This is a sel -a chi ed e sion o an o iginal a icle. This e sion
may di e om he o iginal in pagina ion and ypog aphic de ails.
Au ho (s):
Ti le:
Yea :
Ve sion:
Copy igh :
Righ s:
Righ s u l:
Please ci e he o iginal e sion:
CC BY 4.0
h ps://c ea i ecommons.o g/licenses/by/4.0/
Geome ic Cha ac e iza ion o he Ey ing–K ame s Fo mula
© The Au ho (s) 2023
Published e sion
A elin, Benny; Julin, Vesa; Vii asaa i, Lau i
A elin, B., Julin, V., & Vii asaa i, L. (2023). Geome ic Cha ac e iza ion o he Ey ing–K ame s
Fo mula. Communica ions in Ma hema ical Physics, 404, 401-437.
h ps://doi.o g/10.1007/s00220-023-04845-z
2023
Digi al Objec Iden i ie (DOI) h ps://doi.o g/10.1007/s00220-023-04845-z
Commun. Ma h. Phys. Communica ions in
Ma hema ical
Physics
Geome ic Cha ac e iza ion o he Ey ing–K ame s
Fo mula
Benny A elin1, Vesa Julin2, Lau i Vii asaa i1
1Depa men o Ma hema ics, Uppsala Uni e si y, 751 06 Uppsala, Sweden.
E-mail: benny.a [email p o ec ed]; [email p o ec ed]
2Depa men o Ma hema ics and S a is ics, Uni e si y o Jy äskylä, P.O. Box 35, 40014 Jy askyla, Finland.
E-mail: [email p o ec ed]
Recei ed: 29 Augus 2022 / Accep ed: 2 Sep embe 2023
© The Au ho (s) 2023
Abs ac : In his pape we conside he mean ansi ion ime o an o e -damped B ow-
nian pa icle be ween local minima o a smoo h po en ial. When he minima and saddles
a e non-degene a e his is in he low noise egime exac ly cha ac e ized by he so called
Ey ing–K ame s law and gi es he mean ansi ion ime as a quan i y depending on
he cu a u e o he minima and he saddle. In his pape we ind an ex ension o he
Ey ing–K ame s law gi ing an uppe bound on he mean ansi ion ime when bo h he
minima/saddles a e degene a e ( la ) while a he same ime co e ing mul iple saddles a
he same heigh . Ou main con ibu ion is a new sha p cha ac e iza ion o he capaci y
o wo local minima as a a io o wo geome ic quan i ies, i.e., he minimal cu and he
geodesic dis ance.
Con en s
1. In oduc ion .................................
1.1 Assump ions and s a emen o he main esul s .............
2. P elimina ies .................................
2.1 Po en ial heo y .............................
2.2 Classical poin wise es ima es ......................
3. Technical Lemmas ..............................
3.1 Rough es ima es o po en ials .....................
3.2 Laplace asymp o ics o log-conca e unc ions .............
4. P oo s o Theo ems 1 and 2 .........................
P oo o Theo em 1 .............................
Pa allel case .................................
Acknowledgemen s ................................
B. A elin, V. Julin, L. Vii asaa i
1. In oduc ion
In his pape we in es iga e he so called me as able exi imes o he s ochas ic di e -
en ial equa ion
dX =−∇F(X )d +√ε
2dB
,(1.1)
whe e Fis a smoo h po en ial wi h many local minima and εis a small numbe .
The main ques ion o me as abili y is o de e mine how much ime he p ocess (1.1)
akes om going om one local minima o ano he one. We call hese he me as able exi
imes. This ques ion has a ich his o y and in he double well case wi h non-degene a e
minima and a saddle poin his is cha ac e ized by a o mula called Ey ing–K ame s law
[11,17] which can be s a ed as ollows: Assume ha xand ya e quad a ic local minima
o F, sepa a ed by a unique saddle zwhich is such ha he Hessian has a single nega i e
eigen alue λ1(z). Then he expec ed ansi ion ime τ om x o ysa is ies
Ex[τ]≃ 2π
|λ1(z)||de (∇2F(z))|
de (∇2F(x)) e(F(z)−F(x))/ε,(1.2)
whe e ≃deno es ha he compa ison cons an ends o 1 as ε→0.
The alidi y o he abo e o mula has been s udied, om a quali a i e pe spec i e,
qui e ex ensi ely, s a ing om he wo k o F eidlin and Wen zell. Fo mo e in o ma ion,
see he book [12]. Roughly 15 yea s ago, Bo ie e . al. p oduced a se ies o pape s
[6–9] (see also [5]) which p o ided he i s p oo o (1.2) in he gene al se ing o
Mo se unc ions. Speci ically, hey showed ha he compa ison unc ion is like 1 +
O(ε1/2|log ε|3/2). In hese pape s, hey u ilized he connec ion o classical po en ial
heo y in o de o educe he p oblem o es ima ing me as able exi imes o he p oblem
o es ima ing ce ain capaci ies sha ply. This app oach was la e used in [4] o gene alize
(1.2) o gene al polynomial ype o degene acies.
In his pape we a e in e es ed in es ima ing he me as able exi imes in he case o
gene al ype o degene a e c i ical poin s. This equi es new echniques and e ec i e
no a ion om geome ic unc ion heo y which we will desc ibe below. Ou mo i a ion
comes om he ield o non-con ex op imiza ion whe e we canno expec he min-
ima/saddles o be quad a ic o e en o ha e polynomial g ow h in any di ec ion. In
pa icula , such si ua ions a e well known in he con ex o neu al ne wo ks, whe e he
minima and saddles may be comple ely la in some di ec ions [15]. Fu he mo e, i
seems ha hey a e p e e able, see [19] o a discussion, see also [3] o an explici
example.
The main goal is o es ima e he dependency o he me as able exi imes wi h espec
o he geome y o he po en ial F. In he p oo o (1.2)in[8] his is educed o es ima ing
he a io o he L1no m o he hi ing p obabili y and he capaci y. Thus in o de o
es ima e he me as able exi imes, one needs o p oduce
(1) Es ima es o he L1in eg al o he hi ing p obabili y, i.e. he in eg al o capaci a y
po en ials wi h espec o he Gibbs measu e.
(2) Es ima es o he capaci y i sel , i.e. es ima es o he ene gy o he capaci a y po en-
ials.
The in e es ing poin is ha he in luence o Fon 1 and 2 is in a sense dual. Speci ically,
he shape o minima o Fin luence 1 while he shape o saddles be ween minima
Geome ic Cha ac e iza ion o he Ey ing–K ame s Fo mula
in luence 2. As is well known, he main di icul y is o es ima e 2, which is an in e es ing
opic o i s own.
Ou main con ibu ion is a sha p capaci y es ima e o a e y gene al class o degen-
e a e saddle poin s. In o de o achie e his, we ph ase he p oblem in he language o
geome ic unc ion heo y, whe e he capaci y es ima es a e a cen al opic [13,20]. We
in oduce wo geome ic quan i ies which allow us o es ima e he capaci y in a sha p
and na u al way. As a byp oduc , we see ha in he case o se e al saddle poin s a he
same heigh , he opology dic a es how he local capaci ies add up. He e we esc ic
ou sel es in wo opological cases, which we call he pa allel and he se ial case, and i
u ns ou ha he o mulas o he o al capaci y ha e na u al coun e pa s in elec ical
ne wo ks o capaci o s, see Theo em 1. E en in he con ex o non-degene a e saddles,
ou o mulas p o ide a gene aliza ion o he esul o [8] whe e he au ho s conside
only he pa allel case. As we men ioned, we allow he saddle poin s o be degene a e bu
we ha e o assume ha saddles a e non-b anching, see (1.6).
1.1. Assump ions and s a emen o he main esul s. In o de o s a e ou main esul s
we i s need o in oduce ou assump ions on he po en ial F. We also need o in oduce
no a ion om geome ic unc ion heo y which migh seem a he hea y a i s , bu
i u ns ou o be obus enough o us o ea he po en ials wi h possible degene a e
c i ical poin s.
Le us i s in oduce some gene al e minology. We say ha a c i ical poin zo a
unc ion ∈C1(Rn)is a local minimum (maximum) o i (x)≥ (z)( (x)≤
(z)) in a neighbo hood o z.I is no locally cons an a a c i ical poin z, hen zis
a saddle poin i i is no a local minimum / maximum. Fo echnical easons we also
allow saddle poin s o include poin s zwhe e is locally cons an . We say ha a local
minimum a zis p ope i he e exis s a ˆ
δ>0 such ha o e e y 0 <δ<ˆ
δ he e exis s
aρsuch ha
(x)≥ (z)+δ o all x∈∂Bρ(z),
whe e Bρ(z)deno es an open ball wi h adius ρcen e ed a z. When he cen e is a he
o igin we use he sho no a ion Bρ.
Le us hen p oceed o ou assump ions on he po en ial F. Th oughou he pape we
assume ha F∈C2(Rn)and sa is ies he ollowing quad a ic g ow h condi ion
F(x)≥|x|2
C0−C0(1.3)
o a cons an C0≥1. We assume ha e e y local minimum poin zo Fis p ope , as
desc ibed abo e, and ha he e is a con ex unc ion Gz:Rn→Rwhich has a p ope
minimum a 0 wi h G(0)=0 such ha
F(x+z)−F(z)−Gz(x)≤ωGz(x),(1.4)
whe e ω:[0,∞)→[0,∞)is a con inuous and inc easing unc ion wi h
lim
s→0
ω(s)
s=0.(1.5)
B. A elin, V. Julin, L. Vii asaa i
We deno e by δ0 he la ges numbe o which ω(δ) ≤δ
8 o all δ≤4δ0. We de ine a
neighbo hood o he local minimum poin zand δ<δ
0as
Oz,δ := {x∈Rn:Gz(x)<δ}+{z}.
Fo he saddles, we assume ha o e e y saddle poin zo F he e a e con ex
unc ions gz:R→Rand Gz:Rn−1→Rwhich ha e a p ope minimum a 0 wi h
gz(0)=Gz(0)=0, and ha he e exis s an isome y1Tz:Rn→Rnsuch ha , deno ing
x=(x1,x)∈R×Rn−1, i holds
(F◦Tz)(x)−F(z)+gz(x1)−Gz(x)≤ω(gz(x1)) +ω(Gz(x)), (1.6)
whe e ω:[0,∞)→[0,∞)is as in (1.5). The assump ion (1.6) allows he saddle
poin o be degene a e, bu we do no allow hem o ha e many b anches, i.e., he se s
{F<F(z)}∩Bρ(z)canno ha e mo e han wo componen s. No e ha he con ex
unc ions gz,Gzand he isome y Tzdepend on z, while he unc ion ωis he same o
all saddle poin s. We de ine a neighbo hood o he saddle poin zand δ<δ
0as
Oz,δ := Tz{x1∈R:gz(x1)<δ}×{x∈Rn−1:Gz(x)<δ},(1.7)
whe e Tzis he isome y in (1.6). No e ha , since he saddle may be la , we should alk
abou se s a he han poin s. Howe e , we adop he con en ion ha we always choose a
ep esen a i e poin om each saddle (se ) and hus we may label he saddles by poin s
z1,z2,.... Mo eo e , we assume ha he e is a δ1≤δ0such ha o δ<δ
1we ha e ha
i z1and z2a e wo di e en saddle poin s, hen hei neighbo hoods Oz1,3δand Oz2,3δ
de ined in (1.7) a e disjoin . We assume he same o local minima (o mo e p ecisely,
he ep esen a i e poin s o se s o local minima).
Le us hen in oduce he no a ion ela ed o geome ic unc ion heo y [13,20]. Le
us ix wo disjoin se s Aand Bin a domain (open and connec ed se ). We say ha a
smoo h pa h γ:[0,1]→Rnconnec s A and B in he domain i
γ(0)∈A,γ(1)∈Band γ([0,1])⊂.
We deno e he se o all pa hs connec ing Aand Binside as C(A,B;). We ollow he
s anda d no a ion and de ine a dual objec o his by saying ha a smoo h hype su ace
S⊂Rn(possibly wi h bounda y) sepa a es A om B in i e e y pa h γ∈C(A,B;)
in e sec s S. We deno e he se o smoo h hype su aces, sepa a ing Aand Binside 
as S(A,B;). We de ine he geodesic dis ance be ween Aand Bin as
dε(A,B;) := in 1
0|γ( )|eF(γ ( ))
ε:γ∈C(A,B;)(1.8)
and i s dual, which we call he minimal cu ,by
Vε(A,B;) := in S
e−F(x)
εdHn−1(x):S∈S(A,B;).(1.9)
1Recall ha a mapping Tis an isome y i |T(x)−T(y)|=|x−y|.InRn, his implies ha T(x)=Ax +b,
whe e Ais an o hogonal ma ix. Tha is, Tconsis s o ansla ion band a o a ion A.

