Spectral condition, hitting times and Nash inequality

(1)

www.imstat.org/aihp 2014, Vol. 50, No. 4, 1213–1230

DOI:10.1214/13-AIHP560

Spectral condition, hitting times and Nash inequality

Eva Löcherbach

^a

, Oleg Loukianov

^b

and Dasha Loukianova

^c

aUniversité de Cergy-Pontoise, CNRS UMR 8088, Département de Mathématiques, 95 000 Cergy-Pontoise, France.

E-mail:[email protected]

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

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

Received 6 February 2012; revised 21 November 2012; accepted 21 March 2013

Abstract. LetXbe aμ-symmetric Hunt process on a LCCB spaceE. For an open setG⊆E, letτGbe the exit time ofXfromG andA^Gbe the generator of the process killed when it leavesG. Letr:[0,∞[ → [0,∞[andR(t)=_t

0r(s)ds.

We give necessary and sufficient conditions forEμR(τG) <∞in terms of the behavior near the origin of the spectral measure of−A^G. Whenr(t)=t^l,l≥0, by means of this condition we derive the Nash inequality for the killed process.

In the diffusion case this permits to show that the existence of moments of orderl+1 forτ_Gimplies the Nash inequality of orderp=^l_l⁺₊²₁for the whole process. The associated rate of convergence of the semi-group inL²(μ)is bounded byt⁻^(l⁺¹⁾.

Finally, we show for general Hunt processes that the Nash inequality giving rise to a convergence rate of ordert⁻^(l⁺¹⁾of the semi-group implies the existence of moments of orderl+1−εforτ_G, for allε >0.

Résumé. SoitXun processus de Huntμ-symétrique à valeurs dans un espace LCCBE. Pour un ouvertG⊆E, soitτGle temps de sortie deGparXetA^Gle générateur du processus tué lorsqu’il quitteG. Soitr:[0,∞[ → [0,∞[etR(t)=_t

0r(s)ds.

Nous établissons des conditions nécéssaires et suffisantes pour queEμR(τG) <∞. Ces conditions sont données en termes du comportement au voisinage de zéro de la mesure spectrale de−A^GDans le cas our(t)=t^l,l≥0, en utilisant ces conditions, à partir deEμR(τG) <∞nous déduisons l’inégalité de Nash pour le processes tué.

Dans le cas d’un processus de diffusion cela permet de montrer que l’existence des moments d’ordrel+1 pourτGimplique l’inégalité de Nash d’ordrep= ^l_l⁺₊²₁pour le processusX. La vitesse de convergence du semi-groupe dansL²(μ)est donnée par t⁻^(l⁺¹⁾.

Finalement pour un processus de Huntμ-symétrique à valeurs dans un espace LCCB nous montrons que l’inégalité de Nash donnant lieu à la convergence du semi-groupe avec la vitesset⁻^(l⁺¹⁾implique l’existence des moments d’ordrel+1−εpourτ_G, pour toutε >0.

MSC:60J25; 60J35; 60J60

Keywords:Recurrence; Hitting times; Dirichlet form; Nash inequality; Weak Poincaré inequality;α-mixing; Continuous time Markov processes

1. Introduction

In the recent literature on convergence rates for continuous time Markov processes, the link between functional inequalities and the integrability of hitting times has regained a new interest.

The most studied case is undoubtedly the exponential one. It is known since Carmona–Klein [5] that for a very general Markov process with invariant probabilityμand Dirichlet form(E,D(E))onL²(μ), the Poincaré inequality

μ f²

≤CPE(f, f ), f ∈D(E), μ(f )=0,

(2)

implies the exponentialμ-integrability of hitting times of open sets. The converse implication for reversible diffusions can be deduced from the work of Down–Meyn–Tweedie [6]. In the particular case of linear diffusions, a simple proof of the equivalence between Poincaré inequality and exponential integrability of hitting times, with explicit estimations, was given in Loukianov, Loukianova and Song [10]. In a recent preprint by Cattiaux, Guillin and Zitt [4] the authors show that for symmetric hypo-elliptic diffusions in Rⁿ, both Poincaré inequality and exponential integrability of hitting times are equivalent to the existence of Lyapunov functions.

Although the exponential case, at least for diffusion processes, is now fairly well understood, the sub-exponential, and in particular the polynomial one, is less studied. To the best of our knowledge, the first work in this direction was done by Mathieu [11]. For a diffusion driven by a logarithmically decreasing potential, he gives a bound for the first moment of hitting times and relates this bound to some functional inequality.

