Spectral gaps and exponential integrability of hitting times for linear diffusions

www.imstat.org/aihp 2011, Vol. 47, No. 3, 679–698

DOI:10.1214/10-AIHP380

© Association des Publications de l’Institut Henri Poincaré, 2011

Spectral gaps and exponential integrability of hitting times for linear diffusions

Oleg Loukianov

, Dasha Loukianova

and Shiqi Song

aDépartement Informatique, IUT de Fontainebleau, Université Paris Est, route Hurtault, 77300 Fontainebleau, France.

E-mail:oleg.loukianov@u-pec.fr

bDépartement de Mathématiques, Université d’Evry-Val d’Essonne, Bd François Mitterrand, 91025 Evry, France.

E-mail:dasha.loukianova@univ-evry.fr;shiqi.song@univ-evry.fr Received 1 October 2009; revised 18 May 2010; accepted 28 June 2010

Abstract. LetX be a regular continuous positively recurrent Markov process with state spaceR, scale functionS and speed measurem. Fora∈Rdenote

B_a⁺=sup x≥a

m

]x,+∞[

S(x)−S(a) , B_a⁻=sup

x≤a m

]−∞;x[

S(a)−S(x) .

It is well known that the finiteness ofB^±_a is equivalent to the existence of spectral gaps of generators associated withX. We show how these quantities appear independently in the study of the exponential moments of hitting times ofX. Then we establish a very direct relation between exponential moments and spectral gaps, all by improving their classical bounds.

Résumé. SoitX un processus de Markov récurrent positif à trajectoires continues et à valeurs dansR.SoientS sa fonction d’échelle etmsa mesure de vitesse. Poura∈Rnotons

B_a⁺=sup x≥a

m

]x,+∞[

S(x)−S(a) , B_a⁻=sup

x≤a m

]−∞;x[

S(a)−S(x) .

Il est bien connu que la finitude deB_a^±est équivalente à l’existence d’un trou spectral du générateur associé àX.Nous montrons comment ces quantités apparaissent d’une manière indépendante dans l’étude des temps d’atteinte deX.Ensuite nous établissons une relation directe entre les moments exponentiels et le trou spectral, en améliorant en plus leurs encadrements classiques.

MSC:60J25; 60J35; 60J60

Keywords:Recurrence; Linear Markov process; Exponential moments; Hitting times; Poincaré inequality; Spectral gap; Dirichlet form

Introduction

Let(X_t; t≥0)be a regular linear continuous Markov process with the state space R. We assume throughout the paper thatX is positively recurrent and conservative (the killing time is identically +∞). Denote byS(x) a scale function ofXandm(dx)the speed measure associated withS (cf. [23], Chapter VII). Recall thatS is a continuous strictly increasing function andm(dx)is a symmetric measure forX, charging every not empty open set. Moreover, the positive recurrence ofXimplies limx→±∞S(x)= ±∞andm(R) <∞.

In this paper we study some relations between the exponential moments of hitting times ofX, the finiteness of the quantitiesB_a⁺,B_a⁻given by (0.1), the Hardy and Poincaré inequalities for Dirichlet forms associated withX. As a consequence we give a chain of equalities linking all these objects.

Leta∈RandTa=inf{t≥0: Xt=a}be the hitting time ofabyX. The first question we are interested in is the estimation of exponential momentsEx[e^λTa],x∈R,λ >0.

In some particular cases such moments have been well studied. We mention, for example, Ditlevsen [7] for Ornstein–Uhlenbeck process, Giorno et al. [12] for Bessel and Ornstein–Uhlenbeck processes, Deaconu and Wantz [6] for diffusion with strong drift, and the book of Borodin and Salminen [3] for an overview of known formulas.

But we were not able to find in the literature a simple general estimate of exponential moments in terms of the scale functionS(x)and the speed measurem(dx).

In the present paper this question receives a very satisfactory response with the quantities:

⁺_a(x)=m

]x,+∞[

S(x)−S(a) , ⁻_a(x)=m

]−∞;x[

S(a)−S(x) .