Geome ic Cha ac e iza ion o he Ey ing–K ame s Fo mula
Fig. 1. The neighbo hood Oz,δ o he saddle poin zconnec s he se s Uxaand Uxb
Whene e =Rnwe ins ead use he no a ion dε(A,B)and Vε(A,B).He eHkdeno es
he k-dimensional Hausdo measu e. Finally, we de ine he communica ion heigh
be ween he se s Aand Bas
F(A;B):= in
γ∈C(A,B;Rn)sup
∈[0,1]
F(γ ( )).
Le us hen assume ha xaand xba e local minimum poin s and deno e he commu-
nica ion heigh be ween xaand xbas
F(xa;xb):= F({xa};{xb}).
No ice ha F(xa), F(xb)≤F(xa;xb).Fo s∈R, deno e
Us:= {x∈Rn:F(x)<F(xa;xb)+s}.
Assuming ha 0 <δ≤δ1, We no e ha he poin s xaand xblie in di e en componen s
o he se U−δ/3while hey a e in he same componen o he se Uδ/3. We will always
deno e he componen s o U−δ/3con aining he poin s xaand xbby Uxaand Uxb, espec-
i ely. I is impo an o no ice ha i zis a saddle poin and F(z)<F(xa;xb)+δ/3,
hen he neighbo hood Oz,δ de ined in (1.7) in e sec s he se U−δ/3. We will some-
imes call he componen s o he se U−δ/3islands and he neighbo hoods Oz,δ b idges
since we may connec islands wi h b idges, see Fig. 1. (The e minology is ob iously
aken om he Se en B idges o Königsbe g). We say ha he se o saddle poin s
Zxa,xb={z1,...,zN}cha ge capaci y i i is he smalles se wi h he p ope y ha e -
e y γ∈C(Bε(xa), Bε(xa);Uδ/3)in e sec s he b idge Ozi,δ, de ined in (1.7), o some
zi∈Zxa,xb. In pa icula , i holds ha Zxa,xb⊂Uδ/3.
We will ocus on wo di e en opological si ua ions, whe e he saddle poin s in
Zxa,xba e ei he pa allel o in se ies. We say ha he poin s in Zxa,xba e pa allel i o
e e y zi∈Zxa,xb he e is a pa h
γ∈C(Bε(xa), Bε(xb);Uδ/3)
passing only h ough zi. We say ha he poin s in Zxa,xba e in se ies i e e y pa h
γ∈C(Bε(xa), Bε(xb);Uδ/3)passes h ough he b idge Ozi,δ, de ined in (1.7), o all
zi∈Zxa,xb. In o he wo ds, i he poin s in Zxa,xb={z1,...,zN}a e pa allel, hen he
islands occupied by he poin s xaand xb espec i ely a e connec ed wi h Nb idges and
we need o pass only one o ge om xa o xb. I hey a e in se ies, hen we ha e o pass
all Nb idges in o de o ge om xa o xb, see Fig. 2.
B. A elin, V. Julin, L. Vii asaa i
Fig. 2. Le pic u e is he pa allel case and he igh is he se ies case
Recall ha Uxaand Uxbdeno e he islands, i.e., he componen s o U−δ/3, which
con ain he poin s xaand xb. I he poin s in Zxa,xb={z1,...,zN}a e pa allel, hen i
ollows om ou assump ions on F ha we may connec Uxaand Uxbwi h one b idge,
i.e., o e e y zi he se
Uzi,δ := Ozi,δ ∪Uxa∪Uxb(1.10)
is connec ed, again see Fig. 1. Then all pa hs γ∈C(Bε(xa), Bε(xb);Uzi,δ)pass h ough
he b idge Ozi,δ. I he poin s in Zxa,xba e in se ies, hen i is use ul o o de hem
Zxa,xb={z1,...,zN}as ollows. Le us conside a pa h γ∈C(Bε(xa), Bε(xb);Uδ/3)
which passes h ough each poin in Zxa,xbp ecisely once. This means ha he e a e
0< 1<···< N<1 such ha γ( i)=zi,(1.11)
which gi es a na u al o de ing o poin s in Zxa,xb. By he assump ion (1.6), we also de-
duce ha he e a e s1,...,sN−1such ha i<si< i+1 and min{F(γ (si)), F(γ (si+1))}
<F(zi+1)−δ/3. We deno e
γ(si)=xi,x0=xaand xN=xb.(1.12)
The idea is ha hen e e y poin xilie in a di e en island, i.e., componen o U−δ/3
which we deno e by Uxi, see Fig. 2. We may also choose xisuch ha hey a e local
minimum poin s o F. Again i ollows om ou assump ions on F ha he se =
N
i=1Ozi,δ ∪Uxi∪Uxais connec ed.
We a e now eady o s a e ou main esul s. The i s esul is a quan i a i e lowe
bound on he capaci y be ween he se s Bε(xa)and Bε(xb), whe e xaand xba e wo
local minimum poin s o F. Fo a gi en domain ⊂Rnwe de ine he capaci y o wo
disjoin se s A,B⊂wi h espec o he domain as
cap(A,B;) := in ε|∇u|2e−F
εdx :u=1in A,u=0in B.
Abo e, he in imum is aken o e unc ions u∈W1,2
loc (). In he case =Rnwe deno e
cap(A,B)=cap(A,B;Rn)
o sho .
Geome ic Cha ac e iza ion o he Ey ing–K ame s Fo mula
Finally, o unc ions and gwhich depend con inuously on ε>0, we adop he
no a ion
(ε) ≃g(ε)
when he e exis s a cons an Cdepending only on he da a o he p oblem such ha
(1−ˆη(C,ε)) (ε) ≤g(ε) ≤(1+ ˆη(C,ε)) (ε),
whe e ˆη(C,·)is an inc easing and con inuous unc ion ˆη(C,·):[0,∞)→[0,∞)wi h
lims→0ˆη(C,·)=0. In all ou es ima es, he unc ion ˆηis speci ied and depends only
on he unc ion ω om (1.4) and (1.6). In o de o de ine i , we i s le 0 <ε≤δ0/2be
ixed and le ε1(ε) be he unique solu ion o
ω(ε1)ε1=ε. (1.13)
F om he assump ion ha ω(s)<s/2 o s<δ
0we see ha ε<ε
1. Fu he mo e, since
ωis inc easing we ge ha ε1→0asε→0. Now, om he de ini ion o ε1in (1.13)
we see, using lims→0ω(s)
s=0 and ε1→0asε→0, ha
ε1
ε=√ε1
√ω(ε1)→∞ as ε→0.
On he o he hand, again using he same ac s, we see ha
ω(ε1)
ε=√ω(ε1)
√ε1→0asε→0.
Thus
lim
ε→0
ε1
ε=∞ and lim
ε→0
ω(ε1)
ε=0.(1.14)
In he ollowing we will deno e
η(x)=e−1/xxnand (1.15)
ˆη(C,ε)=max ε, η Cε1(ε)
ε.(1.16)
Finally, in ou main heo ems and ou lemmas/p oposi ions beyond Sec .3 he e is
a ball BRwhich con ains all he le el se s o in e es . The exis ence o such a ball is
gi en by he quad a ic g ow h condi ion (1.3). The cons an s in he es ima es in ou main
heo ems and in Sec .3a e unless o he wise s a ed, depending on n,∇FBR,δ,R,C0,
speci ically, his applies o he cons an s in ˆη, and as such, gi es p ecise meaning o
a≃b.
Theo em 1. Assume ha F sa is ies he s uc u al assump ions abo e. Le xaand xbbe
wo local minimum poin s o F and le Zxa,xb={z1,...,zN}be he se o saddle poin s
which cha ge capaci y as de ined abo e, and le 0<δ≤δ1be ixed. The e exis s an
0<ε
0≤δsuch ha i 0<ε≤ε0 he ollowing holds:
B. A elin, V. Julin, L. Vii asaa i
I he poin s in Zxa,xb={z1,...,zN}a e pa allel, hen, using he no a ion Uzi,δ om
(1.10), i holds
cap(Bε(xa), Bε(xb)) ≃
N

