepsilon_{0}$, the following holds:

For $x\in G_{k+1}^{\varepsilon}$ we have

$\displaystyle\mathbb{E}^{x}[\tau_{S_{k}^{\varepsilon}}\mathbb{I}_{\tau_{S_{k}^{\varepsilon}}<\tau_{\Omega^{c}}}]\leq\frac{Ce^{-F(G_{k+1})/\varepsilon}\sum_{x\in
G_{k+1}}|O_{x,\varepsilon}|}{\max_{x\in G_{k},y\in
G_{k+1}^{\varepsilon}}\operatorname{cap}(B_{\varepsilon}(x),B_{\varepsilon}(y);\Omega)}+C\varepsilon^{\alpha/2}.$

Let $x_{a}\in G_{k}$, $x_{b}\in G_{k+1}$ be a pair that maximizes the pairwise
capacity. Then, with the notation of Theorem 1, we get in the parallel case

$\displaystyle\mathbb{E}^{x}[\tau_{S_{k}^{\varepsilon}}\mathbb{I}_{\tau_{S_{k}^{\varepsilon}}<\tau_{\Omega^{c}}}]\leq\varepsilon^{-1}\frac{Ce^{-F(G_{k+1})/\varepsilon}\sum_{x\in
G_{k+1}}|O_{x,\varepsilon}|}{\sum_{i=1}^{N}e^{-F(z_{i})/\epsilon}\frac{\mathcal{H}^{n-1}(\\{G_{z_{i}}<\epsilon\\})}{\mathcal{H}^{1}(\\{g_{z_{i}}<\epsilon\\})}}+C\varepsilon^{\alpha/2},$

and in the series case we get

$\displaystyle\mathbb{E}^{x}[\tau_{S_{k}^{\varepsilon}}\mathbb{I}_{\tau_{S_{k}^{\varepsilon}}<\tau_{\Omega^{c}}}]\leq\frac{C}{\varepsilon}\sum_{x\in
G_{k+1}}\sum_{i=1}^{N}e^{(F(z_{i})-F(G_{k+1}))/\varepsilon}\frac{|O_{x,\varepsilon}|\mathcal{H}^{1}(\\{g_{z_{i}}<\epsilon\\})}{\mathcal{H}^{n-1}(\\{G_{z_{i}}<\epsilon\\})}+C\varepsilon^{\alpha/2}.$

If both the minimas and saddles are non-degenerate points, and there is only
one saddle connecting $x_{a},x_{b}$ (with $F(x_{a})<F(x_{b})$), where
$x_{a},x_{b}$ are the only minimas of $F$, the above estimate coincides with
Eyring-Kramers formula (up to a constant)

$\displaystyle\mathbb{E}^{x}[\tau_{S_{k}^{\varepsilon}}\mathbb{I}_{\tau_{S_{k}^{\varepsilon}}<\tau_{\Omega^{c}}}]\leq
C\frac{1}{\lambda_{1}}\sqrt{\frac{\det(\nabla^{2}F(z))}{\det(\nabla^{2}F(x_{a}))}}e^{(F(z)-F(x_{a}))/\epsilon}.$

Here $\lambda_{1}$ is the first eigenvalue of the Hessian of $F$ at the saddle
$z$, and the additive error in Theorem 2 can be removed for small $\epsilon$
as the right hand side of the above tends to $\infty$ as $\epsilon\to 0$.

##  2\. Preliminaries

The generator of the process 1.1 is the following elliptic operator

$\displaystyle L_{\varepsilon}=-\varepsilon\Delta+\nabla F\cdot\nabla.$ (2.1)

In this section we study the potential and regularity theory associated with
the operator 2.1. We provide the identities and pointwise estimates that we
will need in the course of the proofs. Most of these are standard, but we
provide them adapted to our situation for the reader’s convenience. We note
that in this section we only require that the potential $F$ is of class
$C^{2}$ and satisfy the quadratic growth condition 1.3.

###  2.1. Potential theory

######  Definition 2.1.

Let $\Omega\subset\mathbb{R}^{n}$ be a regular domain and let
$G_{\Omega}(x,y)$ be the Green’s function for $\Omega$, i.e., for every $f\in
C(\Omega)$ the function

$\displaystyle u_{f}=\int_{\Omega}G_{\Omega}(x,y)f(y)dy$

is the solution of the Poisson equation

$\displaystyle\begin{cases}L_{\varepsilon}u_{f}=f&\text{in }\,\Omega\\\
u_{f}=0&\text{on }\,\partial\Omega.\end{cases}$

The natural associated measures are the Gibbs measure
$d\mu_{\varepsilon}=e^{-F/\varepsilon}dx$ and the Gibbs surface measure
$d\sigma_{\varepsilon}=e^{-F/\varepsilon}d\mathcal{H}^{n-1}$.

######  Remark 2.2.

Note that the Green’s function is symmetric w.r.t. the Gibbs measure, i.e.

$\displaystyle G(x,y)e^{-F(x)/\varepsilon}=G(y,x)e^{-F(y)/\varepsilon}.$

We also have the fundamental Green’s identities. Here we assume that $\Omega$
is a Lipschitz domain and denote the inner normal by $n$.

######  Lemma 2.3.

Let $\Omega$ be a smooth domain, $\psi,\phi$ be in $C^{2}(\overline{\Omega})$,
and $G_{\Omega}$ be the Green’s function for $\Omega$. Then the following
Green’s identities holds (Green’s first identity)

$\displaystyle\int_{\Omega}\psi
L_{\varepsilon}\phi-\varepsilon\nabla\psi\cdot\nabla\phi
d\mu_{\varepsilon}=\varepsilon\int_{\partial\Omega}\psi\nabla\phi\cdot
nd\sigma_{\varepsilon},$ (2.2)

and (Green’s second identity)

$\displaystyle\int_{\Omega}\psi L_{\varepsilon}\phi-\phi L_{\varepsilon}\psi
d\mu_{\varepsilon}=\varepsilon\int_{\partial\Omega}\psi\nabla\phi\cdot
n-\phi\nabla\psi\cdot nd\sigma_{\varepsilon}.$ (2.3)

Furthermore, the following (balayage) representation formula holds: for every
$g\in C(\partial\Omega)$ the function

$\displaystyle u(x)=\varepsilon
e^{F(x)/\varepsilon}\int_{\partial\Omega}g\nabla_{y}G_{\Omega}(y,x)\cdot
nd\sigma_{\varepsilon}(y)$ (2.4)

is the solution of the Dirichlet problem

$\displaystyle\begin{cases}L_{\varepsilon}u=0&\text{in }\,\Omega\\\
u=g&\text{on }\,\partial\Omega.\end{cases}$

###### Proof.

Integration by parts gives

$\displaystyle\int_{\Omega}\psi L_{\varepsilon}\phi d\mu_{\varepsilon}$
$\displaystyle=\int_{\Omega}\psi(-\varepsilon\Delta\phi+\nabla
F\cdot\nabla\phi)d\mu_{\varepsilon}$
$\displaystyle=\int_{\Omega}(\varepsilon\nabla\psi\cdot\nabla\phi-\frac{\varepsilon}{\varepsilon}\psi\nabla
F\cdot\nabla\phi+\psi\nabla
F\cdot\nabla\phi)d\mu_{\varepsilon}+\varepsilon\int_{\partial\Omega}\psi\nabla\phi\cdot
nd\sigma_{\varepsilon}$
$\displaystyle=\int_{\Omega}\varepsilon\nabla\psi\cdot\nabla\phi
d\mu_{\varepsilon}+\varepsilon\int_{\partial\Omega}\psi\nabla\phi\cdot
nd\sigma_{\varepsilon}.$

The second Green’s identity follows from the first by applying it twice

$\int_{\Omega}\psi L_{\varepsilon}\phi-\varepsilon\nabla\psi\cdot\nabla\phi
d\mu_{\varepsilon}-\int_{\Omega}\phi
L_{\varepsilon}\psi+\varepsilon\nabla\psi\cdot\nabla\phi d\mu_{\varepsilon}\\\
=\varepsilon\int_{\partial\Omega}\psi\nabla\phi\cdot
nd\sigma_{\varepsilon}-\varepsilon\int_{\partial\Omega}\phi\nabla\psi\cdot
nd\sigma_{\varepsilon}.$

We may now obtain the representation formula for the Dirichlet problem. We
choose $\phi(x)=G_{\Omega}(x,y)$ and obtain by Green’s second identity that

$\displaystyle\psi(x)e^{-F(x)/\varepsilon}-\varepsilon\int_{\partial\Omega}\psi\nabla\phi\cdot
nd\sigma_{\varepsilon}=\int_{\Omega}\phi L_{\varepsilon}\psi
d\mu_{\varepsilon}.$

Now relabeling $x\to y$, we get the representation formula 2.4. ∎

Recall the definition of capacity (variational): for $A,B\subset\Omega$ two
disjoint compact sets

$\displaystyle\operatorname{cap}(A,B;\Omega):=\inf\left(\varepsilon\int_{\Omega}|\nabla
h|^{2}e^{-\frac{F}{\varepsilon}}\,dx\,:\,\,h\geq 1\,\,\text{in }A,u\in
H^{1}_{0}(\Omega\setminus B)\right).$ (2.5)

The extension of capacity to open sets follows in the classical way

$\displaystyle\operatorname{cap}(U,B;\Omega):=\sup\\{\operatorname{cap}(A,B;\Omega)\,:\,\text{$A$
compact and $A\subset U$}\\}.$

It is well known that for bounded sets with regular boundary the continuity of
the capacity implies that
$\operatorname{cap}(U,B;\Omega)=\operatorname{cap}(\overline{U},B;\Omega)$.
The extension w.r.t the second entry follows similarly. The variational
definition of the capacity has many equivalent forms, one that we will need is
the one below:

######  Lemma 2.4.

Let $A,B\subset\Omega$ be two disjoint compact sets. Then the variational
formulation of capacity coincides with the balayage definition, i.e.,

$\displaystyle\operatorname{cap}(A,B;\Omega)=\sup\left\\{\int_{A}e^{-F(y)/\varepsilon}d\mu(y):\operatorname{supp}\mu\subset
A,\int_{\Omega}G_{\Omega\setminus B}(x,y)d\mu(y)\leq 1\right\\}.$

The unique measure which maximizes the above, i.e., satisfying

$\displaystyle\int_{A}e^{-F(y)/\varepsilon}d\mu_{A,B}(y)=\operatorname{cap}(A,B;\Omega),\quad\int_{\Omega}G_{\Omega\setminus
B}(x,y)d\mu(y)\leq 1,$

is called the equilibrium measure $\mu_{A,B}$. The corresponding equilibrium
potential is defined as $h_{A,B}=\int_{\Omega}G_{\Omega\setminus
B}(x,y)d\mu_{A,B}(y)$ and is the minimizer of 2.5.

If in addition $A,B$ are smooth, then we have

$\begin{split}\operatorname{cap}(A,B;\Omega)&=\int_{A}e^{-F(y)/\varepsilon}d\mu_{A,B}(y)=\varepsilon\int|\nabla
h_{A,B}|^{2}d\mu_{\varepsilon}\\\ &=-\varepsilon\int_{\partial A}\nabla
h_{A,B}\cdot nd\sigma_{\varepsilon}.\end{split}$ (2.6)

###### Proof.

The claim follows from the symmetry of the Green’s function, Remark 2.2, and
the strong maximum principle that $h_{A,B}=1$ in $A$, see [1, 10]. From 2.2 we
get that

$\displaystyle\int_{\Omega\setminus
B}h_{A,B}L_{\varepsilon}h_{A,B}-\varepsilon|\nabla
h_{A,B}|^{2}d\mu_{\varepsilon}=\varepsilon\int_{\partial(\Omega\setminus
B)}h_{A,B}\nabla h_{A,B}\cdot nd\sigma_{\varepsilon}.$

Using $h_{A,B}=0$ on $\partial(\Omega\setminus B)$ we see that the right hand
side of the above is zero. Moreover, from $L_{\epsilon}h_{A,B}=\mu_{A,B}$ and
from the definition of $d\mu_{\epsilon}$ we get

$\displaystyle\int_{A}e^{-F(y)/\varepsilon}d\mu_{A,B}(y)=\varepsilon\int_{\Omega}|\nabla
h_{A,B}|^{2}d\mu_{\varepsilon}.$ (2.7)

Note that since $h_{A,B}=1$ in $A$ and $0$ on $\partial(\Omega\setminus B)$,
and since $L_{\epsilon}h_{A,B}=0$ in $\Omega\setminus(A\cup B)$, we have by
the uniqueness of the solution to the Dirichlet problem that $h_{A,B}$
coincides with the variational minimizer of 2.5. This establishes the first
two equalities of 2.6.