More recently, the last chapter of [4] is devoted to the study of the relation between polynomial moments of hitting times and the weak Poincaré inequality

μ f²

≤β(s)E(f, f )+sΦ(f ), s >0, f ∈D(E), μ(f )=0, (1.1) where

Φ:L²(μ)→ [0,∞], Φ(cf )=c²Φ(f ),∀c∈R, f ∈L²(μ). (1.2) For uniformly strongly hypo-elliptic symmetric diffusions onRⁿ, using Lyapunov functions, the authors show that the finiteness of polynomial momentsv_m(x)=Ex(T_U^m),m∈N,T_U =inf{t >0: X_t ∈U}for some bounded open setU together with a local Poincaré inequality implies the weak Poincaré inequality withΦ(f )=(Oscf )²and with rate-generating functionβgiven by

β(s)=C

inf

u:μ v_m₋₁

1+v_m< u

> s ₋1

. (1.3)

It is known since the work of Liggett [9] and its generalization by Röckner and Wang [14], and Wang [16], that the weak Poincaré inequality (1.1) gives rise to theL²-convergence of the semigroup

μ (Ptf )²

≤ξ(t ) Φ(f )+μ f²

, t >0, μ(f )=0, f ∈L²(μ), with speed at least

ξ(t ):=inf

s >0; −(1/2)β(s)logs≤t .

When the weak Poincaré inequality is deduced as a consequence of the finiteness of themth moment of hitting time, one interesting question is the explicit dependence ofξ(t ) onm. Unfortunately, the implicit form ofβ(s) in (1.3) makes it difficult to obtain this dependence.

The aim of the present work is to describe more explicitly an inequality which corresponds to the finiteness of polynomial moments of hitting times. It is well known (see [14]) that in the caseβ(s)=cs¹⁻^pwithp >1 and some c >0, the weak Poincaré inequality (1.1) is equivalent to the following Nash inequality of orderp:

μ f²

≤CE^1/p(f, f )Φ^1/q(f ), f ∈D(E), μ(f )=0,1 p+1

q =1. (1.4)

In this article we show that the finiteness of polynomial (not necessarily integer) moments of hitting times is related to the Nash inequality with explicit relation between the order of the moment, the order of the inequality and the speed of convergence of the semigroup. Our result can be summarized in the following scheme: For any open setG, letτ_G=inf{t≥0:X_t ∈/G},l≥0. Then the following implication holds.

Eμτ_G^l⁺¹<∞for suitableG ⇒ Nash inequality of orderl+2

l+1, (1.5)

(3)

where the functional Φ(f ) depends both onEμτ_G^l⁺¹ and on Osc(f ). On the other hand, for all l >0 and any Φ satisfying (1.2),

Nahs inequality of orderl+2

l+1 ⇒ Eμτ_G^l⁺¹⁻^ε<∞ (1.6)

for allε >0 and for all openGsuch thatμ(G^c) >0. Moreover it is well known since [9] that for symmetric semigroups, the Nash inequality of order ^l_l⁺₊²₁ is equivalent to the following speed of convergence for the semigroup

μ (P_tf )²

≤CΦ(f )t⁻^(l⁺¹⁾, μ(f )=0, f ∈L²(μ).

The implication (1.5) is proved only in the diffusion case, but (1.6) is valid for a very general Markov process. The method to prove the first implication relies on the use of killed processes. We establish a condition for the existence of general hitting time moments in terms of spectral properties of the killed process. This spectral condition generalizes the well known equivalence “exponential moments⇐⇒spectral gap.”

Let us now give the precise statement of our results.Xwill be aμ-symmetric Hunt process on a locally compact separable Hausdorff spaceEwhereμis a bounded Radon measure (wlog we suppose thatμis a probability measure).

For an open set G⊆E, we put P_t^G[A](x)=Px[Xt ∈A;t < τG] for a measurable subset A of E. Denote A^G the infinitesimal generator of (P_t^G) inL²G(μ)= {f ∈L²(μ): f =0 μ-a.s. onG^c} and let (E_ξ^G, ξ ≥0)be its spectral family.

It is known, see e.g. Friedman [7], or Loukianova, Loukianov and Song [10], thatEμexp(λτG) <∞; λ < λ₀, is equivalent to the fact that −A^G has a spectral gap of width at least equal to λ0. It turns out that hitting time moments generated by other functions than the exponential ones are still related with the spectral properties of−A^G in the following sense: Letr:[0,∞[ → [0,∞[,R(t )=t

0r(s)ds and denote byΛr:[0,∞[ → [0,∞]the Laplace transform ofr:

∀ξ≥0, Λr(ξ )= _∞

0

r(t )e⁻^{ξ t}dt. (1.7)

We show in Theorem2.2thatEμR(τG) <∞if and only if the spectral measure of−A^GintegratesΛr:

∀f:G→R,f∞<∞,

[0,∞[Λ_r(ξ )d

E_ξ^Gf, f

<∞.