i=1
cap(Bε(xa), Bε(xb);Uzi,δ). (1.17)
Mo eo e o all i =1,...,N we ha e he es ima e
cap(Bε(xa), Bε(xb);Uzi,δ)≃εVε(Bε(xa), Bε(xb);Uzi,δ)
dε(Bε(xa), Bε(xb);Uzi,δ)e
F(zi)
ε,
whe e dε(Bε(xa), Bε(xb);Uzi,δ)and Vε(Bε(xa), Bε(xb);Uzi,δ)a e de ined in (1.8)and
(1.9).
I he poin s in Zxa,xba e in se ies, hen, using he o de ing z1,...,zN om (1.11)
o he poin s in Zxa,xband he poin s x0,x1,...,xNde ined in (1.12), i holds
1
cap(Bε(xa), Bε(xb)) ≃
N

i=1
1
cap(Bε(xi−1), Bε(xi)),(1.18)
whe e we ha e he es ima e
cap(Bε(xi−1), Bε(xi)) ≃εVε(Bε(xi−1), Bε(xi))
dε(Bε(xi−1), Bε(xi)) e
F(zi)
ε
o all i =1,...,N.
Le us make a ew ema ks on he s a emen o he abo e heo em. Fi s , in he case
o a single saddle Zxa,xb={z} he abo e capaci y es ima e educes o
cap(Bε(xa), Bε(xb)) ≃εVε(Bε(xa), Bε(xb))
dε(Bε(xa), Bε(xb)) eF(z)
ε,
whe e dε(Bε(xa), Bε(xb)) is he geodesic dis ance be ween Bε(xa)and Bε(xa), and
Vε(Bε(xa), Bε(xb)) is he a ea o he ’smalles c oss sec ion’. This is in acco dance wi h
he classical esul on pa allel pla e capaci o s, whe e he capaci y depends linea ly on
he a ea and is in e sely p opo ional o hei dis ance.
The s a emen (1.17), when he saddle poin s a e pa allel, means ha each saddle
poin z1,...,zNcha ges capaci y and he o al capaci y is hei sum. Again he si ua ion
is he same as in he case o pa allel pla e capaci o s wi h capaci y C1,...,CN, whe e
he o al capaci y is he sum
C=C1+···+CN.
On he o he hand, i he pla e capaci o s a e in se ies hei o al capaci y sa is ies
1
C=1
C1
+···+1
CN
which is p ecisely he s a emen in (1.18).
Geome ic Cha ac e iza ion o he Ey ing–K ame s Fo mula
and
⨏B2R(x)|u|pdx1/p
≤Cpin
BR(x)u,
whe e he symbol ⨏deno es he a e age in eg al, and he cons an Cpin addi ion o
abo e depends also on p.
In he non-homogeneous case Lεu= we ha e he ollowing gene aliza ion o Ha -
nack’s inequali y.
Lemma 2.8 Le be a domain and le u ∈C2() be a non-nega i e unc ion sa is ying
Lεu= . Then o any B3R(x)⊂i holds ha
sup
BR(x)
u≤Cin
BR(x)u+R
ε Ln(B2R(x))
o a cons an C =C(n,νR). In pa icula , i ∇FL∞() ≤L, hen o R ≤ε
L he
cons an C is independen o εand we ha e
sup
BR(x)
u≤Cin
BR(x)u+ Ln(B2R(x)).
The Ha nack inequali y in Lemma 2.7 holds also in he case o he punc u ed ball.
Lemma 2.9. Le u ∈C2(B3R(x) {x})be a non-nega i e unc ion sa is ying Lεu=0
in B3R(x) {x}. Then
sup
∂BR(x)
u≤Cin
∂BR(x)u
o a cons an C =C(n,νR).
P oo . By ansla ing he coo dina es we may assume ha x=0. Le x0,y0∈∂BR
be such ha sup∂BR(x)u=u(x0)and in ∂BR(x)u=u(y0). We choose poin s x1,...,
xN−1,xN∈∂BRsuch ha |xi−xi−1|≤R/4 and xN=y0. No e ha he numbe Nis
bounded. Now we may use Ha nack’s inequali y Lemma 2.7 in balls BR/4(xi) o ge
u(xi−1)≤Cu(xi).
We ob ain he claim by applying he abo e o e i=1,...,N.
Lemma 2.10. Le be a domain and le u,h∈C2() be non-nega i e unc ions such
ha Lεu=h and h sa is ies Ha nack’s inequali y wi h cons an c0. Then he unc ion
=u+h sa is ies Ha nack’s inequali y, i.e., o all B3R(x)⊂i holds ha
sup
BR(x)
≤Cin
BR(x)
o a cons an C =C(n,νR,R2/ε, c0). In pa icula , i ∇FL∞() ≤L, hen o
R≤min{ε/L,√ε} he cons an C is independen o ε.