To prove the last equality in 2.6 we insert $h_{A,B}=\phi=\psi$ into 2.2
(Green’s first identity) and get

$\displaystyle\int_{\Omega\setminus(A\cup
B)}h_{A,B}L_{\varepsilon}h_{A,B}-\varepsilon|\nabla
h_{A,B}|^{2}d\mu_{\varepsilon}=\varepsilon\int_{\partial(\Omega\setminus(A\cup
B))}h_{A,B}\nabla h_{A,B}\cdot nd\sigma_{\varepsilon}.$

Since $\operatorname{supp}\mu_{A,B}\subset A$, and $h_{A,B}=0$ on $B$ and $1$
on $A$, we get

$\displaystyle-\varepsilon\int_{\Omega}|\nabla
h_{A,B}|^{2}d\mu_{\varepsilon}=\varepsilon\int_{\partial A}\nabla h_{A,B}\cdot
nd\sigma_{\varepsilon},$ (2.8)

where $n$ is the outward unit normal of $A$. The result now follows from 2.7
and 2.8. ∎

######  Definition 2.5.

Let $\Omega$ be a smooth domain and $A\subset\Omega$. Then we define the
potential of the equilibrium potential as

$\displaystyle w_{A,\Omega}(x)=\int_{\Omega}G_{\Omega\setminus
A}(x,y)h_{A,\Omega^{c}}(y)dy.$

The definition of the potential of the equilibrium potential might seem
technical at first. However, $w_{A,\Omega}$ has a clear probabilistic
interpretation as the expected hitting time of hitting $A$ of a process killed
at $\partial\Omega$. Indeed, the probabilistic interpretation of
$h_{A,\Omega^{c}}$ is $\mathbb{P}(\tau_{A}<\tau_{\Omega^{c}})$ i.e. the
probability of hitting $A$ before ${\Omega^{c}}$. By Dynkin’s formula we see
that then

$\displaystyle w_{A,{\Omega}}(x)$
$\displaystyle=\mathbb{E}^{x}[w_{A,{\Omega}}(X_{\tau_{A\cup{\Omega^{c}}}})]-\mathbb{E}^{x}\left[\int_{0}^{\tau_{A\cup{\Omega^{c}}}}L_{\varepsilon}w_{A,{\Omega}}(X_{t})dt\right]$
$\displaystyle=\int_{0}^{\infty}\mathbb{E}^{x}\left[\mathbb{I}_{t\leq\tau_{A\cup{\Omega^{c}}}}\mathbb{E}^{X_{t}}[\mathbb{I}_{A}(X_{\tau_{A\cup{\Omega^{c}}}})]\right]dt$
$\displaystyle=\int_{0}^{\infty}\mathbb{E}^{x}\left[\mathbb{I}_{t\leq\tau_{A\cup{\Omega^{c}}}}\mathbb{I}_{A}(X_{\tau_{A\cup{\Omega^{c}}}})\right]dt$
$\displaystyle=\mathbb{E}^{x}[\tau_{A}\mathbb{I}_{\tau_{A}<\tau_{\Omega^{c}}}].$

We also have the following integration by parts formula for the potential of
the equilibrium potential:

######  Lemma 2.6.

Let $\Omega$ be a smooth domain, let $A\subset\Omega$ be a smooth set, and
assume that $B_{2\rho}(x)\subset\Omega\setminus A$. Then

$\displaystyle\int_{\partial
B_{\rho}(x)}w_{A,\Omega}(y)e^{-F(y)/\varepsilon}d\mu_{B_{\rho}(x),A}(y)=\int_{\Omega\setminus
A}h_{A,\Omega^{c}}(z)h_{B_{\rho}(x),A}(z)d\mu_{\varepsilon}.$

The above statement looks more familiar if we write it in the formal way as

$\displaystyle\int
w_{A,\Omega}e^{-F/\varepsilon}L_{\varepsilon}h_{B_{\rho},A}=\int
L_{\varepsilon}w_{A,\Omega}e^{-F/\varepsilon}h_{B_{\rho},A}dz.$

###### Proof.

Using the definition of $w_{A,\Omega}$ and Fubini’s theorem

$\displaystyle\int_{\partial B_{\rho}(x)}$ $\displaystyle
w_{A,\Omega}(y)e^{-F(y)/\varepsilon}d\mu_{B_{\rho}(x),A}(y)$
$\displaystyle=\int_{\partial B_{\rho}(x)}\left(\int_{\Omega\setminus
A}G_{\Omega\setminus
A}(y,z)h_{A,\Omega^{c}}(z)dz\right)e^{-F(y)/\varepsilon}d\mu_{B_{\rho}(x),A}(y)$
$\displaystyle=\int_{\Omega\setminus A}h_{A,\Omega^{c}}(z)\int_{\partial
B_{\rho}(x)}G_{\Omega\setminus
A}(y,z)e^{-F(y)/\varepsilon}d\mu_{B_{\rho}(x),A}(y)dz.$

Using the symmetry of the Green’s function, Remark 2.2,

$\displaystyle\int_{\Omega\setminus A}$ $\displaystyle
h_{A,\Omega^{c}}(z)\int_{\partial B_{\rho}(x)}G_{\Omega\setminus
A}(y,z)e^{-F(y)/\varepsilon}d\mu_{B_{\rho}(x),A}(y)dz$
$\displaystyle=\int_{\Omega\setminus
A}h_{A,\Omega^{c}}(z)e^{-F(z)/\varepsilon}\int_{\partial
B_{\rho}(x)}G_{\Omega\setminus A}(z,y)d\mu_{B_{\rho}(x),A}(y)dz.$

Note that $\operatorname{supp}\mu_{B_{\rho}(x),A}\subset\partial B_{\rho}(x)$
and as such

$\displaystyle u(z)=\int_{\Omega}G_{\Omega\setminus
A}(z,y)d\mu_{B_{\rho}(x),A}(y)=\int_{\partial B_{\rho}(x)}G_{\Omega\setminus
A}(z,y)d\mu_{B_{\rho}(x),A}(y)$

solves the equation $L_{\epsilon}u=\mu_{B_{\rho}(x),A}$ in $\Omega$ and $u=0$
on $\partial{\Omega\setminus A}$. Consequently, by the uniqueness result to
the Dirichlet-Poisson problem, we get $u(z)=h_{B_{\rho}(x),A}(z)$. Hence

$\int_{\Omega\setminus
A}h_{A,\Omega^{c}}(z)e^{-F(z)/\varepsilon}\int_{\partial
B_{\rho}(x)}G_{\Omega\setminus A}(z,y)d\mu_{B_{\rho}(x),A}(y)dz\\\
=\int_{\Omega\setminus
A}h_{A,\Omega^{c}}(z)h_{B_{\rho}(x),A}(z)d\mu_{\varepsilon}(z).$

Combining the equalities above yields the result. ∎

###  2.2. Classical pointwise estimates

In this section we recall classical pointwise estimates for functions which
satisfy

$\displaystyle L_{\varepsilon}u=f$

in a domain $\Omega$, where the operator $L_{\varepsilon}$ is defined in 2.1.
First, since we assume that $F\in C^{2}(\mathbb{R}^{n})$, then for all Hölder
continuous $f$ the solutions of the above equation are $C^{2,\alpha}$-regular,
see [14]. However, these regularity estimates depend on $\varepsilon$ and blow
up as $\varepsilon\to 0$. The point is that we may obtain regularity estimates
for constants independent of $\varepsilon$ if we restrict ourselves on small
enough scales. To this aim, for a given domain $\Omega$ we choose a positive
number $\nu$ such that

$\displaystyle\frac{\|\nabla F\|_{L^{\infty}(\Omega)}}{\varepsilon}\leq\nu.$

We have the following two theorems from [14].

######  Lemma 2.7.

Let $\Omega$ be a domain and let $u\in C^{2}(\Omega)$ be a non-negative
function satisfying $L_{\varepsilon}u=0$. Then for any
$B_{3R}(x)\subset\Omega$ it holds that

$\displaystyle\sup_{B_{R}(x)}u\leq C\inf_{B_{R}(x)}u$

for a constant $C=C(n,\nu R)$. In particular, if $\|\nabla
F\|_{L^{\infty}(\Omega)}\leq L$, then for $R\leq\frac{\varepsilon}{L}$ the
constant $C$ is independent of $\varepsilon$. Furthermore, for
$p\in(1,\infty)$ and any number $k$ we have

$\displaystyle\sup_{B_{R}(x)}|u-k|\leq
C_{p}\left(\strokedint_{B_{2R}(x)}|u-k|^{p}dx\right)^{1/p}$

and

$\displaystyle\left(\strokedint_{B_{2R}(x)}|u|^{p}dx\right)^{1/p}\leq
C_{p}\inf_{B_{R}(x)}u,$

where the symbol $\strokedint$ denotes the average integral, and the constant
$C_{p}$ in addition to above depends also on $p$.

In the non-homogeneous case $L_{\varepsilon}u=f$ we have the following
generalization of Harnack’s inequality.

######  Lemma 2.8.

Let $\Omega$ be a domain and let $u\in C^{2}(\Omega)$ be a non-negative
function satisfying $L_{\varepsilon}u=f$. Then for any
$B_{3R}(x)\subset\Omega$ it holds that

$\displaystyle\sup_{B_{R}(x)}u\leq
C\left(\inf_{B_{R}(x)}u+\frac{R}{\varepsilon}\|f\|_{L^{n}(B_{2R}(x))}\right)$

for a constant $C=C(n,\nu R)$. In particular, if $\|\nabla
F\|_{L^{\infty}(\Omega)}\leq L$, then for $R\leq\frac{\varepsilon}{L}$ the
constant $C$ is independent of $\varepsilon$ and we have

$\displaystyle\sup_{B_{R}(x)}u\leq
C\left(\inf_{B_{R}(x)}u+\|f\|_{L^{n}(B_{2R}(x))}\right).$

The Harnack inequality in Lemma 2.7 holds also in the case of the punctured
ball.

######  Lemma 2.9.

Let $u\in C^{2}(B_{3R}(x)\setminus\\{x\\})$ be a non-negative function
satisfying $L_{\varepsilon}u=0$ in $B_{3R}(x)\setminus\\{x\\}$. Then

$\displaystyle\sup_{\partial B_{R}(x)}u\leq C\inf_{\partial B_{R}(x)}u$

for a constant $C=C(n,\nu R)$.

###### Proof.

By translating the coordinates we may assume that $x=0$. Let
$x_{0},y_{0}\in\partial B_{R}$ be such that $\sup_{\partial
B_{R}(x)}u=u(x_{0})$ and $\inf_{\partial B_{R}(x)}u=u(y_{0})$. We choose
points $x_{1},\dots,x_{N-1},x_{N}\in\partial B_{R}$ such that
$|x_{i}-x_{i-1}|\leq R/4$ and $x_{N}=y_{0}$. Note that the number $N$ is
bounded. Now we may use Harnack’s inequality Lemma 2.7 in balls
$B_{R/4}(x_{i})$ to get

$\displaystyle u(x_{i-1})\leq Cu(x_{i}).$

We obtain the claim by summing over $i=1,\dots,N$. ∎

######  Lemma 2.10.

Let $\Omega$ be a domain and let $u,h\in C^{2}(\Omega)$ be non-negative
functions such that $L_{\varepsilon}u=h$ and $h$ satisfies Harnack’s
inequality with constant $c_{0}$. Then the function $v=u+h$ satisfies
Harnack’s inequality, i.e., for all $B_{3R}(x)\subset\Omega$ it holds that

$\displaystyle\sup_{B_{R}(x)}v\leq C\inf_{B_{R}(x)}v$

for a constant $C=C(n,\nu R,R^{2}/\varepsilon,c_{0})$. In particular, if
$\|\nabla F\|_{L^{\infty}(\Omega)}\leq L$, then for
$R\leq\min\\{\varepsilon/L,\sqrt{\varepsilon}\\}$ the constant $C$ is
independent of $\varepsilon$.

###### Proof.

Again we may assume that $x=0$. Using Lemma 2.8 and Harnack’s inequality for
$h$ yields

$\displaystyle\sup_{B_{R}}u$ $\displaystyle\leq
C\inf_{B_{R}}u+\frac{CR}{\varepsilon}\|h\|_{L^{n}(B_{2R})}$ $\displaystyle\leq
C\inf_{B_{R}}u+\frac{CR}{\varepsilon}|B_{R}|^{1/n}\inf_{B_{R}}h$
$\displaystyle\leq C\inf_{B_{R}}u+\frac{CR^{2}}{\varepsilon}\inf_{B_{R}}h.$

Now, using Harnack’s inequality for $h$ again, we obtain

$\displaystyle\sup_{B_{R}}v$ $\displaystyle\leq\sup_{B_{R}}u+\sup_{B_{R}}h\leq
C\inf_{B_{R}}u+C\inf_{B_{R}}h\leq C\inf_{B_{R}}v$

for a constant $C$ as in the statement. This proves the claim. ∎

The Harnack’s inequality in Lemma 2.7 implies Hölder continuity for solutions
of $L_{\varepsilon}u=0$.

######  Lemma 2.11.

Let $u\in C^{2}(B_{3R}(x))$ be a function such that for any constant $c$, for
which $v=u+c$ is non-negative, the function $v$ satisfies Harnack’s inequality
with constant $C_{0}$, independent of $c$. Then there exists $C=C(C_{0})>1$
and $\alpha=\alpha(C_{0})\in(0,1)$ such that, for all $\rho\leq R$, it holds
that

$\displaystyle\operatorname{osc}_{B_{\rho}(x)}u\leq
C\left(\frac{\rho}{R}\right)^{\alpha}\operatorname{osc}_{B_{R}(x)}u.$

In particular, if $u,h\in C^{2}(\Omega)$ are non-negative functions such that
$L_{\varepsilon}u=h$ and $h$ satisfies Harnack’s inequality with constant
$C_{0}$, then $u+h$ satisfies the estimate above.

###### Proof.

The proof follows verbatim from the classical proof of Moser, see [14, Theorem
8.22]. ∎

##  3\. Technical lemmas

In this section we provide some preliminary results for the proofs of the main
theorems. We recall that we assume that the potential $F$ satisfies the
structural assumptions from Section 1.1, and that from this moment on our
constants are allowed to depend on the data, see paragraph after 1.16.

###  3.1. Rough estimates for potentials

In this subsection we provide estimates for the capacitary potential
$h_{A,B}$, when $A$ and $B$ are two disjoint closed sets. The first estimate
is the so called renewal estimate of [8]. In order to trace dependencies of
constants, we provide a proof.