This condition on the spectral measure will be called in the sequel ther-spectral condition.Then we show how we can derive in a very elementary way the Nash inequality for the killed processX^Gwith the help of the spectral condition specified byr(t )=t^l (Proposition2.6). In this case the corresponding rate of transience of the killed process, i.e. the rate of convergence ofP_t^Gto zero, is given byt⁻^(l⁺¹⁾. All this is the content of Section2, which is entirely devoted to the study of the killed process.

In Section3 we address the question how the polynomial spectral condition for the killed process (equivalently the existence of polynomial moments of hitting times) can be used to derive the Nash inequality for the non-killed process. In this section, our method applies only in the case when the Dirichlet form is local, i.e. in the diffusion case, in the sense thatX has a.s. continuous trajectories. But we do not need to suppose that the process is driven by a stochastic differential equation.

In the one-dimensional diffusion case, from the existence of polynomial moments of orderl+1, l≥0, we derive the Nash inequality specified byp=^l_l⁺₊²₁without any further assumptions.

The multidimensional diffusion case is treated as well. Here we need an additional non-degeneracy condition on the diffusion. Like in [4], we have to suppose that a local Poincaré inequality on some small domain holds. At the end of this section we provide the example of a multidimensional diffusion for which our result holds.

Finally, in Section4we study the implication “Nash inequality ⇒polynomial moments.” The Nash inequality gives an explicitα-mixing rate of the process, and then the main idea is to use this mixing rate in order to obtain a deviation inequality to estimatePμ(τ_G> t). This nice idea is borrowed from Cattiaux and Guillin [3]. As a consequence, for alll >0, the Nash inequality of orderp=^l_l⁺₊²₁implies the existence of the polynomial moments of hitting times of orderl+1−ε, for anyε >0. Note that the main tool of this section is the bound on the variance of the sum of strictly mixing variables. This bound is only valid forl >0, hence this is a technical hypothesis for the last section.

Note also that this last section is valid for general Hunt processes.

(4)

2. Killed process

2.1. Modulated moments and spectral condition for the killed process

Consider a Hunt processX on a locally compact separable Hausdorff spaceEin the sense of Fukushima, Oshima, Takeda [8]. Letμ be a Radon measure onE. Suppose that μis bounded (wlogμ is supposed to be a probability measure), everywhere dense inE, and thatXis aμ-symmetric process. Let(P_t)_t_≥₀be the transition semigroup ofX with associated Dirichlet form(E,D(E))onL²(μ). We suppose thatEis regular. Denote byPxthe law of the process Xstarting fromx∈E. For an open setG⊆E, setτ_G=inf{t≥0:X_t∈/G}the exit time ofXfromG. Throughout this section we supposeτ_G<∞Pμ-almost surely. Introduce

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

X^G_t =

X_t, 0≤t < τ_G, Δ, t≥τ_G,

where we adjoin an extra pointΔto the spaceE.Δplays the role of a cemetery, and we putP_t^G(Δ,·)=δ_Δ. Any measurable function defined onEis extended in a natural way toE∪ {Δ}by definingf (Δ)=0. Then, according to [8],X^Gis a Hunt process on the state spaceG∪Δ, symmetric with respect to the measureIG·μ(dx), with transition semi-group(P_t^G)_t_≥₀ onL²_G(μ)= {f ∈L²(μ): f =0 μ-a.s. onG^c}. We writeA^Gfor the infinitesimal generator of (P_t^G)inL²G(μ): With

D A^G

:

u∈L²G(μ): lim

t→0

P_t^Gu−u

t exists inL²G(μ)

,

we defineA^Gf for anyf∈D(A^G)as follows A^Gf =lim

t→0

P_t^Gf−f

t inL²G(μ).

A^G is a self-adjoint negative operator. Let us denote by (·,·) the scalar product inL²G(μ) and by(E^G_ξ, ξ ≥0)the spectral family of−A^G.

(E_ξ^G, ξ≥0)is a right-continuous and increasing family of projection operators such that for any bounded Borel measurable functionf defined on[0,∞[,the operatorf (−A^G)is given by

f

−A^G u=

[0,∞[f (ξ )dE_ξ^Gu, u∈L²G(μ).