Namely, letλ⁺_a be the supremum ofλ >0 such thatExe^λTa <∞for somex > a (hence for allx > a, see the

“all-or-none” property, Proposition1.2). Respectively, letλ⁻_a be the supremum ofλsuch thatExe^λTa <∞for some (all)x < a. Put

B_a⁺=sup

x≥a

⁺_a(x), B_a⁻=sup

x≤a

⁻_a(x) (0.1)

and

C_a^±=lim inf

x→±∞^±_a(x).

Our first result (see Section1, Theorem1.1) asserts that 1

4Ba^±

≤λ^±_a ≤ 1 4Ca^±

∧ 1

B_a^±, (0.2)

whereB_a⁺orB_a⁻can eventually be infinite. The positivity ofλ^±_a is thereby equivalent to the finiteness ofB_a^±. More- over, ifλ^±_a >0 for somea∈R, it is so for alla∈R.

As it turns out, the importance of the quantitiesB_a^±is well-known in some different context. Using Krein’s method, Kac and Krein [15] and Kotani and Watanabe [17] have shown that the spectral gapsγ_a⁺ (respectivelyγ_a⁻) of the generator ofXkilled when it exits]a,+∞[(resp.]−∞, a[) satisfy

1 4Ba^±

≤γ_a^±≤ 1

B_a^±. (0.3)

We see therefore that the positivity ofλ^±_a is equivalent to this ofγ_a^±, a fact that can actually be derived e.g. from the works of Down et al. [8] and Bakry [1] or Rökner and Wang [24]. But comparing (0.2) and (0.3) leads to the more explicit conjectureλ^±_a =γ_a^±. This identity would not be surprising, since for exit times from a bounded domainDit is well known from the works of Khasminskii [16] and Friedman [10]. Namely, ifτDis the exit time fromDandX^D is a process killed atτ_D, then the equality holds between the widthγ_D of the spectral gap of the generator ofX^D and the supremumλ_D ofλsuch thatExe^λτD<∞for allx∈D. However, the PDE methods of [16] requireExτ to be bounded, which is not the case in general, and in particular forD= ]a,∞[.

What was surprising, is that we have not found in the literature an analogue of Khasminskii identity for unbounded domains. So in the second section of this article we firstly show (Theorem2.1) in the setting ofm-symmetric Hunt processes that

λD=γD ifλD=sup

λ: Exe^λτD∈L¹(mID)

>0.

The proof is based on the spectral calculus, available thanks to the symmetry of the generator of X^D. Since this symmetry is automatically fulfilled in dimension one, we get for anya∈R,

γ_a⁺=λ⁺_a and γ_a⁻=λ⁻_a. (0.4)

Notice that we cannot establish such kind of equality directly onR, because our process is conservative and the exit time fromRis identically infinite. But it is well known that the spectral gapγ of am-symmetric operator can be characterized in terms of Poincaré inequality for its Dirichlet form. We recall in the sequel of Section2that

γ_a^±= 1

A^±_a and γ= 1 c_P,

whereA^±_a andc_P are the best possible constants satisfying _∞

a

F (x)−F (a)2

m(dx)≤A⁺_{a ∞}

a

dF dS

2

(t)dS(t), _a

−∞

F (x)−F (a)2

m(dx)≤A⁻_{a a}

−∞

dF dS

2

(t)dS(t) and

_+∞

−∞

F (x)−m(F ) m(R)

2

m(dx)≤c_{P +∞}

−∞

dF dS

2

(x)dS(x) for allF in an appropriate functional spaceF.

A classical Mukenhoupt result [22] yields B_a⁺≤A⁺_a ≤4B_a⁺ and B_a⁻≤A⁻_a ≤4B_a⁻.

Looking at (0.2) and (0.4), we see that the lower bounds above can actually be replaced by 4C_a^±∨B_a^±, which are sometimes more precise. They also allow to recover (in our settings) the case of equalityA^±_a =4B_a^±studied in Miclo [21].

Further,c_P can be easily related toA^±_a by sup

a

A⁺_a ∧A⁻_a

≤c_P ≤inf

a

A⁺_a ∨A⁻_a

which yields sup

a

B_a⁺∧B_a⁻

≤cP ≤4 inf

a

B_a⁺∨B_a⁻ .