######  Lemma 3.1.

Let $\Omega$ be a smooth domain, let $A,B\subset\Omega$ be disjoint smooth
sets, and consider $h_{A,B}$ as the capacitary potential in $\Omega$. Assume
that $B_{4\varrho}(x)\subset(\Omega\setminus(A\cup B))$, and that
$r\leq\min\left\\{\frac{\varepsilon}{\|\nabla
F\|_{L^{\infty}(B_{2\varrho}(x))}},\varrho\right\\}$. Then there exists a
constant $C=C(n,\nu)>1$ such that

$\displaystyle h_{A,B}(x)$ $\displaystyle\leq
C\frac{\operatorname{cap}(B_{r}(x),A;\Omega)}{\operatorname{cap}(B_{r}(x),B;\Omega)}.$

###### Proof.

Again, without loss of generality, we may assume that $x=0$. Since $h_{A,B\cup
B_{r}}=h_{A,B}$ on $\partial(\Omega\setminus(A\cup B))$, we can use 2.4 to
represent $h_{A,B}$ as follows

$\displaystyle h_{A,B}(z)=\varepsilon
e^{F(z)/\varepsilon}\int_{\partial(\Omega\setminus(A\cup B))}h_{A,B\cup
B_{r}}\nabla G_{\Omega\setminus(A\cup B)}(y,z)\cdot
nd\sigma_{\varepsilon}(y).$ (3.1)

Now by Green’s second identity 2.3 in $\Omega\setminus(A\cup B\cup B_{r})$ and
3.1 we see that, for $z\in\Omega$,

$h_{A,B}(z)=h_{A,B\cup B_{r}}(z)\\\ -\varepsilon
e^{F(z)/\varepsilon}\int_{\partial B_{r}}G_{\Omega\setminus(A\cup
B)}(y,z)\nabla h_{A,B\cup B_{r}}(y)\cdot nd\sigma_{\varepsilon}(y),$ (3.2)

where $n$ is the inward unit normal of $B_{r}$. First note that by 2.3 we can
identify the equilibrium measure as

$\displaystyle\mu_{B\cup B_{r},A}=-\varepsilon\nabla h_{B\cup B_{r},A}\cdot
nd\sigma_{\varepsilon}=\varepsilon\nabla h_{A,B\cup B_{r}}\cdot
nd\sigma_{\varepsilon}.$

Using that $h_{A,B\cup B_{r}}=1-h_{B\cup B_{r},A}$, together with the above
and 3.2, we get for $z\in\overline{B_{r}}$ (since $h_{A,B\cup B_{r}}(z)=0$)
that

$\displaystyle h_{A,B}(z)$ $\displaystyle=\int_{\partial
B_{r}}G_{\Omega\setminus(A\cup B)}(y,z)e^{(F(z)-F(y))/\varepsilon}d\mu_{B\cup
B_{r},A}(y).$ (3.3)

First note that $\mu_{B_{r}\cup B,A}\lvert_{\partial B_{r}}$ is an admissible
measure for $\operatorname{cap}(B_{r},A;\Omega)$, which follows from the fact
that by the comparison principle, the potentials for ordered measures are
ordered and the support of $\mu_{B_{r}\cup B,A}\lvert_{\partial B_{r}}$ is in
$\overline{B_{r}}$. To bound $h_{A,B}$ from above, note that by the balayage
representation of capacity (see Lemma 2.4) and the above, we obtain

$\displaystyle\int_{\partial B_{r}}e^{-F(y)/\varepsilon}d\mu_{B\cup
B_{r},A}(y)\leq\operatorname{cap}(B_{r},A;\Omega).$

Applying the above to 3.3 gives, for $z\in B_{r}$,

$\displaystyle h_{A,B}(z)\leq\sup_{y\in\partial B_{r}}G_{\Omega\setminus(A\cup
B)}(y,z)e^{F(z)/\varepsilon}\operatorname{cap}(B_{r},A;\Omega).$ (3.4)

It remains to bound the Green’s function. For $z\in\overline{B_{r}}$ we have
by 2.4, 2.2 and 2.6 that

$\displaystyle 1$ $\displaystyle=h_{B_{r},A\cup B}(z)=\int_{\partial
B_{r}}G_{\Omega\setminus(A\cup B)}(z,y)d\mu_{B_{r},A\cup B}(y)$ (3.5)
$\displaystyle=\int_{\partial B_{r}}G_{\Omega\setminus(A\cup
B)}(y,z)e^{(F(z)-F(y))/\varepsilon}d\mu_{B_{r},A\cup B}(y)$
$\displaystyle\geq\inf_{\partial B_{r}}G_{\Omega\setminus(A\cup
B)}(y,z)e^{F(z)/\varepsilon}\operatorname{cap}(B_{r},A\cup B;\Omega)$
$\displaystyle\geq\inf_{\partial B_{r}}G_{\Omega\setminus(A\cup
B)}(y,z)e^{F(z)/\varepsilon}\operatorname{cap}(B_{r},B;\Omega).$

Now putting together 3.4, 3.5 and 2.9 we are done. ∎

The result below is a version of the rough capacity bound of [8], but we give
a simplified proof. We will later use a similar argument in the proof of
Theorem 1.

######  Lemma 3.2.

Let $D\subset B_{R}$ be a smooth closed set. Let $x\in B_{R}\setminus D$ be
such that $B_{4\rho}(x)\subset B_{R}\setminus D$, for $\rho\leq\varepsilon$.
Then there exists constants $q_{1},q_{2}\in\mathbb{R}$ and $C>1$ such that

$\displaystyle\frac{1}{C}\varepsilon^{q_{1}}\rho^{n-1}e^{-F(x;D)/\varepsilon}\leq\operatorname{cap}(B_{\rho}(x),D)\leq
C\varepsilon\rho^{q_{2}}e^{-F(x;D)/\varepsilon}.$

###### Proof.

We assume without loss of generality that $F(B_{\rho};D)=0$, since the
quantities can always be scaled back. Consider
$\gamma\in\mathcal{C}(B_{\rho}(x),D;B_{R})$ (i.e. a curve connecting
$B_{\rho}(x)$ and $D$ inside $\Omega$) such that $\sup_{t}F(\gamma(t))\leq
C\varepsilon$ and let $u(z)=h_{D,B_{\rho}(x)}(z)$. We first note by Lemma 2.4
that

$\displaystyle\operatorname{cap}(B_{\rho}(x),D)=\varepsilon\int|\nabla
u|^{2}e^{-F(y)/\varepsilon}dy.$

Fix an $n-1$ dimensional disk $D_{\rho}$ of radius $\rho$. Then by Cauchy-
Schwarz

$\displaystyle\int_{\Omega}|\nabla
u|^{2}e^{-F(y)/\varepsilon}dy\geq\int_{0}^{1}\int_{D_{\rho}}\left|\left\langle\frac{\dot{\gamma}}{|\dot{\gamma}|},\nabla
u(\gamma(t)+z)\right\rangle\right|^{2}|\dot{\gamma}|d\sigma_{\varepsilon}(z)dt.$

By the fundamental theorem of calculus and Cauchy-Schwarz, we have for a fixed
point $z\in D_{\rho}$ that

$\displaystyle 1$
$\displaystyle=u(\gamma(1))-u(\gamma(0))=\int_{0}^{1}\frac{d}{dt}u(\gamma(t)+z)dt$
$\displaystyle=\int_{0}^{1}\frac{d}{dt}u(\gamma(t)+z)\frac{\sqrt{|\dot{\gamma}|}}{\sqrt{|\dot{\gamma}|}}e^{-F(\gamma(t)+z)/(2\varepsilon)}e^{F(\gamma(t)+z)/(2\varepsilon)}dt$
$\displaystyle\leq\left(\int_{0}^{1}|\frac{d}{dt}u(\gamma(t)+z)|^{2}\frac{1}{|\dot{\gamma}|}e^{-F(\gamma(t)+z)/\varepsilon}dt\right)^{1/2}\left(\int|\dot{\gamma}|e^{F(\gamma(t)+z)/\varepsilon}dt\right)^{1/2}.$

From the above we get

$\displaystyle\int_{\Omega}|\nabla u|^{2}e^{-F(y)/\varepsilon}dy$
$\displaystyle\geq\int_{0}^{1}\int_{D_{\rho}}\left|\left\langle\frac{\dot{\gamma}}{|\dot{\gamma}|},\nabla
u(\gamma(t)+z)\right\rangle\right|^{2}|\dot{\gamma}|d\sigma_{\varepsilon}(z)dt$
$\displaystyle\geq\int_{D_{\rho}}\left(\int_{0}^{1}|\dot{\gamma}|e^{F(\gamma(t)+z)/\varepsilon}dt\right)^{-1}d\sigma_{\varepsilon}(z).$

Now since $F$ is Lipschitz in $B_{R}$ and $F(B_{\rho};D)=0,$ we know that
there exists a constant $C(\gamma)$ such that, for $z\in D_{\rho}$ and
$\rho<2\varepsilon$,

$\displaystyle\int_{0}^{1}|\dot{\gamma}|e^{F(\gamma(t)+z)/\varepsilon}dt\leq
C(\gamma).$ (3.6)

In the above the constant $C$ depends on the length of $\gamma$, which can be
assumed to be bounded. To see this, take an $\epsilon$ neighborhood of
$\gamma$, $E_{\varepsilon}$ and consider a set of balls
$\bigcup_{i}B_{\varepsilon}(y_{i})\supset E_{\varepsilon}$ such that
$\bigcup_{i}B_{\varepsilon}(y_{i})\subset E_{C_{1}\varepsilon}$ (for some
large $C_{1}$), the maximal number of such balls needed is
$C_{1}R^{n}/\epsilon^{n}$. If we construct a piecewise linear curve
$\gamma_{\varepsilon}$ connecting the center of each ball in the covering,
this curve will be inside $E_{C_{1}\varepsilon}$ and its length will be
bounded by $2C_{1}R^{n}/\varepsilon^{n-1}$. This newly constructed curve can
be mollified to achieve a smooth curve without increasing the length by more
than a factor. From the above and the Lipschitz continuity of $F$ it is clear
that $\sup_{t}F(\gamma_{\varepsilon}(t))\leq C\varepsilon$, and as such we can
replace $\gamma$ with $\gamma_{\varepsilon}$ in the above and get from 3.6
that there is a constant $C>1$ depending only on the data such that

$\displaystyle\int_{0}^{1}|\dot{\gamma}|e^{F(\gamma(t)+z)/\varepsilon}dt\leq\varepsilon^{1-n}C.$

This implies that for a new constant $C$ we have

$\displaystyle\int_{\Omega}|\nabla u|^{2}e^{-F(y)/\varepsilon}dy\geq
C\varepsilon^{n-1}\rho^{n-1},$

which completes the proof of the lower bound after rescaling our potential
$F$.

To prove the upper bound we have two possible cases: In the case when
$F(x;D)=F(x)$ we can take a cutoff function
$\chi_{B_{\rho}(x)}\leq\phi\leq\chi_{B_{2\rho}(x)}$ where $|\nabla\phi|\leq
C/\rho$ as a competitor in the variational formulation of capacity 2.5. Then

$\displaystyle\int_{\Omega}|\nabla\phi|^{2}e^{-F(y)/\varepsilon}dx=\int_{B_{2\rho}(x)}|\nabla\phi|^{2}e^{-F(y)/\varepsilon}dx\leq
C\rho^{n-2}.$

In the case where $F(x;D)>F(x)$, consider the set $\hat{D}=\\{z\in
B_{R}:F(z)\leq F(x;D)\\}$ and let $\hat{D}_{1}$ be the component that
intersects $D$. We set $\widetilde{D}=(\hat{D}_{1}\cup D)\setminus
B_{4\rho}(x)$. By the Lipschitz continuity, we know that
$\inf_{\widetilde{D}}F>-C\rho$. We take
$\chi_{\widetilde{D}+B_{\rho}}\leq\phi\leq\chi_{\widetilde{D}+B_{2\rho}}$,
where $|\nabla\phi|\leq C/\rho$, and get

$\displaystyle\int_{\Omega}|\nabla\phi|^{2}e^{-F(y)/\varepsilon}dx$
$\displaystyle=\int_{(\widetilde{D}+B_{2\rho})\setminus\widetilde{D}}|\nabla\phi|^{2}e^{-F(y)/\varepsilon}dx$
$\displaystyle\leq C|(\widetilde{D}+B_{2\rho})\setminus\widetilde{D}|.$

Again, the upper bound follows from rescaling the potential $F$ as in the case
of the lower bound. This completes the whole proof. ∎

######  Lemma 3.3.

Let $A,B\subset B_{R}$ be smooth disjoint sets, and let $x\in B_{R}$ be such
that $B_{\varepsilon}(x)\subset B_{R}\setminus(A\cup B)$,
$\varepsilon\in(0,1)$. Then, if $F(x;B)\leq F(x;A)$, there exists constants
$q$ and $C$ such that

$\displaystyle h_{A,B}(x)\leq
C\varepsilon^{q}e^{-(F(x;A)-F(x;B))/\varepsilon}.$

###### Proof.

Let $L:=\|\nabla F\|_{L^{\infty}(B_{R})}$. By combining Lemmas 3.1 and 3.2
with $R=\varepsilon$, $r=\min\\{\varepsilon/L,\varepsilon\\}$ yields the
result. ∎

######  Remark 3.4.

By relabeling $A,B$ to $B,A$ and using the fact that $h_{A,B}=1-h_{B,A}$, we
get that if the reverse inequality holds, i.e. $F(x;B)>F(x;A)$, then

$\displaystyle 1-h_{A,B}(x)\leq
C\varepsilon^{q}e^{-(F(x;B)-F(x;A))/\varepsilon}.$

######  Lemma 3.5.

Let $\Omega$ be a smooth domain and let $x_{a},x_{b}\in\Omega\subset B_{R}$ be
two local minimum points of $F$. Fix $0<\delta<\delta_{1}$ and assume that
$U_{-\delta/3}=\\{x:F(x)<F(x_{a};x_{b})-\delta/3\\}\subset\Omega$. Then there
exists an $\varepsilon_{0}\in(0,1)$ and a constant $C=C>1$ such that, for any
$0\leq\varepsilon\leq\varepsilon_{0}$ for which
$B_{3\varepsilon}(x_{a}),B_{3\varepsilon}(x_{b})\subset U_{-\delta/3}$, the
following holds: If $U_{i}$ is a component of $U_{-\delta/3}$, then

$\displaystyle\underset{U_{i}}{\operatorname{osc}\,}h_{B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b})}\leq
C\varepsilon.$

###### Proof.

Consider any component $U_{i}$ of $U_{-\delta/3}$. We note that we can take
$\varepsilon$ small enough depending on the Lipschitz constant of $F$ in
$B_{R}$ and $\delta$ such that there exists a Lipschitz domain $D_{i}$
satisfying

$\displaystyle U_{i}+B_{\varepsilon}\subset D_{i}\subset U_{-\delta/4}.$

For simplicity, denote $u:=h_{B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b})}$.
Since $D_{i}$ is Lipschitz we may use the Poincaré inequality to get

$\displaystyle\int_{D_{i}}|u-u_{D_{i}}|^{2}dx\leq C\int_{D_{i}}|\nabla
u|^{2}dx.$

Using that $D_{i}\subset U_{-\delta/4}$ together with Lemma 2.4

$\displaystyle\int_{D_{i}}|\nabla u|^{2}dx$ $\displaystyle\leq
e^{\sup_{D_{i}}F/\varepsilon}\int_{D_{i}}|\nabla
u|^{2}e^{-F(x)/\varepsilon}dx$
$\displaystyle\leq\varepsilon^{-1}e^{\sup_{D_{i}}F/\varepsilon}\operatorname{cap}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b});\Omega).$

Using the definition of $U_{-\delta/4}$ and Lemma 3.2, we get

$\displaystyle\int_{D_{i}}|u-u_{D_{i}}|^{2}dx\leq
C\varepsilon^{q_{1}}e^{-\delta/4\varepsilon}$ (3.7)

for some constant $q_{1}\in\mathbb{R}$. Now, for any $x_{0}\in U_{i}$ we have
by Lemma 2.7 that

$\displaystyle\sup_{B_{\varepsilon}(x_{0})}|u-u_{D_{i}}|^{2}$
$\displaystyle\leq
C\left(\strokedint_{B_{2\varepsilon}}|u-u_{D_{i}}|^{2}dx\right)$

which together with 3.7 gives

$\displaystyle\sup_{B_{\varepsilon}(x_{0})}|u-u_{D_{i}}|^{2}\leq
C\varepsilon^{q_{1}-n}e^{-\delta/4\varepsilon}.$

Since $x_{0}$ was an arbitrary point in $U_{i}$ we conclude that there exists
$\varepsilon_{0}\in(0,1)$ depending only on the data such that if
$\varepsilon<\varepsilon_{0}$, the claim holds. ∎

We conclude this subsection with an estimate relating the value of the
potential of the equilibrium potential to the ratio of the $L^{1}$ norm of the
equilibrium potential and the capacity.