B. A elin, V. Julin, L. Vii asaa i
P oo . Again we may assume ha x=0. Using Lemma 2.8 and Ha nack’s inequali y
o hyields
sup
BR
u≤Cin
BR
u+CR
εhLn(B2R)
≤Cin
BR
u+CR
ε|BR|1/nin
BR
h
≤Cin
BR
u+CR2
εin
BR
h.
Now, using Ha nack’s inequali y o hagain, we ob ain
sup
BR
≤sup
BR
u+sup
BR
h≤Cin
BR
u+Cin
BR
h≤Cin
BR
o a cons an Cas in he s a emen . This p o es he claim. 
The Ha nack’s inequali y in Lemma 2.7 implies Hölde con inui y o solu ions o
Lεu=0.
Lemma 2.11. Le u ∈C2(B3R(x)) be a unc ion such ha o any cons an c, o which
=u+c is non-nega i e, he unc ion sa is ies Ha nack’s inequali y wi h cons an
C0, independen o c. Then he e exis s C =C(C0)>1and α=α(C0)∈(0,1)such
ha , o all ρ≤R, i holds ha
oscBρ(x)u≤Cρ
Rα
oscBR(x)u.
In pa icula , i u,h∈C2() a e non-nega i e unc ions such ha Lεu=h and h
sa is ies Ha nack’s inequali y wi h cons an C0, hen u +h sa is ies he es ima e abo e.
P oo . The p oo ollows e ba im om he classical p oo o Mose , see [14, Theo em
8.22]. 
3. Technical Lemmas
In his sec ion we p o ide some p elimina y esul s o he p oo s o he main heo ems.
We ecall ha we assume ha he po en ial Fsa is ies he s uc u al assump ions om
Sec .1.1, and ha om his momen on ou cons an s a e allowed o depend on he da a,
see pa ag aph a e (1.16).
3.1. Rough es ima es o po en ials. In his subsec ion we p o ide es ima es o he
capaci a y po en ial hA,B, when Aand Ba e wo disjoin closed se s. The i s es ima e
is he so called enewal es ima e o [8]. In o de o ace dependencies o cons an s, we
p o ide a p oo .
Lemma 3.1. Le be a smoo h domain, le A,B⊂be disjoin smoo h se s, and
conside hA,Bas he capaci a y po en ial in . Assume ha B4(x)⊂( (A∪B)),
and ha ≤min ε
∇FL∞(B2(x)) ,
. Then he e exis s a cons an C =C(n,ν) > 1
such ha
hA,B(x)≤Ccap(B (x), A;)
cap(B (x), B;) .
Geome ic Cha ac e iza ion o he Ey ing–K ame s Fo mula
P oo . Again, wi hou loss o gene ali y, we may assume ha x=0. Since hA,B∪B =
hA,Bon ∂( (A∪B)), we can use (2.4) o ep esen hA,Bas ollows
hA,B(z)=εeF(z)/ε ∂( (A∪B))
hA,B∪B ∇G (A∪B)(y,z)·ndσε(y). (3.1)
Now by G een’s second iden i y (2.3)in (A∪B∪B )and (3.1) we see ha , o
z∈,
hA,B(z)=hA,B∪B (z)
−εeF(z)/ε ∂B
G (A∪B)(y,z)∇hA,B∪B (y)·ndσε(y), (3.2)
whe e nis he inwa d uni no mal o B . Fi s no e ha by (2.3) we can iden i y he
equilib ium measu e as
μB∪B ,A=−ε∇hB∪B ,A·ndσε=ε∇hA,B∪B ·ndσε.
Using ha hA,B∪B =1−hB∪B ,A, oge he wi h he abo e and (3.2), we ge o z∈B
(since hA,B∪B (z)=0) ha
hA,B(z)=∂B
G (A∪B)(y,z)e(F(z)−F(y))/εdμB∪B ,A(y). (3.3)
Fi s no e ha μB ∪B,A|∂B is an admissible measu e o cap(B ,A;), which ol-
lows om he ac ha by he compa ison p inciple, he po en ials o o de ed measu es
a e o de ed and he suppo o μB ∪B,A|∂B is in B . To bound hA,B om abo e, no e ha
by he balayage ep esen a ion o capaci y (see Lemma 2.4) and he abo e, we ob ain
∂B
e−F(y)/εdμB∪B ,A(y)≤cap(B ,A;).
Applying he abo e o (3.3) gi es, o z∈B ,
hA,B(z)≤sup
y∈∂B
G (A∪B)(y,z)eF(z)/ε cap(B ,A;). (3.4)
I emains o bound he G een’s unc ion. Fo z∈B we ha e by (2.4) and (2.6) and
Rema k 2.2 ha
1=hB ,A∪B(z)=∂B
G (A∪B)(z,y)dμB ,A∪B(y)
=∂B
G (A∪B)(y,z)e(F(z)−F(y))/εdμB ,A∪B(y)
≥in
∂B
G (A∪B)(y,z)eF(z)/ε cap(B ,A∪B;)
≥in
∂B
G (A∪B)(y,z)eF(z)/ε cap(B ,B;).
(3.5)
Now pu ing oge he (3.4) and (3.5) and Lemma 2.9 we a e done. 
The esul below is a e sion o he ough capaci y bound o [8], bu we gi e a
simpli ied p oo . We will la e use a simila a gumen in he p oo o Theo em 1.
B. A elin, V. Julin, L. Vii asaa i
Lemma 3.2. Le D ⊂BRbe a smoo h closed se . Le x ∈BR D be such ha B4ρ(x)⊂
BR D, o ρ≤ε. Then he e exis s cons an s q1,q2∈Rand C >1such ha
1
Cεq1ρn−1e−F(x;D)/ε ≤cap(Bρ(x), D)≤Cερq2e−F(x;D)/ε.
P oo . We assume wi hou loss o gene ali y ha F(Bρ;D)=0, since he quan i ies
can always be scaled back. Conside γ∈C(Bρ(x), D;BR)(i.e. a cu e connec ing
Bρ(x)and Dinside BR) such ha sup F(γ ( )) ≤Cεand le u(z)=hD,Bρ(x)(z).We
i s no e by Lemma 2.4 ha
cap(Bρ(x), D)=ε|∇u|2e−F(y)/εdy.
Fix an n−1 dimensional disk Dρo adius ρ. Then by Cauchy-Schwa z
BR|∇u|2e−F(y)/εdy ≥1
0Dρ˙γ
|˙γ|,∇u(γ ( )+z)
2
|˙γ|dσε(z)d .
By he undamen al heo em o calculus and Cauchy-Schwa z, we ha e o a ixed
poin z∈Dρ ha
1=u(γ (1)) −u(γ (0)) =1
0
d
d u(γ ( )+z)d
=1
0
d
d u(γ ( )+z)√|˙γ|
√|˙γ|e−F(γ ( )+z)/(2ε)eF(γ ( )+z)/(2ε)d
≤1
0|d
d u(γ ( )+z)|21
|˙γ|e−F(γ ( )+z)/εd 1/2|˙γ|eF(γ ( )+z)/εd 1/2
.
F om he abo e we ge
BR|∇u|2e−F(y)/εdy ≥1
0Dρ˙γ
|˙γ|,∇u(γ ( )+z)
2
|˙γ|dσε(z)d
≥Dρ1
0|˙γ|eF(γ ( )+z)/εd −1
dσε(z).
Now since Fis Lipschi z in BRand F(Bρ;D)=0,we know ha he e exis s a cons an
C(γ ) such ha , o z∈Dρand ρ<2ε,
1
0|˙γ|eF(γ ( )+z)/εd ≤C(γ ). (3.6)
In he abo e he cons an Cdepends on he leng h o γ, which can be assumed o
be bounded. To see his, ake an εneighbo hood o γ,Eεand conside disjoin balls
Bε(yi)⊂Eεsuch ha Eε⊂iB5ε(yi)gi en e.g. by he Vi ali co e ing lemma. The
numbe o such balls is a mos C1Rn/εn, o a dimensional cons an C1. I we cons uc
a piecewise linea cu e γεconnec ing he cen e o each ball in he co e ing, his cu e
will be inside E10εand i s leng h will be bounded by C2Rn/εn−1, o a dimensional
cons an C2. This newly cons uc ed cu e can be molli ied o achie e a smoo h cu e
wi hou inc easing he leng h by mo e han a ac o . F om he abo e and he Lipschi z
Geome ic Cha ac e iza ion o he Ey ing–K ame s Fo mula
con inui y o Fi is clea ha sup F(γε( )) ≤Cε, and as such we can eplace γwi h
γεin he abo e and ge om (3.6) ha he e is a cons an C>1 depending only on he
da a such ha
1
0|˙γ|eF(γ ( )+z)/εd ≤ε1−nC.
This implies ha o a new cons an Cwe ha e
BR|∇u|2e−F(y)/εdy ≥Cεn−1ρn−1,
which comple es he p oo o he lowe bound a e escaling ou po en ial F.
To p o e he uppe bound we ha e wo possible cases: In he case when F(x;D)=
F(x)we can ake a cu o unc ion χBρ(x)≤φ≤χB2ρ(x)whe e |∇φ|≤C/ρ as a
compe i o in he a ia ional o mula ion o capaci y (2.5). Then
BR|∇φ|2e−F(y)/εdx =B2ρ(x)|∇φ|2e−F(y)/εdx ≤Cρn−2.
In he case whe e F(x;D)>F(x), conside he se ˆ
D={z∈BR:F(z)≤F(x;D)}
and le ˆ
D1be he componen ha in e sec s D.Wese 
D=(ˆ
D1∪D) B4ρ(x).By he
Lipschi z con inui y, we know ha in 
DF>−Cρ.We akeχ
D+Bρ≤1−φ≤χ
D+B2ρ,
whe e |∇φ|≤C/ρ, and ge
BR|∇φ|2e−F(y)/εdx =(
D+B2ρ) 
D|∇φ|2e−F(y)/εdx
≤Cρ−2|(
D+B2ρ) 
D|.
Again, he uppe bound ollows om escaling he po en ial Fas in he case o he lowe
bound. This comple es he whole p oo . 
Lemma 3.3. Le A,B⊂BRbe smoo h disjoin se s, and le x ∈BRbe such ha
Bε(x)⊂BR (A∪B),ε∈(0,1). Then, i F(x;B)≤F(x;A), he e exis s cons an s
q and C such ha
hA,B(x)≤Cεqe−(F(x;A)−F(x;B))/ε.
P oo . Le L:= ∇FL∞(BR). By combining Lemmas 3.1 and 3.2 wi h R=ε, =
min{ε/L,ε}yields he esul . 
Rema k 3.4. By elabeling A,B o B,Aand using he ac ha hA,B=1−hB,A,we
ge ha i he e e se inequali y holds, i.e. F(x;B)>F(x;A), hen
1−hA,B(x)≤Cεqe−(F(x;B)−F(x;A))/ε.
Lemma 3.5. Le be a smoo h domain and le xa,xb∈⊂BRbe wo local minimum
poin s o F. Fix 0<δ<δ
1and assume ha U−δ/3={x:F(x)<F(xa;xb)−δ/3}⊂
. Then he e exis s an ε0∈(0,1)and a cons an C =C>1such ha , o any
0≤ε≤ε0 o which B3ε(xa), B3ε(xb)⊂U−δ/3, he ollowing holds: I Uiis a
componen o U−δ/3, hen
osc
Ui
hBε(xa),Bε(xb)≤Cε.
B. A elin, V. Julin, L. Vii asaa i
P oo . Conside any componen Uio U−δ/3. We no e ha we can ake εsmall enough
depending on he Lipschi z cons an o Fin BRand δsuch ha he e exis s a Lipschi z
domain Disa is ying
Ui+Bε⊂Di⊂U−δ/4.
Fo simplici y, deno e u:= hBε(xa),Bε(xb). Since Diis Lipschi z we may use he Poinca é
inequali y o ge
Di|u−uDi|2dx ≤CDi|∇u|2dx.
Using ha Di⊂U−δ/4 oge he wi h Lemma 2.4
Di|∇u|2dx ≤esupDiF/ε Di|∇u|2e−F(x)/εdx
≤ε−1esupDiF/ε cap(Bε(xa), Bε(xb);).
Using he de ini ion o U−δ/4and Lemma 3.2, we ge
Di|u−uDi|2dx ≤Cεq1e−δ/4ε(3.7)
o some cons an q1∈R.Now, o anyx0∈Uiwe ha e by Lemma 2.7 ha
sup
Bε(x0)|u−uDi|2≤C⨏B2ε|u−uDi|2dx
which oge he wi h (3.7)gi es
sup
Bε(x0)|u−uDi|2≤Cεq1−ne−δ/4ε.
Since x0was an a bi a y poin in Uiwe conclude ha he e exis s ε0∈(0,1)depending
only on he da a such ha i ε<ε
0, he claim holds. 
We conclude his subsec ion wi h an es ima e ela ing he alue o he po en ial o
he equilib ium po en ial o he a io o he L1no m o he equilib ium po en ial and he
capaci y.
Lemma 3.6. Le be a smoo h domain and le A be a smoo h open se and
conside wA, as he po en ial o he equilib ium po en ial in (see De ini ion 2.5).
Le x ∈be a c i ical poin o F such ha B3√ε(x)⊂. Then he e exis s a cons an
C>1such ha o ρ<√εwe ha e
wA,(x)−Cρ
√εα
(wA,(x)+1)≤hA,chBρ(x),Adμε
cap(Bρ(x), A;)
≤wA,(x)+Cρ
√εα
(wA,(x)+1).