In the works of Bobkov and Götze [2] and Malrieu and Roberto [20] it was shown that these inequalities can be replaced by

1

2B_me⁺ ∨B_me⁻ ≤c_P ≤4

B_me⁺ ∨B_me⁻ ,

wheremeis a median ofm(see [20], Theorem 6.6.2). At the end of Section2we give another refinement of the bounds oncP. After proving thatA^±_a areS-Hölder ina, we show that there exists a pointc∈Rsuch thatA⁺_c =A⁻_c =cP, which yields

B_c⁺∨B_c⁻∨4C_c⁺∨4C_c⁻≤c_P ≤4

B_c⁺∧B_c⁻ .

The pointcis, however, unknown (except for obvious symmetric cases) and worths further investigations.

Finally, the last section contains some examples to illustrate the above-mentioned results.

1. Exponential integrability of hitting times

In this section we study the exponential moments of hitting timesT_a. Fora∈R, denote λ⁺_a =sup

λ≥0:∀x > a,Exe^λTa <∞

; and

λ⁻_a =sup

λ≥0:∀x < a,Exe^λTa <∞ .

As we will see (Proposition1.2), an important “all-or-none” property holds for anyλ >0:

∃x > a, Exe^λTa<∞ ⇐⇒ ∀x > a, Exe^λTa <∞, the same being true forx < a.

Recall the definitions B_a⁺=sup

x≥a

m

]x,+∞[

S(x)−S(a) , B_a⁻=sup

x≤a

m

]−∞;x[

S(a)−S(x) and

C_a⁺=lim inf

x→∞ m

]x,+∞[

S(x)−S(a) , C_a⁻=lim inf

x→−∞m

]−∞;x[

S(a)−S(x) . The main result of this section is

Theorem 1.1. For alla∈R, 1

4Ba⁺

≤λ⁺_a ≤ 1 4Ca⁺

∧ 1

B_a⁺ and 1 4Ba⁻

≤λ⁻_a ≤ 1 4Ca⁻

∧ 1 B_a⁻ with the convention1/∞ =0.

In the sequel we often prove only assertions concerningB_a⁺andλ⁺_a, since the proofs of their “left” counterparts are completely similar.

1.1. Kac formula

The Kac formula, first derived in [13,14] for linear Brownian motion, then generalized in [5] and in [9], permits to calculate the moments ofA_v= ₀^Tv(X_t)dt for a functionvof a Markov process(X)and a suitable random timeT. In our proof we need a particular case of this formula, wherev=1 andT is an exit time from an interval or a hitting time.

Fora < x < bconsider T_a,b=inf

t≥0;X_t∈ ]/ a, b[ .

The Green potential kernel on[a, b]is given by (see e.g. [23], Chapter VII) G(a, b, x, y)=(S(b)−S(x∨y))(S(x∧y)−S(a))

S(b)−S(a) .

This kernel defines the Green operator Gf (x)=

_b

a

G(a, b, x, y)f (y)dm(y).

Notice that sinceG(a, b, x, a)=G(a, b, x, b)=0, the integration interval may or may not includeaandb.

With the help of this operator we can calculate the moments ofT_a,busing Kac formula ExT_a,bⁿ =n

_b

a

G(a, b, x, y)EyT_a,bⁿ⁻¹dm(y)=n!Gⁿ1(x). (1.1) To obtain an analogous formula for the moments of hitting times, recall that limt→±∞S(t)= ±∞, and consider the limits ofG(a, b, x, y)whena→ −∞(resp.b→ ∞):

G(−∞, b, x, ξ )= S(b)−S(ξ )

, x≤ξ≤b, S(b)−S(x)

, −∞< ξ≤x,

G(a,+∞, x, ξ )= S(ξ )−S(a)

, a≤ξ≤x, S(x)−S(a)

, x≤ξ <∞.

Taking monotone limits in (1.1), we get a formula for thenth moment of hitting times (see also [18]):

ExT_bⁿ=n b

−∞G(−∞, b, x, ξ )EξT_bⁿ⁻¹dm(ξ ) ifx < b,

(1.2) ExT_aⁿ=n

_+∞

a

G(a,+∞, x, ξ )EξT_aⁿ⁻¹dm(ξ ) ifx > a.