######  Lemma 3.6.

Let $\Omega$ be a smooth domain and let $A\Subset\Omega$ be a smooth open set
and consider $w_{A,\Omega}$ as the potential of the equilibrium potential in
$\Omega$ (see Definition 2.5). Let $x\in\Omega$ be a critical point of $F$
such that $B_{3\sqrt{\varepsilon}}(x)\subset\Omega$. Then there exists a
constant $C>1$ such that for $\rho<\sqrt{\varepsilon}$ we have

$\displaystyle
w_{A,\Omega}(x)-C\bigg{(}\frac{\rho}{\sqrt{\varepsilon}}\bigg{)}^{\alpha}(w_{A,\Omega}(x)+1)$
$\displaystyle\leq\frac{{\displaystyle\int}h_{A,\Omega^{c}}h_{B_{\rho}(x),A}d\mu_{\varepsilon}}{\operatorname{cap}(B_{\rho}(x),A;\Omega)}$
$\displaystyle\leq
w_{A,\Omega}(x)+C\bigg{(}\frac{\rho}{\sqrt{\varepsilon}}\bigg{)}^{\alpha}(w_{A,\Omega}(x)+1).$

###### Proof.

From Lemma 2.6 we get

$\displaystyle\int_{\partial
B_{\rho}(x)}w_{A,\Omega}(y)e^{-F(y)/\varepsilon}d\mu_{B_{\rho}(x),A}(y)=\int_{(A\cup
B)^{c}}h_{A,\Omega^{c}}(z)h_{B_{\rho}(x),A}(z)d\mu_{\varepsilon}.$

We can estimate the left hand side as

$\displaystyle w_{A,\Omega}(x)-\operatorname{osc}_{B_{\rho}}w_{A,\Omega}$
$\displaystyle\leq\inf_{B_{\rho}(x)}w_{A,\Omega}\leq\frac{\int_{\partial
B_{\rho}(x)}e^{-F(y)/\varepsilon}w_{A,\Omega}d\mu_{B_{\rho}(x),A}(y)}{\int_{\partial
B_{\rho}(x)}e^{-F(y)/\varepsilon}d\mu_{B_{\rho}(x),A}(y)}$
$\displaystyle\leq\sup_{B_{\rho}(x)}w_{A,\Omega}\leq
w_{A,\Omega}(x)+\operatorname{osc}_{B_{\rho}}w_{A,\Omega}.$

We want to estimate the oscillation of $w_{A,\Omega}$ which we do by
considering

$\displaystyle\operatorname{osc}w_{A,\Omega}=\operatorname{osc}(w_{A,\Omega}+h_{A,\Omega^{c}}-h_{A,\Omega^{c}})\leq\operatorname{osc}(w_{A,\Omega}+h_{A,\Omega^{c}})+\operatorname{osc}(h_{A,\Omega^{c}}).$

Now, the oscillation of $w_{A,\Omega}+h_{A,\Omega^{c}}$ and $h_{A,\Omega^{c}}$
can estimated by Lemma 2.11 for $\rho\leq\frac{1}{C}\sqrt{\varepsilon}$. That
is,

$\displaystyle\operatorname{osc}_{B_{\rho}}(w_{A,\Omega}+h_{A,\Omega^{c}})+\operatorname{osc}_{B_{\rho}}(h_{A,\Omega^{c}})\leq$
$\displaystyle
C\left(\frac{\rho}{\sqrt{\varepsilon}}\right)^{\alpha}\sup_{B_{\sqrt{\varepsilon}}}(w_{A,\Omega}+h_{A,\Omega^{c}})$
$\displaystyle+C\left(\frac{\rho}{\sqrt{\varepsilon}}\right)^{\alpha}\sup_{B_{\sqrt{\varepsilon}}}(h_{A,\Omega^{c}}).$

We apply Lemma 2.7 to replace the supremums on the right hand side with the
value at $x$ as both $w_{A,\Omega}+h_{A,\Omega^{c}}$ and $h_{A,\Omega^{c}}$
satisfies the Harnack inequality (see Lemma 2.10). That is,

$\displaystyle\operatorname{osc}_{B_{\rho}}(w_{A,\Omega}+h_{A,\Omega^{c}})+\operatorname{osc}_{B_{\rho}}(h_{A,\Omega^{c}})$
$\displaystyle\leq
C\left(\frac{\rho}{\sqrt{\varepsilon}}\right)^{\alpha}(w_{A,\Omega}(x)+h_{A,\Omega^{c}}(x))$
$\displaystyle\leq
C\left(\frac{\rho}{\sqrt{\varepsilon}}\right)^{\alpha}(w_{A,\Omega}(x)+1).$

It is easily seen that the above can be extended to $\rho\leq\sqrt{\epsilon}$
by applying Lemma 2.10 again and by enlarging the constant $C$. The proof is
completed by using 2.6 and collecting the estimates above. ∎

###  3.2. Laplace asymptotics for log-concave functions

The assumptions 1.4 and 1.6 ensure that near critical points the potential $F$
is well approximated by convex functions. Therefore we will need basic
estimates for log-concave functions, which rather surprisingly we did not find
in the literature.

######  Lemma 3.7.

Assume $G:\mathbb{R}^{n}\to\mathbb{R}$ is a convex function which has a proper
minimum at the origin and $G(0)=0$. Then there exists a constant $C=C(n)>1$
such that

$\displaystyle\frac{1}{C}|\\{G<\varepsilon\\}|\leq\int_{\mathbb{R}^{n}}e^{-\frac{G}{\varepsilon}}\,dx\leq
C|\\{G<\varepsilon\\}|.$ (3.8)

Moreover, there is a constant $C=C(n)$ such that for all $\Lambda>1$, we have

$\displaystyle\int_{\\{G<\Lambda\varepsilon\\}}e^{-\frac{G}{\varepsilon}}\,dx\geq(1-\eta(C\Lambda^{-1}))\int_{\mathbb{R}^{n}}e^{-\frac{G}{\varepsilon}}\,dx,$
(3.9)

with $\eta$ as in 1.15.

###### Proof.

By approximation we may assume that $G$ is smooth. The lower bound in 3.8
follows immediately from

$\displaystyle\int_{\mathbb{R}^{n}}e^{-\frac{G}{\varepsilon}}\,dx\geq\int_{\\{G<\varepsilon\\}}e^{-\frac{G}{\varepsilon}}\,dx\geq
e^{-1}|\\{G<\varepsilon\\}|.$

To prove the upper bound in 3.8 we first show that, for all $t>0$, it holds

$\displaystyle|\\{G<2t\\}|\leq 2^{n}|\\{G<t\\}|.$ (3.10)

In order to prove 3.10 it is enough to consider only the case $t=1$ (the
general case follows by considering $\widetilde{G}=G/t$). Denote
$E_{1}=\\{G<1\\}$ and $E_{2}=\\{G<2\\}$. Hence our goal is to show

$\displaystyle E_{2}\subset 2E_{1}=\\{2x:x\in E_{1}\\}.$

Fix $\hat{x}\in\partial E_{1}$ and define $g(t)=G(t\hat{x})$ for $t\geq 0$. By
our assumptions, $g(t)$ is a smooth convex function satisfying $g(0)=0$ and
$g(1)=1$. As such, both $g$, $g^{\prime}$ are increasing functions from which
we can conclude that $g^{\prime}(1)\geq 1$. Now, by the fundamental theorem of
calculus,

$\displaystyle g(2)-g(1)=\int_{1}^{2}g^{\prime}(t)\,dt\geq 1$

which gives $g(2)\geq 2$. This means that for all $\hat{x}\in\partial E_{1}$
we have $G(2\hat{x})\geq 2$. That is, we have $E_{2}\subset 2E_{1}$. Thus

$\displaystyle|E_{2}|\leq|2E_{1}|\leq 2^{n}|E_{1}|$

and 3.10 follows. Iterating 3.10 gives

$\displaystyle|\\{G<2^{j}\varepsilon\\}|\leq 2^{jn}|\\{G<\varepsilon\\}|$

and hence

$\displaystyle|\\{G<\varrho\varepsilon\\}|\leq(2\varrho)^{n}|\\{G<\varepsilon\\}|$
(3.11)

for all $\varrho\geq 1$. We conclude the proof of the upper bound in 3.8 by
using 3.11 as

$\displaystyle\int_{\mathbb{R}^{n}}e^{-\frac{G}{\varepsilon}}\,dx$
$\displaystyle\leq\sum_{j=0}^{\infty}\int_{\\{j\varepsilon\leq
G<(j+1)\varepsilon\\}}e^{-\frac{G}{\varepsilon}}\,dx$
$\displaystyle\leq\sum_{j=0}^{\infty}|\\{j\varepsilon\leq
G<(j+1)\varepsilon\\}|e^{-j}$ $\displaystyle\leq
2^{n}\sum_{j=0}^{\infty}e^{-j}(j+1)^{n}|\\{G<\varepsilon\\}|\leq
C(n)|\\{G<\varepsilon\\}|.$

It remains to prove 3.9. Fix $\Lambda>1$. Then, for every
$x\in\\{G\geq\Lambda\varepsilon\\}$, it holds

$\displaystyle e^{-\frac{G(x)}{\varepsilon}}=e^{-\frac{\Lambda
G(x)}{\Lambda\varepsilon}}=\left(e^{-\frac{G(x)}{\Lambda\varepsilon}}\right)^{\Lambda}=\left(e^{-\frac{G(x)}{\Lambda\varepsilon}}\right)^{\Lambda-1}e^{-\frac{G(x)}{\Lambda\varepsilon}}\leq
e^{-\Lambda+1}e^{-\frac{G(x)}{\Lambda\varepsilon}}.$ (3.12)

Therefore we have, by 3.12, 3.8 and 3.11,

$\begin{split}\int_{\\{G\geq\Lambda\varepsilon\\}}e^{-\frac{G}{\varepsilon}}\,dx&\leq
e^{-\Lambda+1}\int_{\\{G\geq\Lambda\varepsilon\\}}e^{-\frac{G}{\Lambda\varepsilon}}\,dx\leq
e^{-\Lambda+1}\int_{\mathbb{R}^{n}}e^{-\frac{G}{\Lambda\varepsilon}}\,dx\\\
&\leq Ce^{-\Lambda}|\\{G<\Lambda\varepsilon\\}|\\\ &\leq
Ce^{-\Lambda}\Lambda^{n}|\\{G<\varepsilon\\}|\\\ &\leq
Ce^{-\Lambda}\Lambda^{n}\int_{\mathbb{R}^{n}}e^{-\frac{G}{\varepsilon}}\,dx\end{split}$
(3.13)

and the inequality 3.9 follows by using 1.15. ∎

######  Lemma 3.8.

Assume $G:\mathbb{R}^{n}\to\mathbb{R}$ is a function which has a proper global
minimum at the origin and $G(0)=0$. Furthermore, assume there is a constant
$C_{0}$ such that, for all $a>0$ and $\varepsilon>0$, it holds that

$\displaystyle\int_{G>a}e^{-G/\varepsilon}dx<C_{0}e^{-a/\varepsilon}.$ (3.14)

If there is a level $\varepsilon_{0}>0$ such that $G$ is convex on the
component of $\\{G(x)<\varepsilon_{0}\\}$ that contains $0$, then there is an
$\varepsilon_{1}(n,|\\{G<\varepsilon_{0}/2\\}|)<\varepsilon_{0}$ and a
constant $C=C(C_{0},n)>1$ such that, for all $\varepsilon<\varepsilon_{1}$, it
holds that

$\displaystyle
C^{-1}|\\{G<\varepsilon\\}|\leq\int_{\mathbb{R}^{n}}e^{-\frac{G}{\varepsilon}}\,dx\leq
C|\\{G<\varepsilon\\}|.$

###### Proof.

Since $G$ is convex in the level set $\\{G<\varepsilon_{0}\\}$, we know that
the level set $\\{G\leq\varepsilon_{0}/2\\}$ is convex and as such we can
extend the function $G$ outside that level set to a globally convex function.
This allows us to apply Lemma 3.7 and obtain

$\displaystyle\frac{1}{C}|\\{G<\varepsilon\\}|\leq\int_{\\{G<\varepsilon_{0}/2\\}}e^{-\frac{G}{\varepsilon}}\,dx\leq
C|\\{G<\varepsilon\\}|.$ (3.15)

Now, split the integral as

$\displaystyle\int
e^{-G/\varepsilon}dx=\int_{G\leq\varepsilon_{0}/2}e^{-G/\varepsilon}dx+\int_{G>\varepsilon_{0}/2}e^{-G/\varepsilon}dx.$

From 3.15 it follows that it suffices to bound the second integral on the
right hand side. Using 3.14 for $a=\varepsilon_{0}/2$ we get

$\displaystyle\int_{G>\varepsilon_{0}/2}e^{-G/\varepsilon}dx\leq
C_{0}e^{-(\varepsilon_{0}/2)/\varepsilon}.$

Since $G$ is convex in the level set $\\{G<\varepsilon_{0}/2\\}$, which is
again convex, it follows that we can construct a conical function
$\widetilde{G}$ as follows: For any
$\hat{x}\in\partial\\{G<\varepsilon_{0}/2\\}$ define
$\widetilde{G}(t\hat{x}/\|\hat{x}\|)=t\varepsilon_{0}/2$. The level sets of
$\widetilde{G}$ satisfy, for $\varepsilon<\varepsilon_{0}/2$,

$\displaystyle|\\{\widetilde{G}<\varepsilon\\}|\leq|\\{G<\varepsilon\\}|.$

However,

$\displaystyle|\\{\widetilde{G}<\epsilon\\}|=\left(\frac{\varepsilon}{\varepsilon_{0}/2}\right)^{n}|\\{\widetilde{G}<\varepsilon_{0}/2\\}|=\left(\frac{\varepsilon}{\varepsilon_{0}/2}\right)^{n}|\\{G<\varepsilon_{0}/2\\}|.$

Now, we can choose
$\varepsilon_{1}(n,C_{0},|\\{G<\varepsilon_{0}/2\\}|)<\varepsilon_{0}/2$ such
that for $\varepsilon<\varepsilon_{1}$ we have

$\displaystyle
e^{-(\varepsilon_{0}/2)/\varepsilon}\leq\left(\frac{\varepsilon}{\varepsilon_{0}}\right)^{n}|\\{G<\varepsilon_{0}/2\\}|.$

This means that for $\varepsilon<\varepsilon_{1}$ we also have

$\displaystyle\int_{G>\varepsilon_{0}/2}e^{-G/\varepsilon}dx\leq
C_{0}|\\{G<\varepsilon\\}|$

which together with 3.15 completes the proof. ∎

We conclude this section with the following technical lemma which is useful
when we study the potential near critical points.