Geome ic Cha ac e iza ion o he Ey ing–K ame s Fo mula
P oo . F om Lemma 2.6 we ge
∂Bρ(x)
wA,(y)e−F(y)/εdμBρ(x),A(y)=(A∪B)c
hA,c(z)hBρ(x),A(z)dμε.
We can es ima e he le hand side as
wA,(x)−oscBρwA, ≤in
Bρ(x)wA, ≤∂Bρ(x)e−F(y)/εwA,dμBρ(x),A(y)
∂Bρ(x)e−F(y)/εdμBρ(x),A(y)
≤sup
Bρ(x)
wA, ≤wA,(x)+osc
BρwA,.
We wan o es ima e he oscilla ion o wA, which we do by conside ing
osc wA, =osc(wA, +hA,c−hA,c)≤osc(wA, +hA,c)+osc(hA,c).
Now, he oscilla ion o wA, +hA,cand hA,ccan es ima ed by Lemma 2.11 o
ρ≤1
C√ε. Tha is,
oscBρ(wA, +hA,c)+osc
Bρ(hA,c)≤Cρ
√εα
sup
B√ε
(wA, +hA,c)
+Cρ
√εα
sup
B√ε
(hA,c).
We apply Lemma 2.7 o eplace he sup emums on he igh hand side wi h he alue a
xas bo h wA, +hA,cand hA,csa is ies he Ha nack inequali y (see Lemma 2.10).
Tha is,
oscBρ(wA, +hA,c)+osc
Bρ(hA,c)≤Cρ
√εα
(wA,(x)+hA,c(x))
≤Cρ
√εα
(wA,(x)+1).
I is easily seen ha he abo e can be ex ended o ρ≤√εby applying Lemma 2.10 again
and by enla ging he cons an C. The p oo is comple ed by using (2.6) and collec ing
he es ima es abo e. 
3.2. Laplace asymp o ics o log-conca e unc ions. The assump ions (1.4) and (1.6)
ensu e ha nea c i ical poin s he po en ial Fis well app oxima ed by con ex unc-
ions. The e o e we will need basic es ima es o log-conca e unc ions, which a he
su p isingly we did no ind in he li e a u e.
Lemma 3.7. Assume G :Rn→Ris a con ex unc ion which has a p ope minimum a
he o igin and G(0)=0. Then he e exis s a cons an C =C(n)>1such ha
1
C|{G<ε}| ≤ Rn
e−G
εdx ≤C|{G<ε}|.(3.8)
B. A elin, V. Julin, L. Vii asaa i
Mo eo e , he e is a cons an C =C(n)such ha o all >1, we ha e
{G<ε}
e−G
εdx ≥(1−η(C−1)) Rn
e−G
εdx,(3.9)
wi h ηas in (1.15).
P oo . By app oxima ion we may assume ha Gis smoo h. The lowe bound in (3.8)
ollows immedia ely om
Rn
e−G
εdx ≥{G<ε}
e−G
εdx ≥e−1|{G<ε}|.
To p o e he uppe bound in (3.8) we i s show ha , o all >0, i holds
|{G<2 }| ≤ 2n|{G< }|.(3.10)
In o de o p o e (3.10) i is enough o conside only he case =1 ( he gene al case
ollows by conside ing 
G=G/ ). Deno e E1={G<1}and E2={G<2}. Hence
ou goal is o show
E2⊂2E1={2x:x∈E1}.
Fix ˆx∈∂E1and de ine g( )=G( ˆx) o ≥0. By ou assump ions, g( )is a smoo h
con ex unc ion sa is ying g(0)=0 and g(1)=1. As such, bo h g,ga e inc easing
unc ions om which we can conclude ha g(1)≥1. Now, by he undamen al heo em
o calculus,
g(2)−g(1)=2
1
g( )d ≥1
which gi es g(2)≥2. This means ha o all ˆx∈∂E1we ha e G(2ˆx)≥2. Tha is, we
ha e E2⊂2E1. Thus
|E2|≤|2E1|≤2n|E1|
and (3.10) ollows. I e a ing (3.10)gi es
|{G<2jε}| ≤ 2jn|{G<ε}|
and hence
|{G<ε}| ≤ (2)n|{G<ε}| (3.11)
o all ≥1. We conclude he p oo o he uppe bound in (3.8)byusing(3.11)as
Rn
e−G
εdx ≤∞