The summation overnyields a formula for exponential moments:

Exexp(λTb)=1+λ _b

−∞G(−∞, b, x, ξ )Eξexp(λTb)dm(ξ ), x < b,

(1.3) Exexp(λTa)=1+λ

_+∞

a

G(a,+∞, x, ξ )Eξexp(λTa)dm(ξ ), x > a.

Remark. The expressions(1.2)–(1.3)are always defined,since all functions therein are positive.

The following proposition will be referred to as “all-or-none” property in the sequel:

Proposition 1.2 (All-or-none). Leta∈Randλ >0.The following properties are equivalent:

• for somex > a,Exexp(λTa) <∞,

• for allx > a,Exexp(λTa) <∞,

• _a^+∞Eξexp(λTa)dm(ξ ) <∞.

The same holds forx < a.

Proof. Observe thatG(a,+∞, x, ξ )≡const>0 forξ > x, and thatE_•exp(λTa)is increasing on]a,∞[. Using the exponential Kac formula we then see that forx > a,Exexp(λTa) <∞if and only if Eξexp(λTa)ism-integrable on ]a,∞[. In this case, since mcharges every interval ofR, Eξexp(λTa) <∞ for allξ > a by monotonicity of

Eξexp(λTa).

1.2. Exit time from an interval

It is known for a while (Khasminskii condition, see [9]) that the exit timeT_a,bfrom a bounded interval[a, b]admits an exponential moment for someλ >0. Let

λ_a,b=sup

λ: ∀x∈ ]a, b[,Exexp(λTa,b) <∞ .

In this subsection we establish some upper and lower bounds forλ_a,b. The upper bound will be particularly important in the proof of the Theorem1.1.

Lemma 1.3. λ_a,b≥1/Ca,b,where C_a,b= 1

S(b)−S(a) b

a

S(b)−S(y)

S(y)−S(a) m(dy).

Proof. Observe that

G(a, b, x, y)≤(S(b)−S(y))(S(y)−S(a)) S(b)−S(a) . Iff is positive and bounded, then

Gf (x)≤ 1 S(b)−S(a)

_b

a

S(b)−S(y)

S(y)−S(a)

f (y)m(dy)≤C_a,bf_∞.

It follows thatExT_a,bⁿ /n! =Gⁿ1(x)≤C_a,bⁿ , whenceExe^λTa,b<∞forλ <1/Ca,b. Makingb→ ∞, we getExe^λTa<∞for allx > aif

1

λ > C_a,∞= _∞

a

S(y)−S(a) m(dy),

but the last integral is generally infinite, so this inequality does not provide a satisfactory lower bound forλ⁺_a. Observe, however, that whenCa,∞<∞, we getExe^λTa ≤(1−λCa,∞)⁻¹ for allx > a andλ <1/Ca,∞(we postpone an example for the last section).

To find an upper bound forλ_a,b, fix some interval[a, b] ⊂ ]a, b[.Defineκ₁>0 andκ₂>0 by S

a

−S(a)=κ₁ S

b

−S a

, S(b)−S b

=κ₂ S

b

−S a

(1.4) and denote

c= κ₁κ₂ 1+κ₁+κ₂

S b

−S a

m a, b

.

Lemma 1.4. Ifλ≥1/cthen for allx∈ [a, b],Exe^λTa,b= ∞.In particular,λ_a,b≤1/cfor any choice of[a, b].

Proof. Observe that for allx, yin[a, b],

G(a, b, x, y)≥(S(b)−S(b))(S(a)−S(a))

S(b)−S(a) = κ₁κ₂ 1+κ₁+κ₂

S b

−S a

. It follows that for allx∈ [a, b]

ExT_a,b≥ _b

a

G(a, b, x, y)dm(y)≥ κ₁κ₂ 1+κ₁+κ₂

S b

−S a

m a, b

=c.

By inductionExT_a,bⁿ ≥n!cⁿ, as seen from ExT_a,bⁿ ≥n

_b

a

G(a, b, x, y)EyT_a,bⁿ⁻¹dm(y)≥n(n−1)!cⁿ⁻¹ _b

a

G(a, b, x, y)dm(y)=n!cⁿ.