######  Lemma 3.9.

Assume $G:\mathbb{R}^{n}\to\mathbb{R}$ is a convex function which has a proper
minimum at the origin and $G(0)=0$. Let $\omega:[0,\infty)\to[0,\infty)$ be as
in 1.4 and 1.6. Then for all $\delta\leq\delta_{0}$, we have

$\displaystyle\int_{\\{G<\delta\\}}e^{-\frac{G(x)}{\varepsilon}}e^{\frac{\pm\omega(G(x))}{\varepsilon}}\,dx\simeq\int_{\mathbb{R}^{n}}e^{-\frac{G(x)}{\varepsilon}}\,dx.$

###### Proof.

Denote $\Lambda_{\varepsilon}=\frac{\varepsilon_{1}}{\varepsilon}$ with
$\varepsilon_{1}$ as in 1.13. From 1.14 we know that
$\Lambda_{\varepsilon}\to\infty$ as $\varepsilon\to 0$. Now, by 3.9 in Lemma
3.7 and 1.14, we get

$\displaystyle\int_{\\{G<\varepsilon_{1}\\}}e^{-\frac{G(x)}{\varepsilon}}e^{\frac{\omega(G(x))}{\varepsilon}}\,dx\simeq\int_{\\{G<\Lambda_{\varepsilon}\varepsilon\\}}e^{-\frac{G(x)}{\varepsilon}}\,dx\simeq\int_{\mathbb{R}^{n}}e^{-\frac{G(x)}{\varepsilon}}\,dx.$
(3.16)

The lower bound follows immediately from this. In order to prove the upper
bound, note that $\omega(s)\leq s/2$ for all $s\leq\delta_{0}$ by assumption.
Therefore we can repeat the argument in 3.13 to get

$\displaystyle\int_{\\{\varepsilon_{1}<G<\delta\\}}e^{-\frac{G(x)}{\varepsilon}}e^{\frac{\omega(G(x))}{\varepsilon}}\,dx\leq\int_{\\{G>\Lambda_{\varepsilon}\varepsilon\\}}e^{-\frac{G(x)}{2\varepsilon}}\,dx\leq\eta(C\Lambda_{\varepsilon}^{-1})\int_{\mathbb{R}^{n}}e^{-\frac{G}{\varepsilon}}\,dx,$

which together with 3.16 yields the upper bound. ∎

##  4\. Proofs of Theorem 1 and Theorem 2

In this section we prove the capacity estimate in Theorem 1 and exit time
estimate in Theorem 2. Before we begin, we would like to remind the reader
that, as in Section 3, we will assume that $F$ satisfies our structural
assumptions and that all constants depend on the data, see the paragraph after
1.16.

We first study the geometric quantities $d_{\varepsilon}(A,B;\Omega)$ and
$V_{\varepsilon}(A,B;\Omega)$ defined in 1.8 and 1.9 and give a more explicit,
but less geometric, characterization. The characterization for the geodesic
distance $d_{\varepsilon}(A,B;\Omega)$ turns out to be much easier than for
the separating surface $V_{\varepsilon}(A,B;\Omega)$ and therefore we prove it
first.

######  Proposition 4.1.

Assume that $x_{a}$ and $x_{b}$ are local minimum points of $F$, let $U_{a}$
and $U_{b}$ be the islands, i.e., the components of the set $U_{-\delta/3}$,
containing $B_{\varepsilon}(x_{a})$ and $B_{\varepsilon}(x_{b})$ respectively.
Assume that $z$ is a saddle point in $Z_{x_{a},x_{b}}$, such that the bridge
$O_{z,\delta}$ connects $U_{a}$ and $U_{b}$, and denote $\Omega=U_{a}\cup
U_{b}\cup O_{z,\delta}$. Then it holds for $g_{z}$ given in 1.6 that

$\displaystyle
d_{\varepsilon}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b});\Omega)\simeq
e^{\frac{F(z)}{\varepsilon}}\int_{\mathbb{R}}e^{-\frac{g_{z}(x_{1})}{\varepsilon}}\,dx_{1}.$

###### Proof.

Begin by denoting $g=g_{z}$ and let us first prove the lower bound, i.e.,

$d_{\varepsilon}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b});\Omega)\geq(1-\hat{\eta}(C\varepsilon))e^{\frac{F(z)}{\varepsilon}}\int_{\mathbb{R}}e^{-\frac{g(x_{1})}{\varepsilon}}\,dx_{1}.$
(4.1)

To this aim we choose a smooth curve
$\gamma\in\mathcal{C}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b});\Omega)$
which, by assumptions, intersects the bridge $O_{z,\delta}$. We may choose the
coordinates in $\mathbb{R}^{n}$ such that $z=0$ and

$\displaystyle
O_{\delta}=O_{z,\delta}=\\{x_{1}:g(x_{1})<\delta\\}\times\\{x^{\prime}\in\mathbb{R}^{n-1}:G(x^{\prime})<\delta\\}.$

Moreover, by changing the potential from $F$ to $F-F(z)$ we may assume that
$F(0)=0$. Note that then it holds $U_{a},U_{b}\subset\\{F<-\delta/3\\}$ as
$F(z)\geq F(x_{a};x_{b})$.

Let us fix $s\in\mathbb{R}$ such that $g(s)<\frac{\delta}{10}$ and denote
$\Gamma_{s}=\\{s\\}\times\\{G<\delta\\}\subset O_{\delta}$. By the assumption
1.6 we have

$\displaystyle F>-\frac{\delta}{4}\qquad\text{on }\,\Gamma_{s}.$

In particular, since $U_{a},U_{b}\subset\\{F<-\delta/3\\}$, then the surface
$\Gamma_{s}$ does not intersect $U_{a}$ or $U_{b}$. Thus we conclude that
every
$\gamma\in\mathcal{C}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b});\Omega)$
intersects $\Gamma_{s}$, i.e.,
$\Gamma_{s}\in\mathcal{S}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b});\Omega)$.
Let us denote the projection to the $x_{1}$-axis by
$\pi_{1}:\mathbb{R}^{n}\to\mathbb{R}$, i.e. $\pi_{1}(x)=x_{1}$. From the
previous discussion we conclude that $s\in\pi_{1}\big{(}\gamma([0,1])\cap
O_{\delta}\big{)}$. This holds for every $s\in\\{g<\delta/10\\}$, and
therefore

$\displaystyle\\{g<\delta/10\\}\subset\pi_{1}\big{(}\gamma([0,1])\cap
O_{\delta}\big{)}.$ (4.2)

Now the assumption 1.6 implies that, in the set $O_{\delta}$, it holds that

$\begin{split}F(x)&\geq-g(x_{1})-\omega(g(x_{1}))+G(x^{\prime})-\omega(G(x^{\prime}))\\\
&\geq-g(x_{1})-\omega(g(x_{1}))+\frac{1}{2}G(x^{\prime})\geq-g(x_{1})-\omega(g(x_{1})).\end{split}$
(4.3)

Then for $\gamma_{1}=\pi_{1}(\gamma)$ we have by 4.2 and Lemma 3.9 that

$\displaystyle\int_{\\{t:\gamma(t)\in
O_{\delta}\\}}|\gamma^{\prime}|e^{\frac{F(\gamma)}{\varepsilon}}\,dt$
$\displaystyle\geq\int_{\\{t:\gamma(t)\in
O_{\delta}\\}}|\gamma_{1}^{\prime}|e^{\frac{-g(\gamma_{1})}{\varepsilon}}e^{\frac{-\omega(g(\gamma_{1}))}{\varepsilon}}\,dt$
$\displaystyle\geq\int_{\\{g<\delta/10\\}}e^{\frac{-g(x_{1})}{\varepsilon}}e^{-\frac{\omega(g(x_{1}))}{\varepsilon}}\,dx_{1}$
$\displaystyle\geq(1-\eta(C\varepsilon))\int_{\mathbb{R}}e^{-\frac{g(x_{1})}{\varepsilon}}\,dx_{1},$

proving 4.1.

To prove the upper bound, i.e.

$\displaystyle
d_{\varepsilon}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b});\Omega)\leq(1+\hat{\eta}(C\varepsilon))\int_{\mathbb{R}}e^{-\frac{g(x_{1})}{\varepsilon}}\,dx_{1}$

with $\hat{\eta}$ as in 1.16. we denote by $c_{-}<0<c_{+}$ the numbers such
that $g(c_{-})=g(c_{+})=\delta$. We first connect the points $x_{1}=(c_{-},0)$
and $x_{2}=(c_{+},0)$ by a segment $\gamma_{0}(t)=tx_{1}+(1-t)x_{2}$. Then it
holds by the assumption 1.6 and by Lemma 3.9 that

$\displaystyle\int_{\gamma_{0}}e^{-\frac{F(\gamma_{0})}{\varepsilon}}\,dt\leq\int_{\\{g<\delta\\}}e^{\frac{-g(x_{1})}{\varepsilon}}e^{\frac{\omega(g(x_{1}))}{\varepsilon}}\,dx_{1}\leq(1+\hat{\eta}(C\Lambda_{\varepsilon}^{-1}))\int_{\mathbb{R}}e^{-\frac{g(x_{1})}{\varepsilon}}\,dx_{1}.$

We then connect $x_{a}$ to $x_{1}$ and $x_{2}$ to $x_{b}$ with smooth curves
$\gamma_{1},\gamma_{2}\subset\\{x\in\Omega:F(x)<-\delta/3\\}$. Since it holds
$g(t)\leq C|t|$ we have $|\\{g<\varepsilon\\}|\geq c\,\varepsilon$. Therefore
it holds by Lemma 3.7 that

$\displaystyle\int_{\gamma_{i}}|\gamma_{i}^{\prime}|e^{-\frac{F(\gamma_{i})}{\varepsilon}}\,dt$
$\displaystyle\leq
e^{\frac{-\delta}{3\varepsilon}}\int_{\gamma_{i}}|\gamma_{i}^{\prime}|\,dt\leq
Ce^{\frac{-\delta}{3\varepsilon}}$
$\displaystyle\leq\hat{\eta}(C\varepsilon)|\\{g<\varepsilon\\}|\leq\hat{\eta}(C\varepsilon)\int_{\mathbb{R}}e^{-\frac{g(x_{1})}{\varepsilon}}\,dx_{1}.$

The constant in the last expression depends on the length of $\gamma_{i}$. We
can use a similar argument as in the proof of Lemma 3.2 to bound the length of
the curve. This time, we will however consider coverings with balls of size
comparable to $\delta$, as we are in the level set $\\{F<-\delta/3\\}$ we have
some room to replace our curve with another curve which has a length depending
on $\delta$ and $R$, while still retaining the same upper bound as above.

The upper bound now follows by joining the paths $\gamma_{1},\gamma_{0}$ and
$\gamma_{2}$, thus, constructing a competitor for the geodesic length. ∎

We need to prove similar result to Proposition 4.1, but for the separating
surface. This turns out to be trickier than the previous result for paths.

######  Proposition 4.2.

Assume that $x_{a},x_{b},z,U_{a},U_{b},O_{\delta}$ and $\Omega$ are as in
Proposition 4.1. Then it holds for $G_{z}$ from 1.6 that

$\displaystyle
V_{\varepsilon}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b});\Omega)\simeq
e^{-\frac{F(z)}{\varepsilon}}\int_{\mathbb{R}^{n-1}}e^{-\frac{G_{z}(x^{\prime})}{\varepsilon}}\,dx^{\prime}.$

###### Proof.

Denote $G_{z}=G$ for short. As in the proof of Proposition 4.1 we may assume
that $z=0$, $F(0)=0$ and that

$\displaystyle
O_{\delta}=O_{z,\delta}=\\{x_{1}:g(x_{1})<\delta\\}\times\\{x^{\prime}\in\mathbb{R}^{n-1}:G(x^{\prime})<\delta\\}.$

Let us begin by proving the upper bound. In the proof of Proposition 4.1 we
already observed that the surface $\Gamma_{0}=\\{0\\}\times\\{G<\delta\\}$ is
in the family of separating surfaces
$\Gamma_{0}\in\mathcal{S}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b});\Omega)$.
Therefore the assumption 1.6 and 3.9 together with the definition of
$V_{\varepsilon}$ imply

$\displaystyle\begin{split}V_{\varepsilon}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b});\Omega)&\leq\int_{\Gamma_{0}}e^{-\frac{F}{\varepsilon}}\,d\mathcal{H}^{n-1}\leq\int_{\\{G<\delta\\}}e^{-\frac{G}{\varepsilon}}e^{\frac{\omega(G)}{\varepsilon}}\,dx^{\prime}\\\
&\simeq\int_{\mathbb{R}^{n-1}}e^{-\frac{G(x^{\prime})}{\varepsilon}}\,dx^{\prime}.\end{split}$

The upper bound follows directly from this. Moreover by Lemma 3.7 it holds
that

$V_{\varepsilon}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b});\Omega)\leq
C|\\{G<\varepsilon\\}|\leq C.$ (4.4)

In order to prove the lower bound we fix a small $t>0$ and choose a smooth
hypersurface
$S\in\mathcal{S}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b});\Omega)$ such
that

$\displaystyle\int_{S}e^{-\frac{F}{\varepsilon}}\,d\mathcal{H}^{n-1}\leq
V_{\varepsilon}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b});\Omega)+t.$

Then $S$ divides the domain $\Omega$ into two different components, from which
we denote the component containing $x_{a}$ by $\hat{U}_{a}$. Note that then
$\partial\hat{U}_{a}\cap\Omega\subset S$. Denote $\rho=\varepsilon^{2}$. We
use an idea from [16] and instead of studying the set $\hat{U}_{a}$, we study
the density

$\displaystyle v_{\rho}(x):=\frac{|B_{\rho}(x)\cap\hat{U}_{a}|}{|B_{\rho}|}$

which can be written as a convolution,
$v_{\rho}(x)=\frac{1}{|B_{\rho}|}(\chi_{\hat{U}_{a}}*\chi_{B_{\rho}})$. To see
why studying $v_{\rho}$ is relevant, we need some setup that we will present
next.