j=0{jε≤G<( j+1)ε}
e−G
εdx
≤∞

j=0|{jε≤G<(j+1)ε}|e−j
≤2n∞

j=0
e−j(j+1)n|{G<ε}| ≤ C(n)|{G<ε}|.
Geome ic Cha ac e iza ion o he Ey ing–K ame s Fo mula
I emains o p o e (3.9). Fix >1. Then, o e e y x∈{G≥ε}, i holds
e−G(x)
ε=e−G(x)
ε =e−G(x)
ε =e−G(x)
ε −1
e−G(x)
ε ≤e−+1e−G(x)
ε .(3.12)
The e o e we ha e, by (3.8), (3.11) and (3.12),
{G≥ε}
e−G
εdx ≤e−+1 {G≥ε}
e−G
ε dx ≤e−+1 Rn
e−G
ε dx
≤Ce−|{G<ε}|
≤Ce−n|{G<ε}|
≤Ce−nRn
e−G
εdx
(3.13)
and he inequali y (3.9) ollows by using (1.15). 
Lemma 3.8. Assume G :Rn→Ris a unc ion which has a p ope global minimum a
he o igin and G(0)=0. Fu he mo e, assume he e is a cons an C0such ha , o all
a>0and ε>0, i holds ha
G>a
e−G/εdx <C0e−a/ε.(3.14)
I he e is a le el ε0>0such ha G is con ex on he componen o {G(x)<ε
0} ha
con ains 0, hen he e is an ε1(n,|{G<ε
0/2}|)<ε
0and a cons an C =C(C0,n)>1
such ha , o all ε<ε
1, i holds ha
C−1|{G<ε}| ≤ Rn
e−G
εdx ≤C|{G<ε}|.
P oo . Since Gis con ex in he le el se {G<ε
0}, we know ha he le el se {G≤ε0/2}
is con ex and as such we can ex end he unc ion Gou side ha le el se o a globally
con ex unc ion. This allows us o apply Lemma 3.7 and ob ain
1
C|{G<ε}| ≤ {G<ε0/2}
e−G
εdx ≤C|{G<ε}|.(3.15)
Now, spli he in eg al as
e−G/εdx =G≤ε0/2
e−G/εdx +G>ε0/2
e−G/εdx.
F om (3.15) i ollows ha i su ices o bound he second in eg al on he igh hand side.
Using (3.14) o a=ε0/2 we ge
G>ε0/2
e−G/εdx ≤C0e−(ε0/2)/ε.
Since Gis con ex in he le el se {G<ε
0/2}, which is again con ex, i ollows ha
we can cons uc a conical unc ion 
Gas ollows: Fo any ˆx∈∂{G<ε
0/2}de ine

G( ˆx/ˆx)= ε0/2. The le el se s o 
Gsa is y, o ε<ε
0/2,
|{
G<ε}| ≤ |{G<ε}|.
B. A elin, V. Julin, L. Vii asaa i
Howe e ,
|{
G<ε}| = ε
ε0/2n
|{
G<ε
0/2}| = ε
ε0/2n
|{G<ε
0/2}|.
Now, we can choose ε1(n,C0,|{G<ε
0/2}|)<ε
0/2 such ha o ε<ε
1we ha e
e−(ε0/2)/ε ≤ε
ε0n
|{G<ε
0/2}|.
This means ha o ε<ε
1we also ha e
G>ε0/2
e−G/εdx ≤C0|{G<ε}|
which oge he wi h (3.15) comple es he p oo . 
We conclude his sec ion wi h he ollowing echnical lemma which is use ul when
we s udy he po en ial nea c i ical poin s.
Lemma 3.9. Assume G :Rn→Ris a con ex unc ion which has a p ope minimum a
he o igin and G(0)=0. Le ω:[0,∞)→[0,∞)be as in (1.4)and (1.6). Then o
all δ≤δ0, we ha e {G<δ}
e−G(x)
εe±ω(G(x))
εdx ≃Rn
e−G(x)
εdx.
P oo . Deno e ε=ε1
εwi h ε1as in (1.13). F om (1.14) we know ha ε→∞as
ε→0. Now, by (3.9) in Lemma 3.7 and (1.14), we ge
{G<ε1}
e−G(x)
εeω(G(x))
εdx ≃{G<εε}
e−G(x)
εdx ≃Rn
e−G(x)
εdx.(3.16)
The lowe bound ollows immedia ely om his. In o de o p o e he uppe bound, no e
ha ω(s)≤s/2 o all s≤δ0by assump ion. The e o e we can epea he a gumen in
(3.13) o ge
{ε1<G<δ}
e−G(x)
εeω(G(x))
εdx ≤{G>εε}
e−G(x)
2εdx ≤η(C−1
ε)Rn
e−G
εdx,
which oge he wi h Lemma 3.16 yields he uppe bound. 
4. P oo s o Theo ems 1and 2
In his sec ion we p o e he capaci y es ima e in Theo em 1and exi ime es ima e in
Theo em 2. Be o e we begin, we would like o emind he eade ha , as in Sec .3,we
will assume ha Fsa is ies ou s uc u al assump ions and ha all cons an s depend on
he da a, see he pa ag aph a e (1.16).
We i s s udy he geome ic quan i ies dε(A,B;) and Vε(A,B;) de ined in (1.8)
and (1.9) and gi e a mo e explici , bu less geome ic, cha ac e iza ion. The cha ac e -
iza ion o he geodesic dis ance dε(A,B;) u ns ou o be much easie han o he
sepa a ing su ace Vε(A,B;) and he e o e we p o e i i s .
In he ollowing wo p oposi ions we will i s ix wo local minimum poin s o F,
say xaand xb. Thei communica ion heigh F(xa;xb)de ines he island U−δ/3which we
ecall a e componen s o {F<F(xa;xb)−δ/3}. We i s s udy he geodesic dis ance
be ween Bε(x1)and Bε(x2), whe e x1and x2a e wo local minima o F.
Geome ic Cha ac e iza ion o he Ey ing–K ame s Fo mula
This yields he lowe bound when he saddle poin s a e pa allel.
Fo he uppe bound, we only gi e a ske ch o he a gumen as i is ai ly s aigh -
o wa d. The idea is o con uc a compe i o hin he a ia ional cha ac e iza ion o he
capaci y, see (2.5). Le us i s de ine hin he se Uδ/3={F<F(xa;xb)+δ/3}. Since
he saddle poin s Zxa,xb={z1,...,zN}a e pa allel, i ollows ha he poin s xaand xb
lie in di e en componen s o he se

U=Uδ/3
N

i=1
Ozi,δ,
whe e Ozi,δ is de ined in (1.8). Deno e he componen s o 
Ucon aining xaand xbby

Uxaand 
Uxb, espec i ely. We de ine i s
h=1in
Uxaand h=0in
Uxb.
Le us nex ix a saddle poin zi∈Zxa,xb. As be o e, we may again assume ha
zi=0,F(0)=0
and
Oδ:= O0,δ ={x1∈R:g(x1)<δ}×{x∈Rn−1:G(x)<δ}.
Mo eo e , we may assume ha

Uxa∩∂Oδ⊂{x1>0}and 
Uxb∩∂Oδ⊂{x1<0}.
Le τ−<0<τ
+be numbe s such ha g(τ−)=g(τ+)=δ/100. We de ine h(x)=ϕ(x1)
in Oδsuch ha he unc ion ϕ:[τ−,τ
+]→Ris a solu ion o he o dina y di e en ial
equa ion
d
ds ϕ(s)eg(s)
ε=0in(τ−,τ
+)
wi h bounda y alues ϕ(τ−)=0 and ϕ(τ+)=1. No e ha hen
ϕ(s)eg(s)
ε=τ+
τ−
e−g(x1)
εdx1−1
.
We ex end ϕin o Rby se ing ϕ(s)=0 o s≤τ−and ϕ(s)=1 o s≥τ+. I ollows
ha o he unc ion hwe ha e, by cons uc ion, Lemma 3.9, and an a gumen simila
o he one leading o (4.13), ha
Oδ|∇h|2e−F(x)
εdx
≤{g<δ}|ϕ(x1)|2e
g(x1)
εe
ω(g(x1))
εdx1{G<δ}
e−G(x)
εeω(G(x))
εdx
≤(1+ ˆη(C,ε))R
e−g(x1)
εdx1−1Rn−1
e−G(x)
εdx.
By epea ing he cons uc ion o e e y saddle poin zi∈Zxa,xb, we ob ain a unc ion
which is de ined in Uδ/3. We deno e his unc ion by h:Uδ/3→R. No e ha now o h