HenceExe^λTa,b = ∞pourλ≥1/candx∈ [a, b].

At this stage let us mention a theorem of Carmona–Klein [4]

“spectral gap”⇒Exe^λTU <∞,

whereUis a set of positive invariant measure. The formulation of this theorem is somewhat confusing, since it does not precise thatλdepends onU. In fact, the following corollary shows that the propertyExe^λTa <∞can not hold simultaneously for all(x, a)with a commonλ >0.

Corollary 1.5. ∀λ >0,∀x∈R,there exista < xandb > xsuch thatExe^λTa,b=Exe^λTa =Exe^λTb = ∞.

Proof. Fixλ >0 and x∈R. Put, for example, κ1=κ2=1 and chose [a, b] and[a, b]in such a way that x∈ [a, b] ⊂ ]a, b[and the equalities (1.4) hold. Then

1

c= 3

(S(b)−S(a))m([a, b])< λ

as soon as (S(b)−S(a))m([a, b]) >3λ, which can always be achieved takinga orblarge enough. Hence, ac- cording to Lemma1.4,Exe^λTa,b= ∞and therebyExe^λTa=Exe^λTb = ∞for suchaandb.

1.3. Hitting time moments

In this subsection we will prove the Theorem1.1:

1 4Ba⁺

≤λ⁺_a ≤ 1 4Ca⁺

∧ 1

B_a⁺ and 1 4Ba⁻

≤λ⁻_a ≤ 1 4Ca⁻

∧ 1 B_a⁻.

The proofs of two parts being completely similar, we only give one forλ⁺_a. It will be split in a number of propositions.

Proposition 1.6. ∀a∈R, λ⁺_a ≤_B¹+

a ,where1/∞ =0.

Proof. Fix somea∈R. From Lemma1.4we deduce that fora < a< x < b< b, for allx∈ [a, b],Exe^λTa = ∞if λ≥1/c, where

1

c= 1+κ1+κ2

κ1κ2(S(b)−S(a))m([a, b]).

Now fixaandband makek₂→ ∞(sob→ ∞), then 1

c→ 1

κ₁(S(b)−S(a))m([a, b])= 1

(S(a)−S(a))m([a, b]). We conclude thatExe^λTa= ∞forx∈ [a, b]and

λ > 1

(S(a)−S(a))m([a, b]).

It follows by the “all-or-none” Proposition1.2thatExe^λTa = ∞for anyx > a, alla,bandλas above. Observing that

sup

a,b

S a

−S(a) m

a, b

=sup

a>a

S a

−S(a) m

a,∞

=B_a⁺

by the continuity ofS, we getExe^λTa = ∞for any λ > inf

a,b

1

(S(a)−S(a))m([a, b])= 1 B_a⁺.

The inequalityλ⁺_a ≤1/B_a⁺is thereby proved.

The bounds(4C_a⁺)⁻¹≥λ⁺_a ≥(4B_a⁺)⁻¹require more work. To simplify the notations we putB_a⁺=B. Letb≥aand define forx > aandf ≥0 two positive linear operators,JbandKb:

K_bf (x)= x

a

S(y)−S(a)

f (y)1b≤ym(dy), J_bf (x)=

S(x)−S(a) 1b≤x

_∞

x

f (y)m(dy), where _x^yis understood as _]x,y]. Notice that

Gf =J_af+K_af≥J_bf +K_bf=G(f1_[b,∞[).

Put

C(b)=inf

y>bJb1(y), B(b)=sup

y>b

Jb1(y), soB=B(a).

Proposition 1.7. We have n

l=0

a_n,lC^l(b)K_b^n−l1(x)≤(J_b+K_b)ⁿ1(x)≤ n

l=0

a_n,lB^l(b)K_b^n−l1(x) wherea_n,l≥0satisfy

a_n,l=0 ifl <0orl > n, a_0,0=1, a_n+1,l=

i≤l

a_n,i.