$U_{b}$$U_{a}$$x_{b}$$B_{\varepsilon}(x_{b})$$x_{a}$$B_{\varepsilon}(x_{a})$$O_{\delta}$$O_{\delta}$$\hat{O}$
Figure 3. The bridge $O_{\delta}$ connects the sets $U_{a}$ and $U_{b}$. The
smaller bridge $\hat{O}$ has its lateral boundaries inside $U_{a}\cup U_{b}$.

We choose a subset $\hat{O}$ of the bridge $O_{\delta}$ as

$\displaystyle\hat{O}:=\\{x_{1}:g(x_{1})<\delta\\}\times\\{x^{\prime}\in\mathbb{R}^{n-1}:G(x^{\prime})<\delta/100\\},$

see Fig. 3, and denote its lateral boundaries by $\Gamma_{-}$ and
$\Gamma_{+}$, i.e.,

$\displaystyle\\{g=\delta\\}\times\\{G<\delta/100\\}=\Gamma_{-}\cup\Gamma_{+}.$

Now since $z\in Z_{x_{a},x_{b}}$, we have $F(z)\leq F(x_{a};x_{b})+\delta/3$.
Using this and 1.6 we deduce that
$\Gamma_{-}\cup\Gamma_{+}\subset\\{F<F(x_{a};x_{b})-\delta/3\\}$ and therefore
$\Gamma_{-}\cup\Gamma_{+}\subset U_{a}\cup U_{b}$. Moreover, by relabeling we
may assume that $\Gamma_{+}\subset U_{a}$ and $\Gamma_{-}\subset U_{b}$.
Furthermore, by the Lipschitz-continuity of $F$ we have $|F(x)-F(y)|\leq
c\varepsilon^{2}$ for all $y\in B_{\rho}(x)$. Note also that for all
$x\in\hat{O}$ and $y\in B_{\rho}(x)$ it holds $x-y\in\Omega$.

We will now relate $v_{\rho}$ to the surface integral of $S$ as follows:
Recall that the set $\hat{U}_{a}$ has smooth boundary in $\Omega$ and thus its
characteristic function is a BV-function. In particular, the derivative
$|\nabla\chi_{\hat{U}_{a}}|$ is a Radon measure in $\Omega$ and

$\displaystyle\int_{\Omega}|\nabla\chi_{\hat{U}_{a}}|e^{-\frac{F}{\varepsilon}}\,dx=\int_{\partial\hat{U}_{a}\cap\Omega}e^{-\frac{F}{\varepsilon}}\,d\mathcal{H}^{n-1}\leq\int_{S}e^{-\frac{F}{\varepsilon}}\,d\mathcal{H}^{n-1}.$
(4.5)

Using the definition of $v_{\rho}$ and the Lipschitzness of $F$ inside
$B_{\rho}(x)$, we may thus estimate

$\displaystyle\begin{split}\int_{\hat{O}}|\nabla
v_{\rho}|e^{-\frac{F(x)}{\varepsilon}}\,dx&\leq\frac{1}{|B_{\rho}|}\int_{\hat{O}}e^{-\frac{F(x)}{\varepsilon}}\int_{\mathbb{R}^{n}}|\nabla\chi_{\hat{U}_{a}}(y)||\chi_{B_{\rho}}(x-y)|\,dydx\\\
&\leq\frac{1}{|B_{\rho}|}\int_{\hat{O}}e^{c\varepsilon}\int_{\mathbb{R}^{n}}e^{-\frac{F(y)}{\varepsilon}}|\nabla\chi_{\hat{U}_{a}}(y)||\chi_{B_{\rho}}(x-y)|\,dydx\\\
&\leq(1+C\varepsilon)\int_{\Omega}|\nabla\chi_{\hat{U}_{a}}(y)|e^{-\frac{F(y)}{\varepsilon}}\,dy.\end{split}$
(4.6)

Putting together 4.5 and 4.6 we see that it is enough to establish a lower
bound on the integral of $|\nabla v_{\rho}|$ in $\hat{O}$. In order to achieve
this, we first claim that for all $x$ such that $B_{\rho}(x)\subset\Omega$ we
have, when $\epsilon$ is small,

$v_{\rho}(x)\geq 1-C\varepsilon\,\,\text{ for }\,x\in U_{a}\quad\text{and
}\quad v_{\rho}(x)\leq C\varepsilon\,\,\text{ for }\,x\in U_{b}.$ (4.7)

We now complete the proof of the lower bound, using 4.7, followed by the proof
of 4.7. Assume now 4.7. Then we can use the fundamental theorem of calculus to
get that for all $x^{\prime}\in\\{G<\delta/100\\}$

$1-2C\varepsilon\leq\int_{\\{g<\delta\\}}\partial_{x_{1}}v_{\rho}(x_{1},x^{\prime})\,dx_{1}.$
(4.8)

Now, arguing as in 4.3 we conclude that

$\displaystyle F(x)\leq G(x^{\prime})+\omega(G(x^{\prime}))\qquad\text{for
}\,x\in O_{\delta}.$ (4.9)

Multiplying and dividing with $e^{-\frac{F(x)}{\varepsilon}}$ inside the
integral in 4.8 and using 4.9 we get

$\displaystyle(1-2C\varepsilon)e^{-(G(x^{\prime})+\omega(G(x^{\prime})))/\epsilon}\leq\int_{\\{G<\delta/100\\}}\int_{\\{g<\delta\\}}|\partial_{x_{1}}v_{\rho}(x_{1},x^{\prime})|e^{-F(x)/\epsilon}\,dx_{1}.$

Integrating over $x^{\prime}\in\\{G<\delta/100\\}$ we obtain

$\displaystyle\begin{split}(1-2C\varepsilon)&\int_{\\{G<\delta/100\\}}e^{-\frac{G(x^{\prime})}{\varepsilon}}e^{-\frac{\omega(G(x^{\prime}))}{\varepsilon}}\,dx^{\prime}\\\
&\leq\int_{\\{G<\delta/100\\}}\int_{\\{g<\delta\\}}|\partial_{x_{1}}v_{\rho}(x_{1},x^{\prime})|e^{-\frac{F(x)}{\varepsilon}}\,dx_{1}dx^{\prime}\\\
&\leq\int_{\hat{O}}|\nabla
v_{\rho}|e^{-\frac{F(x)}{\varepsilon}}\,dx.\end{split}$

The lower bound on the integral on the right hand side follows by Lemma 3.9,
i.e. we have

$(1-\eta(\varepsilon))\int_{\mathbb{R}^{n-1}}e^{-\frac{G(x^{\prime})}{\varepsilon}}\,dx^{\prime}\leq\int_{\hat{O}}|\nabla
v_{\rho}|e^{-\frac{F(x)}{\varepsilon}}\,dx.$ (4.10)

Now, assuming 4.7, we may use 4.10, 4.5 and 4.6 to get the lower bound from

$\displaystyle\begin{split}(1-\eta(\varepsilon))\int_{\mathbb{R}^{n-1}}e^{-\frac{G(x^{\prime})}{\varepsilon}}\,dx^{\prime}&\leq(1+C\varepsilon)\int_{S}e^{-\frac{F}{\varepsilon}}\,d\mathcal{H}^{n-1}\\\
&\leq(1+C\varepsilon)\big{(}V_{\varepsilon}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b});\Omega)+t\big{)}\end{split}$

as $t$ is arbitrarily small. Thus we obtain the lower bound, and hence in
order to complete the proof it remains to prove 4.7. For this, we fix $x\in
U_{a}\cup U_{b}$ such that $B_{\rho}(x)\subset\Omega$. By the relative
isoperimetric inequality (also called Dido’s problem, see for instance [2,
Theorem 3.40] or [18]) and by $\rho=\varepsilon^{2}$ it holds

$\displaystyle\begin{split}\mathcal{H}^{n-1}\big{(}\partial\hat{U}_{a}\cap
B_{\rho}(x)\big{)}&\geq
c\min\big{\\{}|B_{\rho}(x)\cap\hat{U}_{a}|^{\frac{n-1}{n}},|B_{\rho}(x)\setminus\hat{U}_{a}|^{\frac{n-1}{n}}\big{\\}}\\\
&\geq
c\,\varepsilon^{2(n-1)}\min\big{\\{}v_{\rho}(x),1-v_{\rho}(x)\big{\\}}^{\frac{n-1}{n}}.\end{split}$

On the other hand, since $x\in\\{F<-\delta/3\\}$ and thus
$B_{\rho}(x)\subset\\{F<-\delta/4\\}$, we have by 4.4 that

$\displaystyle\begin{split}\mathcal{H}^{n-1}\big{(}\partial\hat{U}_{a}\cap
B_{\rho}(x)\big{)}&\leq
e^{-\frac{\delta}{4\varepsilon}}\int_{\partial\hat{U}_{a}\cap
B_{\rho}(x)}e^{-\frac{F}{\varepsilon}}\,d\mathcal{H}^{n-1}\\\ &\leq
e^{-\frac{\delta}{4\varepsilon}}\int_{S}e^{-\frac{F}{\varepsilon}}\,d\mathcal{H}^{n-1}\\\
&\leq
e^{-\frac{\delta}{4\varepsilon}}\big{(}V_{\varepsilon}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b});\Omega)+t\big{)}\\\
&\leq Ce^{-\frac{\delta}{4\varepsilon}}.\end{split}$

By combining the two inequalities above we obtain 4.7 which completes the
whole proof. ∎

### Proof of Theorem 1

We consider parallel case and series case separately.

#### Parallel case:

Assume that the saddle points in $F_{x_{a},x_{b}}=\\{z_{1},\dots,z_{N}\\}$ are
parallel, see Fig. 2. Let us fix a saddle point $z_{i}\in F_{x_{a},x_{b}}$ and
recall the definition of the bridge $O_{z_{i},\delta}$ in 1.7. As before, by
considering $F-F(z_{i})$ instead of $F$, we may assume

$\displaystyle z_{i}=0,\quad F(0)=0$

and

$\displaystyle
O_{\delta}:=O_{0,\delta}=\\{x_{1}\in\mathbb{R}:g(x_{1})<\delta\\}\times\\{x^{\prime}\in\mathbb{R}^{n-1}:G(x^{\prime})<\delta\\}.$

We also recall the notation $U_{-\delta/3}=\\{F<F(x_{a};x_{b})-\delta/3\\}$.
We denote the island, i.e., the component of $U_{-\delta/3}$, which contains
the point $x_{a}$ by $U_{a}$ and the island which contains the point $x_{b}$
by $U_{b}$. Since the saddle points are parallel, the bridge $O_{\delta}$
connects the islands $U_{a}$ and $U_{b}$ which means that the set
$\Omega=U_{a}\cup U_{b}\cup O_{\delta}$ is open and connected. By Lemma 3.5 we
have $\text{osc}_{U_{a}}(h_{A,B})+\text{osc}_{U_{b}}(h_{A,B})\leq
C\varepsilon$. Since $h_{A,B}=1$ in $B_{\varepsilon}(x_{a})\subset U_{a}$ and
$h_{A,B}=0$ in $B_{\varepsilon}(x_{b})\subset U_{b}$, it follows that

$h_{A,B}\geq 1-C\varepsilon\,\,\text{in }\,U_{a}\quad\text{and}\quad
h_{A,B}\leq C\varepsilon\,\,\text{in }\,U_{b}.$ (4.11)

Let us choose a subset $\hat{O}$ of the bridge $O_{\delta}$ as in the proof of
Proposition 4.2 (see Fig. 3), i.e.

$\displaystyle\hat{O}:=\\{x_{1}:g(x_{1})<\delta\\}\times\\{x^{\prime}\in\mathbb{R}^{n-1}:G(x^{\prime})<\delta/100\\},$

and denote its lateral boundaries by $\Gamma_{-}\subset\\{x_{1}<0\\}$ and
$\Gamma_{+}\subset\\{x_{1}>0\\}$, i.e.,

$\displaystyle\\{g=\delta\\}\times\\{G<\delta/100\\}=\Gamma_{-}\cup\Gamma_{+}.$

Then by using 1.6 and arguing as in the proof of Proposition 4.2 we deduce
that $\Gamma_{-},\Gamma_{+}\subset\\{F<F(x_{a};x_{b})-\delta/3\\}$ and we may
assume $\Gamma_{+}\subset U_{a}$ and $\Gamma_{-}\subset U_{b}$. Therefore, by
4.11, we have that $h_{A,B}\leq C\varepsilon$ on $\Gamma_{-}$ and $h_{A,B}\geq
1-C\varepsilon$ on $\Gamma_{+}$. Now, by the fundamental theorem of calculus
and Cauchy-Schwarz inequality, it holds that

$\begin{split}1-2C\varepsilon&\leq\int_{\\{g<\delta\\}}\partial_{x_{1}}h_{A,B}(x)\,dx_{1}=\int_{\\{g<\delta\\}}\partial_{x_{1}}h_{A,B}(x)e^{-\frac{F(x)}{2\varepsilon}}e^{\frac{F(x)}{2\varepsilon}}\,dx_{1}\\\
&\leq\left(\int_{\\{g<\delta\\}}|\nabla
h_{A,B}(x)|^{2}e^{-\frac{F(x)}{\varepsilon}}\,dx_{1}\right)^{\frac{1}{2}}\left(\int_{\\{g<\delta\\}}e^{\frac{F(x)}{\varepsilon}}\,dx_{1}\right)^{\frac{1}{2}}.\end{split}$
(4.12)

Let us next estimate the last term above. By assumption 1.6 we have, for
$x\in\hat{O}$,

$\displaystyle
F(x)\leq-g(x_{1})+\omega(g(x_{1}))+G(x^{\prime})+\omega(G(x^{\prime})).$