B. A elin, V. Julin, L. Vii asaa i
he es ima e (4.15) is op imal. Mo eo e , his Lipschi z con inuous. We ex end h o Rn
wi hou inc easing he Lipschi z cons an L, e.g., by de ining
h(x)=sup
y∈Uδ/3h(y)−L|x−y| o x∈Rn Uδ/3.
This inally leads o he uppe bound comple ing he p oo o he pa allel case, while we
lea e he inal de ails on he uppe bound o he eade .
Se ies case Assume ha he saddle poin s Zxa,xb={z1,...,zN}a einse ies,seeFig.2.
We use he o de ing as in (1.11) and deno e he poin s xias in (1.12). We also ix he
islands, Uxi−1and Uxi(componen s o {F<F(xa;xb)−δ/3}), which a e connec ed
by he b idge Ozi,δ. Again we may assume ha zi=0, F(0)=0 and ha
Oδ=Ozi,δ ={y1:g(y1)<δ}×{y:G(y)<δ}.
By Lemma 3.5 we ha e oscUxi−1(hA,B)+osc
Uxi(hA,B)≤Cε. The e o e he e a e
numbe s ci−1,cisuch ha
|hA,B−ci−1|≤Cεin Uxi−1and |hA,B−ci|≤Cεin Uxi.
Then, using he undamen al heo em o calculus as in (4.13), we ob ain
|ci−1−ci|−2Cε≤{g<δ}|∇hA,B(y)|2e−F(y)
εdy11
2{g<δ}
eF(y)
εdy11
2
o
(y1,y)∈{g<δ}×{G<δ/100}.
Mo eo e , a guing as in (4.14), we ha e
{g<δ}
eF(y)
εdy1≤(1+ ˆη(C,ε))eG(y)
εeω(G(y))
εdε(xi−1,xi).
These oge he imply
{g<δ}|∇hA,B(y)|2e−F(y)
εdy1≥(1−ˆη(C, ε))(ci−1−ci)2
dε(xi−1,xi)e−G(y)
εe−ω(G(y))
ε.
By in eg a ing o e y∈{G<δ/100}we ha e, by Fubini’s heo em, Lemma 3.9, and
P oposi ion 4.2, ha
Oδ|∇hA,B|2e−F(y)
εdy ≥(1−ˆη(C, ε))(ci−1−ci)2Vε(xi−1,xi)
dε(xi−1,xi).
By epea ing he a gumen o e e y saddle zi∈Zxa,xband using he ac ha he se s
Ozi,δ a e disjoin we ob ain
Rn|∇hA,B|2e−F(y)
εdy ≥(1−ˆη(C,ε))
N

i=1
(ci−1−ci)2Vε(xi−1,xi)
dε(xi−1,xi).
Geome ic Cha ac e iza ion o he Ey ing–K ame s Fo mula
Recall ha he numbe s cia e he app oxima e alues o hA,Bin he componen s Uxi.
The e o e we may choose hem such ha 1 =c0and cN=0. By deno ing yi=ci−1−ci
and ai=Vε(xi−1,xi)
dε(xi−1,xi)we may w i e
N

i=1
(ci−1−ci)2Vε(xi−1,xi)
dε(xi−1,xi)=
N

i=1
aiy2
i,
whe e we ha e a cons ain N
i=1yi=1. By a s anda d op imiza ion a gumen (using
Lag ange mul iplie s) we ge ha unde such a cons ain i holds ha
N