Proof. For anyf we have J_bK_bf (x)=

S(x)−S(a) 1b≤x

_∞

x

dm(y) _y

a

S(u)−S(a)

f (u)1b≤udm(u)

=

S(x)−S(a) 1b≤x

_∞

x

dm(y) _x

a

S(u)−S(a)

f (u)1b≤udm(u) +

S(x)−S(a) 1b≤x

_∞

x

dm(y) y

x

S(u)−S(a)

f (u)1b≤udm(u)

=J_b1(x)Kbf (x)+

S(x)−S(a) 1b≤x

_∞

x

f (u)dm(u)

S(u)−S(a) 1b≤u

_∞

u

dm(y)

=Jb1(x)Kbf (x)+Jb(f Jb1)(x)

whence

C(b)(K_bf +J_bf )(x)≤J_bK_bf (x)≤B(b)(K_bf+J_bf )(x),

the inequalities being trivially true forx≤b. By induction, we easily see that the following inequalities hold for all n∈N:

C(b)

Cⁿ(b)+Cⁿ⁻¹(b)Kb1(x)+ · · · +K_bⁿ1(x)

≤J_bK_bⁿ1(x)≤B(b)

Bⁿ(b)+Bⁿ⁻¹(b)K_b1(x)+ · · · +K_bⁿ1(x) .

Now notice that forn=0,(J_b+K_b)ⁿ1(x)=1=a_0,0. By induction again, for anyx > a, (J_b+K_b)ⁿ⁺¹1(x)≤(J_b+K_b)

l≥0

a_n,lB^l(b)K_b^n−l1(x)

=J_b

l≥0

a_n,lB^l(b)K_b^n−l1(x)+

l≥0

a_n,lB^l(b)K_b^n−l+11(x)

≤B(b)

l≥0

a_n,lB^l(b)

n−l

i=0

B^n−l−i(b)K_bⁱ1(x)+

l≥0

a_n,lB^l(b)K_b^n−l+11(x)

=

i≥0

Bⁿ⁺¹⁻ⁱ(b)K_bⁱ1(x)

l≤n−i

a_n,l+

i≥0

a_n,n+1−iBⁿ⁺¹⁻ⁱ(b)K_bⁱ1(x)

=

i≥0

Bⁿ⁺¹⁻ⁱ(b)K_bⁱ1(x)

l≤n+1−i

a_n,l=

j≥0

B^j(b)K_b^n+1−j1(x)

l≤j

a_n,l

=

j≥0

an+1,jB^j(b)K_b^n+1−j1(x).

The lower bound is proved in the same way.

An explicit formula fora_n,l will now be derived.

Lemma 1.8. For0≤l≤n,an,l=C_n^l_+l−C_n^l−₊¹_l,withC_n⁻¹=0.This implies 4ⁿ

n+2

√ 8

π(n+2) n k=0

a_n,k=a_n+1,n≤4ⁿ, whence

nlim→∞

√na_n+1,n=4.

Proof. The expression ofan,lfollows by induction from the recursive definition ofan,land the equality C⁰_n+C_n¹₊₁+ · · · +C_n^l_+l=C_n^l_+l+1.

The second assertion follows then easily by Stirling’s formula.

Theorem 1.9. Recall that B=B_a⁺=sup

x>a

S(x)−S(a) m

]x,∞]

and

C_a⁺=lim inf

x→∞

S(x)−S(a) m

]x,∞]

.

• If0≤λ < (4B_a⁺)⁻¹then for allx > a,Exe^λTadm(x) <∞.

• Ifλ > (4C_a⁺)⁻¹then for allx > a,Exe^λTadm(x)= ∞.

Proof. Putfb= _b^∞f (z)m(dz)forb≥a. We have forf ≥0 Kbfb=

_∞

b

m(dz) z

b

S(u)−S(a)

f (u)m(du)= _∞

b

f (u)m(du)

S(u)−S(a)

∞ u

m(dz), whence

C(b)fb≤ K_bfb≤B(b)fb. (1.5)

The first point of the theorem being obviously true forB= ∞, we can suppose thatB=B(a) <∞. Combining the inequality (1.5) forb=awith the Proposition1.7and Lemma1.8, we can write

1 n!