Therefore, by Lemmas 3.9 and 4.1, we can estimate

$\begin{split}\int_{\\{g<\delta\\}}e^{\frac{F(x)}{\varepsilon}}\,dx_{1}&\leq(1+\hat{\eta}(C\varepsilon))e^{\frac{G(x^{\prime})}{\varepsilon}}e^{\frac{\omega(G(x^{\prime}))}{\varepsilon}}\int_{\\{g<\delta\\}}e^{-\frac{g(x_{1})}{\varepsilon}}e^{\frac{\omega(g(x_{1}))}{\varepsilon}}\,dx_{1}\\\
&\leq(1+\hat{\eta}(C\varepsilon))e^{\frac{G(x^{\prime})}{\varepsilon}}e^{\frac{\omega(G(x^{\prime}))}{\varepsilon}}\int_{\mathbb{R}}e^{-\frac{g(x_{1})}{\varepsilon}}\,dx_{1}\\\
&\leq(1+\hat{\eta}(C\varepsilon))e^{\frac{G(x^{\prime})}{\varepsilon}}e^{\frac{\omega(G(x^{\prime}))}{\varepsilon}}d_{\varepsilon}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b});\Omega).\end{split}$
(4.13)

We combine the inequalities 4.12 and 4.13 leading to (for another constant
$C$)

$\displaystyle\int_{\\{g<\delta\\}}|\nabla
h_{A,B}(x)|^{2}e^{-\frac{F(x)}{\varepsilon}}\,dx_{1}\geq\frac{(1-\hat{\eta}(C\varepsilon))}{d_{\varepsilon}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b});\Omega)}e^{-\frac{G(x^{\prime})}{\varepsilon}}e^{-\frac{\omega(G(x^{\prime}))}{\varepsilon}}$

for all $x^{\prime}\in\\{G<\delta/100\\}$. By integrating over
$x^{\prime}\in\\{G<\delta/100\\}$ we have by Fubini’s theorem, Lemmas 3.9 and
4.2, that

$\begin{split}\int_{\hat{O}}|\nabla
h_{A,B}|^{2}e^{-\frac{F(x)}{\varepsilon}}\,dx&\geq\frac{(1-\hat{\eta}(C\varepsilon))}{d_{\varepsilon}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b});\Omega)}\int_{\\{G<\delta/100\\}}e^{-\frac{G(x^{\prime})}{\varepsilon}}e^{-\frac{\omega(G(x^{\prime}))}{\varepsilon}}\,dx^{\prime}\\\
&\geq\frac{(1-\hat{\eta}(C\varepsilon))}{d_{\varepsilon}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b});\Omega)}\int_{\mathbb{R}^{n-1}}e^{-\frac{G(x^{\prime})}{\varepsilon}}\,dx^{\prime}\\\
&\geq(1-\hat{\eta}(C\varepsilon))\frac{V_{\varepsilon}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b});\Omega)}{d_{\varepsilon}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b});\Omega)}.\end{split}$
(4.14)

Therefore, by repeating the argument for every saddle $z_{i}\in
Z_{x_{a},x_{b}}$ and using the fact that the bridges $O_{z_{i},\delta}$ are
disjoint, we obtain after scaling back the potential

$\displaystyle\begin{split}\int_{\mathbb{R}^{n}}|\nabla
h_{A,B}|^{2}e^{-\frac{F(x)}{\varepsilon}}\,dx\geq(1-\hat{\eta}(C\varepsilon))\sum_{i=1}^{N}\frac{V_{\varepsilon}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b});\Omega)}{d_{\varepsilon}(B_{\varepsilon}(x_{a}),B_{\varepsilon}(x_{b});\Omega)}e^{\frac{F(z_{i})}{\varepsilon}}.\end{split}$

This yields the lower bound when the saddle points are parallel.

For the upper bound, we only give a sketch of the argument as it is fairly
straightforward. The idea is to contruct a competitor $h$ in the variational
characterization of the capacity, see 2.5. Let us first define $h$ in the set
$U_{\delta/3}=\\{F<F(x_{a};x_{b})+\delta/3\\}$. Since the saddle points
$Z_{x_{a},x_{b}}=\\{z_{1},\dots,z_{N}\\}$ are parallel, it follows that the
points $x_{a}$ and $x_{b}$ lie in different components of the set

$\displaystyle\widetilde{U}=U_{\delta/3}\setminus\bigcup_{i=1}^{N}O_{z_{i},\delta},$

where $O_{z_{i},\delta}$ is defined in 1.8. Denote the components of
$\widetilde{U}$ containing $x_{a}$ and $x_{b}$ by $\widetilde{U}_{a}$ and
$\widetilde{U}_{b}$, respectively. We define first

$\displaystyle h=1\,\text{ in }\,\widetilde{U}_{a}\quad\text{and}\quad
h=0\,\text{ in }\,\widetilde{U}_{b}.$

Let us next fix a saddle point $z_{i}\in Z_{x_{a},x_{b}}$. As before, we may
again assume that

$\displaystyle z_{i}=0,\quad F(0)=0$

and

$\displaystyle
O_{\delta}:=O_{0,\delta}=\\{x_{1}\in\mathbb{R}:g(x_{1})<\delta\\}\times\\{x^{\prime}\in\mathbb{R}^{n-1}:G(x^{\prime})<\delta\\}.$

Moreover, we may assume that

$\displaystyle\widetilde{U}_{a}\cap\partial
O_{\delta}\subset\\{x_{1}>0\\}\quad\text{
and}\quad\widetilde{U}_{b}\cap\partial O_{\delta}\subset\\{x_{1}<0\\}.$

Let $c_{-}<0<c_{+}$ be numbers such that $g(c_{-})=g(c_{+})=\delta/100$. We
define $h(x)=\varphi(x_{1})$ in $O_{\delta}$ such that the function
$\varphi:[c_{-},c_{+}]\to\mathbb{R}$ is a solution of the ordinary
differential equation

$\displaystyle\frac{d}{ds}\left(\varphi^{\prime}(s)e^{\frac{g(s)}{\varepsilon}}\right)=0\quad\text{in
}\,(c_{-},c_{+})$

with boundary values $\varphi(c_{-})=0$ and $\varphi(c_{+})=1$. We extend
$\varphi$ into $\mathbb{R}$ by setting $\varphi(s)=0$ for $s\leq c_{-}$ and
$\varphi(s)=1$ for $s\geq c_{+}$. It follows that for the function $h$ we
have, by construction, Lemma 3.9, and an argument similar to the one leading
to 4.12, that

$\displaystyle\begin{split}\int_{O_{\delta}}&|\nabla
h|^{2}e^{-\frac{F(x)}{\varepsilon}}\,dx\\\
&\leq\int_{\\{g<\delta\\}}|\varphi^{\prime}(x_{1})|^{2}e^{\frac{g(x_{1})}{\varepsilon}}e^{\frac{\omega(g(x_{1}))}{\varepsilon}}\,dx_{1}\int_{\\{G<\delta\\}}e^{-\frac{G(x^{\prime})}{\varepsilon}}e^{\frac{\omega(G(x^{\prime}))}{\varepsilon}}\,dx^{\prime}\\\
&\leq(1+\hat{\eta}(C\varepsilon))\left(\int_{\mathbb{R}}e^{-\frac{g(x_{1})}{\varepsilon}}\,dx_{1}\right)^{-1}\left(\int_{\mathbb{R}^{n-1}}e^{-\frac{G(x^{\prime})}{\varepsilon}}\,dx^{\prime}\right).\end{split}$

By repeating the construction for every saddle point $z_{i}\in
Z_{x_{a},x_{b}}$, we obtain a function which is defined in $U_{\delta/3}$. We
denote this function by $h:U_{\delta/3}\to\mathbb{R}$. Note that now for $h$
the estimate 4.14 is optimal. Moreover, $h$ is Lipschitz continuous. We extend
$h$ to $\mathbb{R}^{n}$ without increasing the Lipschitz constant $L$, e.g.,
by defining

$\displaystyle h(x)=\sup_{y\in
U_{\delta/3}}\big{(}h(y)-L|x-y|\big{)}\quad\text{for
}\,x\in\mathbb{R}^{n}\setminus U_{\delta/3}.$

This finally leads to the upper bound completing the proof of the parallel
case, while we leave the final details on the upper bound for the reader.

#### Series case:

Assume that the saddle points $Z_{x_{a},x_{b}}=\\{z_{1},\dots,z_{N}\\}$ are in
series, see Fig. 2. We use the ordering as in 1.11 and denote the points
$x_{i}$ as in 1.12. We also fix the islands, $U_{x_{i-1}}$ and $U_{x_{i}}$
(components of $\\{F<F(x_{a};x_{b})-\delta/3\\}$), which are connected by the
bridge $O_{z_{i},\delta}$. Again we may assume that $z_{i}=0$, $F(0)=0$ and
that

$\displaystyle
O_{\delta}=O_{z_{i},\delta}=\\{x_{1}:g(x_{1})<\delta\\}\times\\{x^{\prime}:G(x_{1})<\delta\\}.$

By Lemma 3.5 we have
$\text{osc}_{U_{x_{i-1}}}(h_{A,B})+\text{osc}_{U_{x_{i}}}(h_{A,B})\leq
C\varepsilon$. Therefore there are numbers $c_{i-1},c_{i}$ such that

$\displaystyle h_{A,B}\geq c_{i-1}-C\varepsilon\,\,\text{in
}\,U_{x_{i-1}}\quad\text{and}\quad h_{A,B}\leq c_{i}+C\varepsilon\,\,\text{in
}\,U_{x_{i}}.$

Then, using the fundamental theorem of calculus as in 4.12, we obtain

$\displaystyle\begin{split}|c_{i-1}-c_{i}|-2C\varepsilon\leq\left(\int_{\\{g<\delta\\}}|\nabla
h_{A,B}(x)|^{2}e^{-\frac{F(x)}{\varepsilon}}\,dx_{1}\right)^{\frac{1}{2}}\left(\int_{\\{g<\delta\\}}e^{\frac{F(x)}{\varepsilon}}\,dx_{1}\right)^{\frac{1}{2}}\end{split}$

for

$\displaystyle(x_{1},x^{\prime})\in\\{g<\delta\\}\times\\{G<\delta/100\\}.$

Moreover, arguing as in 4.13, we have

$\displaystyle\int_{\\{g<\delta\\}}e^{\frac{F(x)}{\varepsilon}}\,dx_{1}\leq(1+\hat{\eta}(C\varepsilon))e^{\frac{G(x^{\prime})}{\varepsilon}}e^{\frac{\omega(G(x^{\prime}))}{\varepsilon}}d_{\varepsilon}(x_{i-1},x_{i}).$

These together imply

$\displaystyle\int_{\\{g<\delta\\}}|\nabla
h_{A,B}(x)|^{2}e^{-\frac{F(x)}{\varepsilon}}\,dx_{1}\geq\frac{(1-\hat{\eta}(C\varepsilon))(c_{i-1}-c_{i})^{2}}{d_{\varepsilon}(x_{i-1},x_{i})}e^{-\frac{G(x^{\prime})}{\varepsilon}}e^{-\frac{\omega(G(x^{\prime}))}{\varepsilon}}.$

By integrating over $x^{\prime}\in\\{G<\delta/100\\}$ we have, by Fubini’s
theorem, Lemma 3.9, and Proposition 4.2, that

$\displaystyle\int_{O_{\delta}}|\nabla
h_{A,B}|^{2}e^{-\frac{F(x)}{\varepsilon}}\,dx\geq(1-\hat{\eta}(C\varepsilon))(c_{i-1}-c_{i})^{2}\frac{V_{\varepsilon}(x_{i-1},x_{i})}{d_{\varepsilon}(x_{i-1},x_{i})}.$

By repeating the argument for every saddle $z_{i}\in Z_{x_{a},x_{b}}$ and
using the fact that the sets $O_{z_{i},\delta}$ are disjoint we obtain

$\displaystyle\int_{\mathbb{R}^{n}}|\nabla
h_{A,B}|^{2}e^{-\frac{F(x)}{\varepsilon}}\,dx\geq(1-\hat{\eta}(C\varepsilon))\sum_{i=1}^{N}(c_{i-1}-c_{i})^{2}\frac{V_{\varepsilon}(x_{i-1},x_{i})}{d_{\varepsilon}(x_{i-1},x_{i})}.$

Recall that the numbers $c_{i}$ are the approximate values of $h_{A,B}$ in the
components $U_{x_{i}}$. Therefore we may choose them such that $1=c_{0}$ and
$c_{N}=0$. By denoting $y_{i}=c_{i-1}-c_{i}$ and
$a_{i}=\frac{V_{\varepsilon}(x_{i-1},x_{i})}{d_{\varepsilon}(x_{i-1},x_{i})}$
we may write

$\displaystyle\sum_{i=1}^{N}(c_{i-1}-c_{i})^{2}\frac{V_{\varepsilon}(x_{i-1},x_{i})}{d_{\varepsilon}(x_{i-1},x_{i})}=\sum_{i=1}^{N}a_{i}y_{i}^{2},$

where we have a constraint $\sum_{i=1}^{N}y_{i}=1$. By a standard optimization
argument (using Lagrange multipliers) we get that under such a constraint it
holds that

$\displaystyle\sum_{i=1}^{N}a_{i}y_{i}^{2}\geq\left(\sum_{i=1}^{N}\frac{1}{a_{i}}\right)^{-1}.$

This yields the lower bound in the case when the saddle points are in series.
The upper bound on the other hand follows from a similar argument than in the
parallel case, and we leave the details for the reader. This completes the
proof in the series case, and hence the whole proof.

∎

### Proof of Theorem 2

Let us first recall the notation related to Theorem 2. We assume that the
local minimas $x_{i}$ of $F$ are ordered such that $F(x_{i})\leq F(x_{j})$ if
$i\leq j$, and they are grouped into sets $G_{i}$ such that $x_{i},x_{j}\in
G_{k}$ if $F(x_{i})=F(x_{j})$ and $x\in G_{i}$, $y\in G_{j}$ with $i<j$ if
$F(x)<F(y)$. We also denoted $F(G_{i}):=F(x)$ with $x\in G_{i}$,
$S_{k}=\bigcup_{i=1}^{k}G_{i}$, $G_{k}^{\varepsilon}=\bigcup_{x\in
G_{k}}B_{\varepsilon}(x)$, and
$S_{k}^{\varepsilon}=\bigcup_{i=1}^{k}G_{i}^{\varepsilon}$.

The proof of Theorem 2 follows from the following lemma together with Lemma
3.6 and Theorem 1.

######  Lemma 4.3.

Under the assumptions of Theorem 2, there exists constants $C=C(F)>1$ and
$\varepsilon=\varepsilon_{0}(F)\in(0,1)$ such that, for all
$0<\varepsilon\leq\varepsilon_{0}$, we have

$\displaystyle\frac{1}{C}\sum_{x\in G_{k+1}}|O_{x,\varepsilon}|\leq
e^{\frac{F(G_{k+1})}{\varepsilon}}\int
h_{G_{k+1}^{\varepsilon},S_{k}^{\varepsilon}}d\mu_{\epsilon}\leq C\sum_{x\in
G_{k+1}}|O_{x,\varepsilon}|.$

We prove Theorem 2 first, while the proof of Lemma 4.3 is given later on.