i=1
aiy2
i≥N

i=1
1
ai −1
.
This yields he lowe bound in he case when he saddle poin s a e in se ies. The uppe
bound on he o he hand ollows om a simila a gumen han in he pa allel case, and
we lea e he de ails o he eade . This comple es he p oo in he se ies case, and hence
he whole p oo . 
P oo o Theo em 2.Le us i s ecall he no a ion ela ed o Theo em 2. We assume
ha he local minima xio Fa e o de ed such ha F(xi)≤F(xj)i i≤j, and hey a e
g ouped in o se s Gisuch ha xi,xj∈Gki F(xi)=F(xj)and x∈Gi,y∈Gjwi h
i<ji F(x)<F(y). We also deno ed F(Gi):= F(x)wi h x∈Gi,Sk=k
i=1Gi,
Gε
k=x∈GkBε(x), and Sε
k=k
i=1Gε
i.
The p oo o Theo em 2 ollows om he ollowing lemma oge he wi h Lemma 3.6
and Theo em 1.
Lemma 4.3. Unde he assump ions o Theo em 2, he e exis s cons an s C =C(F)>1
and ε=ε0(F)∈(0,1)such ha , o all 0<ε≤ε0, we ha e
1
C
x∈Gk+1 |Ox,ε|≤e
F(Gk+1)
εhGε
k+1,Sε
kdμε≤C
x∈Gk+1 |Ox,ε|.
We p o e Theo em 2 i s , while he p oo o Lemma 4.3 is gi en la e on.
P oo o Theo em 2.Using Lemma 3.6 and choosing ρ=εwe ob ain ha , o εsmall
enough and x∈Gε
k+1, ha
Ex[τSε
k
IτSε
k<τc]≤ChSε
k,chGε
k+1,Sε
kdμε
cap(Gε
k+1,Sε
k)+Cεα/2.
The a io abo e can be es ima ed by using Lemma 4.3 and he mono onici y o he
capaci y. Tha is, he nume a o can be bounded by Lemma 4.3, while o he capaci y
we ha e
cap(Gε
k+1,Sε
k)≥cap(Gε
k+1,Gε
k)≥max
x∈Gk,y∈Gk+1
cap(Bε(x), Bε(y)). (4.16)
The claim o he pa allel and se ies cases now ollows by assuming ha he maximum
is a ained o a pai o minima xa∈Gk,xb∈Gk+1 and applying Theo em 1.
B. A elin, V. Julin, L. Vii asaa i
Fig. 4. Geome ic iew o mul iple minima a same heigh and mul iple saddles a he same heigh
Rema k 4.4. We no e ha in he gene al case, he las inequali y in (4.16) has he op imal
dependence wi h espec o εbu he inequali ies may di e by a cons an . Essen ially
he inequali y is sha p only in he case whe e only one saddle con ibu es o he o al
alue o he capaci y. Hence we ha e he sha p es ima e when saddle poin s a e pa allel
o in se ies, bu in gene al he si ua ion migh be mo e complica ed han ha . We ha e
illus a ed his in Fig. 4, whe e each g ay do is a saddle a he same heigh , and A,B
p oduces Gk. Then he p ecise alue o cap(Gε
k+1,A∪B)is al eady non- i ial o
calcula e.
P oo o Lemma 4.3.Fi s we will p o e a localiza ion es ima e o exponen ial in e-
g als. Conside a se 0 ∈Oand a unc ion such ha (0)=lis a p ope local
minimum and ha is locally con ex a ound 0. Then he e exis s an ε0such ha , o
any ε<ε
0,
O
e− (x)/εdx ≤Ce−l/ε|{ <ε}∩O|.(4.17)
We will i s p o e (4.17) and hen epea edly apply i o p o e Lemma 4.3.
In o de o p o e (4.17), we begin by escaling such ha l=0. Then we ex end
ou side Oas +∞and call his ex ended unc ion ˆ
. We i s p o e
{ˆ
>a}
e−ˆ
(x)/εdx ≤ce−a/ε
which, by he de ini ion o ˆ
, is equi alen o
{ >a}∩O
e− (x)/εdx ≤ce−a/ε.
This now ollows om Lemma 3.8 by using |O|<∞and obse ing ha ˆ
sa is ies he
assump ions o Lemma 3.8. Hence we obse e (4.17).
Conside now he se
U−δ2/3≡{y:F(y)≤F(Gk+1;Sk)−δ2/3}
Geome ic Cha ac e iza ion o he Ey ing–K ame s Fo mula
and le Uibe he componen o U−δ2/3con aining xi. We spli
hGε
k+1,Sε
ke−F/εdx =Uc
−δ2/3
hGε
k+1,Sε
ke−F/εdx +U−δ2/3 Sε
k
hGε
k+1,Sε
ke−F/εdx,
whe e complemen is unde s ood wi h espec o he domain . By assump ions (1.4)
and (1.6)onFi holds ha
F(Gk+1;Sk)≥F(Gk+1)+2
3δ2.(4.18)
Also by he quad a ic g ow h (1.3) we can bound he i s in eg al as
Uc
−δ2/3
hGε
k+1,Sε
ke−F/εdx ≤Ce−(F(Gk+1;Sk)−δ2/3)/ε ≤Ce−δ2
3εe−F(Gk+1)/ε,
which shows ha he i s in eg al is neglible in he inal es ima e. Fo he second in eg al
we u he spli
U−δ2/3 Sε
k
hGε
k+1,Sε
ke−F/εdx =
iUi Sε
k
hGε
k+1,Sε
ke−F/εdx.
We will now conside all he di e en componen s Uidepending on wha minima
hey con ain. We s a wi h he componen s Ui ha do no in e sec Sε
k∪Gε
k+1. Then all
local minima in Uia e la ge han F(Gk+1), and hence om (4.17) we ge ha he e
exis s a cons an Csuch ha
Ui
hGε
k+1,Sε
ke−F/εdx ≤Ce−F(Gk+2)/ε ≤Ce−δ2/εe−F(Gk+1)/ε,
whe e he las inequali y ollows om (1.19). This shows ha also his e m is neglible.
Conside nex he componen Ui ha in e sec s Gε
k+1 bu do no in e sec Sε
k. In his case,
by (4.17) and Lemmas 3.5 and 3.7,weha e
1
C
x∈Gk+1∩Ui|Ox,ε|≤e
F(Gk+1)
εUi
hGε
k+1,Sε
ke−F/εdx ≤C
x∈Gk+1∩Ui|Ox,ε|(4.19)
p o iding us he leading e m ha con ibu es o he inal es ima e.
Conside nex a componen Uisuch ha Ui∩Sε
k= ∅. Since Uiis a componen o
U−δ2/3,i ollows omUi∩Sk= ∅ ha F(y;Sk)≤F(y;Gk+1)−δ2/3≤F(y;Gk+1)
in Ui. The e o e we ha e, by Lemma 3.3,inUi ha
hGε
k+1,Sε
k≤Cεqe−(F(y;Gk+1)−F(y;Sk))/ε.
Hence, o q∈R, we ob ain
Ui
hGε
k+1,Sε
ke−F/εdx ≤εqUi
e−(F(y;Gk+1)−F(y;Sk))/εe−F(y)/εdy.
In o de o compu e he in eg al on he igh hand side we s udy he in imum alue o he
unc ion (y)=F(y;Gk+1)−F(y;Sk)+F(y). Clea ly, he in imum is a ained a an
in e io poin o Ui, deno ed by xi. I ollows ha hen xiis necessa ily a local minimum
poin o F. By abo e conside a ions, we also ha e F(y;Sk)<F(y;Gk+1) o all y∈Ui,
B. A elin, V. Julin, L. Vii asaa i
and hus we may deduce ha xi/∈Gk+1.I nowxi∈Sk, hen F(xi)=F(xi;Sk)and
hus, by he de ini ion o and by (4.18),
in
Ui
(y)= (xi)≥F(xi;Gk+1)≥F(Sk;Gk+1)≥F(Gk+1)+2
3δ2.
I emains o s udy he case whe e xi∈Gj o some j≥k+ 2. In his case we apply
in
Ui
(y)= (xi)=
≥0
!"# $
F(xi;Gk+1)−F(xi;Sk)+F(Gj)≥F(Gk+2)≥F(Gk+1)+δ2,
whe e he las inequali y ollows om (1.19). The e o e we can conclude ha , o δ3=
2
3δ2, i holds ha
Ui
εqe−(F(y;Gk+1)−F(y;Sk))/εe−F(y)/εdy ≤Cεqe−δ3
εe−F(Gk+1)/ε.
Consequen ly, he componen Uisa is ying Ui∩Sε
k= ∅does no con ibu e ei he . The
p oo is hence comple ed by (4.19) and by he ac ha he in eg al o e he emaining
componen s a e neglible whene e εis small enough. 
Acknowledgemen s. We would like o hank Aapo Kau anen and Toni Ikonen o help ul discussion on
geome ic unc ion heo y and o le ing us know he e e ences [13,20].
Funding Open access unding p o ided by Uppsala Uni e si y. B.A. was suppo ed by he Swedish Resea ch
Council dn : 2019-04098. V.J. was suppo ed by he Academy o Finland G an 314227.
Da a a ailabili y Da a sha ing no applicable o his a icle as no da ase s we e gene a ed o analysed du ing
he cu en s udy.
Decla a ions
Con lic o in e es The au ho s decla e ha hey ha e no con lic o in e es .
Open Access This a icle is licensed unde a C ea i e Commons A ibu ion 4.0 In e na ional License, which
pe mi s use, sha ing, adap a ion, dis ibu ion and ep oduc ion in any medium o o ma , as long as you gi e
app op ia e c edi o he o iginal au ho (s) and he sou ce, p o ide a link o he C ea i e Commons licence,
and indica e i changes we e made. The images o o he hi d pa y ma e ial in his a icle a e included in he
a icle’s C ea i e Commons licence, unless indica ed o he wise in a c edi line o he ma e ial. I ma e ial is
no included in he a icle’s C ea i e Commons licence and you in ended use is no pe mi ed by s a u o y
egula ion o exceeds he pe mi ed use, you will need o ob ain pe mission di ec ly om he copy igh holde .
To iew a copy o his licence, isi h p://c ea i ecommons.o g/licenses/by/4.0/.
Publishe ’s No e Sp inge Na u e emains neu al wi h ega d o ju isdic ional claims in published maps and
ins i u ional a ilia ions.
Re e ences
1. Aikawa, H., Essén, M.: Po en ial Theo y-Selec ed Topics. Sp inge , Be lin (1996)
2. Amb osio, L., Fusco, N., Palla a, D.: Func ions o Bounded Va ia ion and F ee Discon inui y P oblems.
Ox o d Uni e si y P ess, New Yo k (2000)
3. A elin, B., Julin, V.: App oxima ion o BV unc ions by neu al ne wo ks: A egula i y heo y app oach
(2020). a Xi :2012.08291
4. Be glund, N., Gen z, B.: The Ey ing–K ame s law o po en ials wi h nonquad a ic saddles. Ma ko
P ocess. Rela . Fields 16(3), 549–598 (2010)
5. Bo ie , A., Den Hollande , F.: Me as abili y: A Po en ial-Theo e ic App oach, ol. 351. Sp inge , Be lin
(2016)

Geome ic Cha ac e iza ion o he Ey ing–K ame s Fo mula
6. Bo ie , A., Eckho , M., Gay a d, V., Klein, M.: Me as abili y in s ochas ic dynamics o diso de ed
mean- ield models. P obab. Theo y Rela . Fields 119(1), 99–161 (2001)
7. Bo ie , A., Eckho , M., Gay a d, V., Klein, M.: Me as abili y and low lying spec a in e e sible Ma ko
chains. Commun. Ma h. Phys. 228(2), 219–255 (2002)
8. Bo ie , A., Eckho , M., Gay a d, V., Klein, M.: Me as abili y in e e sible di usion p ocesses I. Sha p
asymp o ics o capaci ies and exi imes. J. Eu . Ma h. Soc. (JEMS) 6, 399–424 (2004)
9. Bo ie , A., Gay a d, V., Klein, M.: Me as abili y in e e sible di usion p ocesses II. P ecise asymp o ics
o small eigen alues. J. Eu . Ma h. Soc. 7, 69–99 (2005)
10. Ca lesson, L.: Selec ed p oblems on excep ional se s (1967)
11. Ey ing, H.: The ac i a ed complex in chemical eac ions. J. Chem. Phys. 3(2), 107–115 (1935)
12. F eidlin, M.I., Wen zell, A.D.: Random pe u ba ions o dynamical sys ems. T ansla ed om he Russian
by Joseph Szücs. G undleh en de ma hema ischen Wissenscha en [Fundamen al P inciples o Ma he-
ma ical Sciences], ol 260, iii+326 pp. Sp inge , New Yo k (1984)
13. Geh ing, F.W.: Ex emal leng h de ini ions o he con o mal capaci y o ings in space. Mich. Ma h. J.
9, 137–150 (1962)
14. Gilba g, D., T udinge , N.S.: Ellip ic Pa ial Di e en ial Equa ions o Second O de . Sp inge , Be lin
(2001)
15. Hoch ei e , S., Jü gen, S.: Fla minima. Neu al Compu . 9(1), 1–42 (1997)
16. Julin, V.: Gene alized Ha nack inequali y o semilinea ellip ic equa ions. J. Ma h. Pu es Appl. 9(106),
877–904 (2016)
17. K ame s, H.A.: B ownian mo ion in a ield o o ce and he di usion model o chemical eac ions. Physica
7(4), 284–304 (1940)
18. Maz’ya, Vladimi : Lec u es on isope ime ic and isocapaci a y inequali ies in he heo y o Sobole
spaces. Con emp. Ma h. 338, 307–340 (2003)
19. Pe zka, H., Kamp, M., Adilo a, L.: Rela i e la ness and gene aliza ion. Ad . Neu al. In . P ocess. Sys .
34, 18420–18432 (2021)
20. Zieme , W.P.: Ex emal leng h and con o mal capaci y. T ans. Am. Ma h. Soc. 126, 460–473 (1967)
Communica ed by R. Sei inge