_+∞

a

ExT_aⁿdm(x)=Gⁿ1

a≤ n l=0

an,lB^lK_a^n−l1

a

≤ n

l=0

a_n,lB^lB^n−l1a=a_n+1,nBⁿm ]a,∞[

≤4ⁿBⁿm ]a,∞[

,

which implies the first assertion.

In the same way, for anyb≥a, Gⁿ1

b≥ n

l=0

an,lC(b)^lC(b)^n−l1b=an+1,nC(b)ⁿm ]b,∞[

.

Lemma1.8now implies that _b^∞Exe^λTam(dx)= ∞for anyλ > (4C(b))⁻¹. The second point follows by the “all-or-

none” property, since limb→∞C(b)=C_a⁺.

Finally, the Propositions1.6and1.9jointly imply the assertion of Theorem1.1, namely 1

4Ba⁺

≤λ⁺_a ≤ 1 4Ca⁺

∧ 1

B_a⁺ and 1 4Ba⁻

≤λ⁻_a ≤ 1 4C⁻a

∧ 1 B_a⁻, the inequalities concerningλ⁻_a being proved in the same way.

Remark. It is easy to see thatB_a^±<∞implies∀a∈R, B_a^±<∞.So the Theorem1.1yields yet another “all-or- none” property:

∃a∈R, λ^±_a >0 ⇐⇒ ∀a∈R, λ^±_a >0.

The Corollary1.5implies,however,that

alim→∞λ⁻_a = lim

a→−∞λ⁺_a =0.

2. Spectral gap

It turns out that the quantitiesB_a^±have already appeared in the context of the operators theory (see, e.g., [15,17]). In this section we discuss some relations between exponential moments of a diffusion and spectral gaps of associated operators.

We begin with a general remark. Consider a Hunt process X on a Polish space Ein the sense of Fukushima et al. [11]. Letmbe a Radon measure onE. Suppose thatmis bounded andXis am-symmetric process. Denote by (P_t)_t≥0the transition semigroup ofX. Denote byPx, x∈E, the law of the processXissued fromx∈E.

For an open setG⊆E, set τ_G=inf{t >0:X_t ∈/G} the exit time ofXfromG. Introduce

P_t^G[A](x)=Px[X_t∈A;t < τ_G] for measurable subsetAofE, and set

X^G_t =

Xt, 0≤t < τG, , t≥τG.

Then, according to [11],X^Gis a Hunt process on the state spaceG, symmetric with respect to the measureIG·m(dx) with the transition semigroup(P_t^G). IfA^Gdenotes the infinitesimal generator of(P_t^G)inL²(IG·m(dx)),A^Gis a self- adjoint negative operator. Let us denote by(·,·)the scalar product inL²(IG·m(dx))and by(E_ξ, ξ≥0)the spectral family of−A^G.

Recall now the usual properties of the spectral decomposition. For any bounded and continuousf on[0,∞[one can definef (−A^G)by

f

−A^G u=

[0;∞[f (ξ )dEξu, u∈L²

IG·m(dx) , with

f

−A^G u, g

−A^G v

=

[0;∞[f (ξ )g(ξ )d(Eξu, v).

In particular, P_t^G=exp

t A^G

=

[0;∞[e^{−ξ t}dEξ.

Denote by E^G the Dirichlet form associated with −A^G on L²(m). Let H_ξ be the image space ofE_ξ (which is a projection operator). The elements ofH₀are those who satisfyP_t^Gu=ufor allt >0. We know that−A^Ghas a spectral gap at 0 of width at leastγ >0 if and only if the following inequality

γu−E₀u²=γ

]0,∞[d(Eξu, u)≤

]0,∞[ξd(Eξu, u)=E^G(u, u) (2.1)

holds for alluin the domain ofE^G. 2.1. Khasminskii identity

The main theorem of this section concerns the caseτG<∞.

For any bounded non negative functionu(x)∈L²(IG·m(dx)), for allλ >0, 0< N <∞, we have the formulas N

0

e^λtP_t^G1(x)dt=1 λEx

e^λτG∧N−1 , _N

0

e^λtP_t^Gu(x)dt≤ u∞1 λEx

e^λτG∧N−1 . The spectral calculus yields

1 λEx

e^λτG∧N−1

=

[0,∞[

e^{(λ−ξ )N}−1 λ−ξ dEξ1.