###### Proof of Theorem 2.

Using Lemma 3.6 and choosing $\rho=\varepsilon$ we obtain that, for
$\varepsilon$ small enough and $x\in G_{k+1}^{\varepsilon}$, that

$\displaystyle\mathbb{E}^{x}[\tau_{S_{k}^{\varepsilon}}\mathbb{I}_{\tau_{S_{k}^{\varepsilon}}<\tau_{\Omega^{c}}}]\leq
C\frac{{\displaystyle\int}h_{{S_{k}^{\varepsilon}},{\Omega^{c}}}h_{G_{k+1}^{\varepsilon},{S_{k}^{\varepsilon}}}d\mu_{\varepsilon}}{\operatorname{cap}(G_{k+1}^{\varepsilon},{S_{k}^{\varepsilon}})}+C\varepsilon^{\alpha/2}.$

The ratio above can be estimated by using Lemma 4.3 and the monotonicity of
the capacity. That is, the numerator can be bounded by Lemma 4.3, while for
the capacity we have

$\displaystyle\operatorname{cap}(G_{k+1}^{\varepsilon},{S_{k}^{\varepsilon}})\geq\operatorname{cap}(G_{k+1}^{\varepsilon},{G_{k}^{\varepsilon}})\geq\max_{x\in
G_{k},y\in
G_{k+1}}\operatorname{cap}(B_{\varepsilon}(x),{B_{\varepsilon}(y)}).$ (4.15)

The claim for the parallel and series cases now follows by assuming that the
maximum is attained for a pair of minimas $x_{a}\in G_{k}$, $x_{b}\in G_{k+1}$
and applying Theorem 1. ∎

######  Remark 4.4.

We note that in the general case, the last inequality in 4.15 has the optimal
dependence with respect to $\varepsilon$ but the inequalities may differ by a
constant. Essentially the inequality is sharp only in the case where only one
saddle contributes to the total value of the capacity. Hence we have the sharp
estimate when saddle points are parallel or in series, but in general the
situation might be more complicated than that. We have illustrated this in
Fig. 4, where each gray dot is a saddle at the same height, and $A,B$ produces
$G_{k}$. Then the precise value of
$\operatorname{cap}(G_{k+1}^{\varepsilon},A\cup B)$ is already non-trivial to
calculate.

$G_{k+1}^{\varepsilon}$$A$$B$$O_{1}$$O_{2}$$O_{3}$ Figure 4. Geometric view of
multiple minima at same height and multiple saddles at the same height

###### Proof of Lemma 4.3.

First we will prove a localization estimate for exponential integrals.
Consider a set $0\in O$ and a function $f$ such that $f(0)=l$ is a proper
local minimum and that $f$ is locally convex around $0$. Then there exists an
$\varepsilon_{0}$ such that, for any $\varepsilon<\varepsilon_{0}$,

$\displaystyle\int_{O}e^{-f(x)/\varepsilon}dx\leq
Ce^{-l/\varepsilon}|\\{f<\varepsilon\\}\cap O|.$ (4.16)

We will first prove 4.16 and then repeatedly apply it to prove Lemma 4.3.

In order to prove 4.16, we begin by rescaling such that $l=0$. Then we extend
$f$ outside $O$ as $+\infty$ and call this extended function $\hat{f}$. We
first prove

$\displaystyle\int_{\\{\hat{f}>a\\}}e^{-\hat{f}(x)/\varepsilon}dx\leq
ce^{-a/\varepsilon}$

which, by the definition of $\hat{f}$, is equivalent to

$\displaystyle\int_{\\{f>a\\}\cap O}e^{-f(x)/\varepsilon}dx\leq
ce^{-a/\varepsilon}.$

This now follows from Lemma 3.8 by using $|O|<\infty$ and observing that
$\hat{f}$ satisfies the assumptions of Lemma 3.8. Hence we observe 4.16.

Consider now the set

$\displaystyle U_{-\delta_{2}/3}\equiv\\{y:F(y)\leq
F(G_{k+1};S_{k})-\delta_{2}/3\\}$

and let $U_{i}$ be the component of $U_{-\delta_{2}/3}$ containing $x_{i}$. We
split

$\displaystyle\int
h_{G_{k+1}^{\varepsilon},S_{k}^{\varepsilon}}e^{-F/\varepsilon}dx=\int_{U_{-\delta_{2}/3}^{c}}h_{G_{k+1}^{\varepsilon},S_{k}^{\varepsilon}}e^{-F/\varepsilon}dx+\int_{U_{-\delta_{2}/3}\setminus
S_{k}^{\varepsilon}}h_{G_{k+1}^{\varepsilon},S_{k}^{\varepsilon}}e^{-F/\varepsilon}dx,$

where complement is understood with respect to the domain $\Omega$. By
assumptions (1.4) and (1.6) on $F$ it holds that

$F(G_{k+1};S_{k})\geq F(G_{k+1})+\tfrac{2}{3}\delta_{2}.$ (4.17)

Also by the quadratic growth (1.3) we can bound the first integral as

$\displaystyle\int_{U_{-\delta_{2}/3}^{c}}h_{G_{k+1}^{\varepsilon},S_{k}^{\varepsilon}}e^{-F/\varepsilon}dx\leq
Ce^{-(F(G_{k+1};S_{k})-\delta_{2}/3)/\varepsilon}\leq
Ce^{-\frac{\delta_{2}}{3\varepsilon}}e^{-F(G_{k+1})/\varepsilon},$

which shows that the first integral is neglible in the final estimate. For the
second integral we further split

$\displaystyle\int_{U_{-\delta_{2}/3}\setminus
S_{k}^{\varepsilon}}h_{G_{k+1}^{\varepsilon},S_{k}^{\varepsilon}}e^{-F/\varepsilon}dx=\sum_{i}\int_{U_{i}\setminus
S_{k}^{\varepsilon}}h_{G_{k+1}^{\varepsilon},S_{k}^{\varepsilon}}e^{-F/\varepsilon}dx.$

We will now consider all the different components $U_{i}$ depending on what
minimas they contain. We start with the components $U_{i}$ that do not
intersect $S_{k}^{\varepsilon}\cup G_{k+1}^{\varepsilon}$. Then all local
minimas in $U_{i}$ are larger than $F(G_{k+1})$, and hence from 4.16 we get
that there exists a constant $C$ such that

$\displaystyle\int_{U_{i}}h_{G_{k+1}^{\varepsilon},S_{k}^{\varepsilon}}e^{-F/\varepsilon}dx\leq
Ce^{-F(G_{k+2})/\varepsilon}\leq
Ce^{-\delta_{2}/\varepsilon}e^{-F(G_{k+1})/\varepsilon},$

where the last inequality follows from (1.19). This shows that also this term
is neglible. Consider next the component $U_{i}$ that intersects
$G^{\varepsilon}_{k+1}$ but do not intersect $S_{k}^{\varepsilon}$. In this
case, by 4.16 and Lemmas 3.7 and 3.5, we have

$\displaystyle\frac{1}{C}\sum_{x\in G_{k+1}\cap U_{i}}|O_{x,\varepsilon}|\leq
e^{\frac{F(G_{k+1})}{\varepsilon}}\int_{U_{i}}h_{G_{k+1}^{\varepsilon},S_{k}^{\varepsilon}}e^{-F/\varepsilon}dx\leq
C\sum_{x\in G_{k+1}\cap U_{i}}|O_{x,\varepsilon}|$ (4.18)

providing us the leading term that contributes to the final estimate.

Consider next a component $U_{i}$ such that $U_{i}\cap
S_{k}^{\varepsilon}\neq\emptyset$. Since $U_{i}$ is a component of
$U_{-\delta_{2}/3}$, it follows from $U_{i}\cap S_{k}\neq\emptyset$ that
$F(y;S_{k})\leq F(y;G_{k+1})-\delta_{2}/3\leq F(y;G_{k+1})$ in $U_{i}$.
Therefore we have, by Lemma 3.3, in $U_{i}$ that

$\displaystyle h_{G_{k+1}^{\varepsilon},S_{k}^{\varepsilon}}\leq
C\varepsilon^{q}e^{-(F(y;G_{k+1})-F(y;S_{k}))/\varepsilon}.$

Hence, for $q\in\mathbb{R}$, we obtain

$\int_{U_{i}}h_{G_{k+1}^{\varepsilon},S_{k}^{\varepsilon}}e^{-F/\varepsilon}dx\leq\varepsilon^{q}\int_{U_{i}}e^{-(F(y;G_{k+1})-F(y;S_{k}))/\varepsilon}e^{-F(y)/\varepsilon}dy.$

In order to compute the integral on the right hand side we study the infimum
value of the function $f(y)=F(y;G_{k+1})-F(y;S_{k})+F(y)$. Clearly, the
infimum is attained at an interior point of $U_{i}$, denoted by $x_{i}$. It
follows that then $x_{i}$ is necessarily a local minimum point of $F$. By
above considerations, we also have $F(y;S_{k})<F(y;G_{k+1})$ for all $y\in
U_{i}$, and thus we may deduce that $x_{i}\notin G_{k+1}$. If now $x_{i}\in
S_{k}$, then $F(x_{i})=F(x_{i};S_{k})$ and thus, by the definition of $f$ and
by (4.17),

$\displaystyle\inf_{U_{i}}f(y)=f(x_{i})\geq F(x_{i};G_{k+1})\geq
F(S_{k};G_{k+1})\geq F(G_{k+1})+\tfrac{2}{3}\delta_{2}.$

It remains to study the case where $x_{i}\in G_{j}$ for some $j\geq k+2$. In
this case we apply

$\displaystyle\inf_{U_{i}}f(y)=f(x_{i})=\overbrace{F(x_{i};G_{k+1})-F(x_{i};S_{k})}^{\geq
0}+F(G_{j})\geq F(G_{k+2})\geq F(G_{k+1})+\delta_{2},$

where the last inequality follows from (1.19). Therefore we can conclude that,
for $\delta_{3}=\tfrac{2}{3}\delta_{2}$, it holds that

$\int_{U_{i}}\varepsilon^{q}e^{-(F(y;G_{k+1})-F(y;S_{k}))/\varepsilon}e^{-F(y)/\varepsilon}dy\leq
C\varepsilon^{q}e^{-\frac{\delta_{3}}{\varepsilon}}e^{-F(G_{k+1})/\varepsilon}.$

Consequently, the component $U_{i}$ satisfying $U_{i}\cap
S_{k}^{\varepsilon}\neq\emptyset$ does not contribute either. The proof is
hence completed by 4.18 and by the fact that the integral over the remaining
components are neglible whenever $\varepsilon$ is small enough. ∎

## Acknowledgments

B.A. was supported by the Swedish Research Council dnr: 2019-04098. V.J. was
supported by the Academy of Finland grant 314227. We would like to thank Aapo
Kauranen and Toni Ikonen for helpful discussion on geometric function theory
and for letting us know the references [13, 20].

## References

  * [1] Aikawa, H. Essén, M., _Potential Theory - Selected topics_ , 1996.
  * [2] Ambrosio, L., Fusco, N. and Pallara, D., _Functions of bounded variation and free discontinuity problems_ , Oxford University Press, New York, 2000.
  * [3] Avelin, B., Julin, V. (2020). _Approximation of BV functions by neural networks: A regularity theory approach._ arXiv preprint arXiv:2012.08291.
  * [4] N. Berglund, B. Gentz. _The Eyring-Kramers law for potentials with nonquadratic saddles_ , Markov Processes and Related Fields 16.3 (2010): 549–598. Steklov. (POMI) 228 (1996), Veroyatn. i Stat. 1, 31–38, 356; translation in J. Math. Sci. (New York) 93 (1999), n. 3, 270–275.
  * [5] Bovier, A., Den Hollander, F. (2016). _Metastability: a potential-theoretic approach_ (Vol. 351). Springer.
  * [6] Bovier, A., Eckhoff, M., Gayrard, V., Klein, M. (2001). _Metastability in stochastic dynamics of disordered mean-field models._ Probability Theory and Related Fields, 119(1), 99-161.
  * [7] Bovier, A., Eckhoff, M., Gayrard, V., Klein, M. (2002). _Metastability and low lying spectra in reversible markov chains._ Communications in mathematical physics, 228(2), 219-255.
  * [8] Bovier, A., Eckhoff, M., Gayrard, V., Klein, M. _Metastability in reversible diffusion processes I. Sharp asymptotics for capacities and exit times_
  * [9] Bovier, A., Gayrard, V., Klein, M. _Metastability in reversible diffusion processes II. Precise asymptotics for small eigenvalues_
  * [10] Carlesson, L. _Selected problems on exceptional sets_ , 1967.
  * [11] Eyring, H. (1935). _The activated complex in chemical reactions._ The Journal of Chemical Physics, 3(2), 107-115.
  * [12] Freidlin, M. I.; Wentzell, A. D. _Random perturbations of dynamical systems._ Translated from the Russian by Joseph Szücs. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 260. Springer-Verlag, New York, 1984. viii+326 pp. ISBN: 0-387-90858-7
  * [13] Gehring, F. W. (1962). _Extremal length definitions for the conformal capacity of rings in space_. Michigan Math. J. 9 , 137–150.
  * [14] Gilbarg D., Trudinger, N.S. _Elliptic partial differential equations of second order._ Springer-Verlag, Berlin, 2001.
  * [15] Hochreiter, S., and Jürgen S. _Flat minima_ , Neural computation 9.1 (1997): 1-42
  * [16] Julin, V. (2016). _Generalized Harnack inequality for semilinear elliptic equations._ J. Math. Pures Appl. (9) 106, 877–904.
  * [17] Kramers, H. A. (1940). _Brownian motion in a field of force and the diffusion model of chemical reactions._ Physica, 7(4), 284-304.
  * [18] Maz’ya, Vladimir. _Lectures on isoperimetric and isocapacitary inequalities in the theory of Sobolev spaces_ , Contemporary Mathematics 338 (2003): 307-340.
  * [19] Petzka, H., Kamp, M., and Adilova, L.,_Relative flatness and generalization_ , Advances in Neural Information Processing Systems 34 (2021): 18420-18432.
  * [20] Ziemer, W. P. (1967). _Extremal length and conformal capacity_. Trans. Amer. Math. Soc. 126, 460–473.