Hypothesis (λ0). λ0>0and for anyλ < λ0,Ex[e^λτG]is an element ofL¹(IG·m(dx)).

Theorem 2.1. Hypothesis(λ0)is equivalent toE_(λ0−)=0,i.e.−A^Ghas a spectral gap of width at least equal toλ₀. Remark. This equivalence for bounded domains G⊂Rⁿ is well-known since the works of Khasminskii [16] and Friedman[10].However,the proof of[16],Theorem2,makes use of the boundedness ofExτ_GinG,which may not be the case in our general setting.

The proof is divided in two parts.

Lemma 2.2. Hypothesis(λ0)implies _[0,λ

0[dEξ=0,i.e.E_(λ0−)=0.

Proof. Let 0< λ < λ₀. For any bounded non negative functionf (x)∈L²(IG·m(dx)), for allλ >0, 0< N <∞, we can write

f²_∞ λ

E•

e^λτG∧N−1 ,1

≥ _N

0

e^λtP_t^Gfdt, f

=

[0,∞[d(Eξf, f ) _N

0

e^{(λ−ξ )t}dt

≥

[0,λ[d(Eξf, f ) _N

0

e^{(λ−ξ )t}dt

=

[0,λ[

e^{(λ−ξ )N}−1

λ−ξ d(Eξf, f ).

Taking the limit whenN↑ ∞, the preceding computation gives (E_(λ−)f, f )=

[0,λ[d(Eξf, f )=0.

The bounded non negative functions being dense inL²(IG·m(dx)), we conclude thatE_(λ−)=0. Since this holds for

any 0< λ < λ0, the lemma is proved.

Lemma 2.3. Let0< λ < λ₀.Suppose thatE_(λ0−)=0,i.e. _[0,λ

0[dEξ=0.Then,E•[e^λ]is an element ofL¹(IG· m(dx))and therefore Hypothesis(λ0)is true.

Proof. For 0< λ < λ₀we look at the formula 1

λE•

e^λτG∧N−1 ,1

=

[λ0,∞[

1−e^{(λ−ξ )N} ξ−λ

d(Eξ1,1).

LetN↑ ∞. The dominated convergence theorem yields 1

λ E_•

e^λτG−1 ,1

=

[λ0,∞[

1

(ξ−λ)d(Eξ1,1)

≤ 1

(λ₀−λ)(1,1)

= 1

(λ₀−λ)m(G) <∞ whence

Eme^λτG≤ m(G) 1−λ/λ0

. (2.2)

Now, forG= ]a,∞[andG= ]−∞, a[, denote byγ_a⁺andγ_a⁻the spectral gaps of the corresponding operators. In virtue of the “all-or-none” Proposition1.2, the Hypotheses(λ^±_a)are verified, and we obtain

Theorem 2.4. For anya∈R,γ_a^±=λ^±_a,whence 1

4Ba^±

≤γ_a^±≤ 1 4Ca^±

∧ 1 B_a^±. 2.2. Hardy and Poincaré inequalities

The Theorem2.1and the above (in)equalities only make sense under the conditionτ_G<∞. In this subsection we would like to estimate the spectral gap of a (not-killed) diffusionXonG=E=R. To do so, we use the well-known relations between spectral gaps and Poincaré inequalities.

Recall thatS andmare a scale function and the corresponding speed measure ofX. Denote by dS the measure induced byS(x). LetF (x)be a real function onR. We shall write dF dS, if there exists a functionf (x)inL¹(dS) such that

_b

a

f (x)dS(x)=F (b)−F (a) ∀a < b.

The functionf (x)will be denoted ^dF_dS(x). Introduce then the function spaces F=

F∈L²(m): dFdS,dF

dS ∈L²(dS)

, (2.3)

F_]a,∞[=

F∈F: F (x)=0, x≤a , F]−∞,a[=

F∈F: F (x)=0, x≥a .

Theorem 2.5. The diffusionXism-symmetric.The Dirichlet space associated withXis the function spaceFgiven by(2.3),and the Dirichlet form has the expression

E(F, F )= _∞

−∞

dF dS

2

(x)dS(x), F ∈F.