Moreover, for allu∈L²G(μ), Idu=

[0,∞[dE_ξ^Gu, P_t^Gu=exp t A^G

u=

[0,∞[e⁻^{ξ t}dE^G_ξu and

−A^Gu=

[0,∞[

ξdE^G_ξu, u∈D A^G

.

The above integrals can be understood in a weak sense, that is, for allu, v∈L²G(μ), f

−A^G u, g

−A^G v

=

[0,∞[f (ξ )g(ξ )d E_ξ^Gu, v

. (2.1)

(5)

Actually, the bounded variation functionξ→(E_ξ^Gu, u)is only increasing on the spectrum of−A^Gand its discontinuity points are eigenvalues of−A^G. Denote byEGthe Dirichlet form associated with−A^GonL²G(μ). We have

EG(u, v)=

[0,∞[ξd E^G_ξu, v for allu, v∈D(EG)= {f ∈L²G(μ):

[0,∞[ξd(E_ξ^Gf, f ) <∞}. LetH_ξ^Gbe the image space ofE_ξ^G. Proposition 2.1. Under the conditionτG<∞almost surely,we haveH₀^G= {0}.

Proof. H₀^Gis invariant underP_t^Gfor allt >0. Indeed, usingE_λ^GE₀^G=E₀^G∀λ≥0 we see that∀u∈H₀^G,∀t≥0 P_t^Gu=

[0,∞[e⁻^{ξ t}dE_ξ^Gu=e⁰E₀^Gu=u.

For all v≥0, bounded, limt→∞P_t^Gv=limt→∞E[v(X_t)1t <τG] =0 and hence (u, v)=(P_t^Gu, v)=(u, P_t^Gv)→ 0, t→ ∞. Positive bounded functions being dense inL², we conclude thatu=0.

Since E₀^G=0, one has

[0,∞[f (ξ )dE_ξ^Gu=

]0,∞[f (ξ )dE^G_ξu. In what follows we give necessary and sufficient conditions for the existence of arbitrary moments ofτ_Gin terms of the behavior near the origin of the spectral measure dE_ξ^G. Let r:[0,+∞[ → [0,+∞[be some measurable function, and denoteΛ_r:[0,∞[ → [0,∞] its Laplace transform:

∀ξ≥0, Λr(ξ )= _∞

0

r(t )e⁻^{ξ t}dt. (2.2)

Instead of hitting time moments, we consider more generally modulated moments defined by_τ_G

0 r(t )f (X_t)dt. Denote byBb(G)⊂L²G(μ)the space of Borel-measurable and bounded real functions which vanishμ-almost surely onG^c. LetR(t )=t

0r(s)dsand · 1:= · L¹(μ).

Theorem 2.2. The following four conditions are equivalent:

1. EμR(τG) <∞;

2. For allf ∈Bb(G),x→f (x)×Ex

τG

0 r(t )f (Xt)dt∈L¹(IG·μ(dx));

3. For allf ∈Bb(G),

[0,∞[Λ_r(ξ )d(E_ξ^Gf, f ) <∞;

4. For allf ∈Bb(G),_∞

0 r(t )P_{t /2}^G f²dt <∞.

Moreover,for anyf∈Bb(G), f ×E_·

_τ_G

0

r(t )f (X_t)dt 1

=

[0,∞[Λ_r(ξ )d

E^G_ξf, f

= _∞

0

r(t )P_{t /2}^G f²dt. (2.3)

Definition 2.3. In the sequel the condition3 of Theorem2.2will be called the r-spectral condition for the killed process.

Proof of Theorem2.2. The equivalence 1 ⇐⇒ 2 is obvious. The following calculus yields 2 ⇐⇒ 3 ⇐⇒ 4 and the equality (2.3) for positive bounded functions.

f,E·

_τ_G

0

r(t )f (X_t)dt

=

f, _∞

0

r(t )P_t^Gf (·)dt

= _∞

0

r(t )

f, P_t^Gf dt

= _∞

0

r(t )P_{t /2}^G f²dt= _∞

0

r(t )

f,

[0,∞[e⁻^{ξ t}dE_ξ^Gf

dt

(6)

= _∞

0

r(t )

[0,∞[e⁻^{ξ t}d

E_ξ^Gf, f dt=

[0,∞[

_∞

0

r(t )e⁻^{ξ t}dtd

E_ξ^Gf, f

=

[0,∞[Λ_r(ξ )d

E_ξ^Gf, f .

IfEμR(τG) <∞andf is only bounded, we have Ex

_τ_G

0

r(t )f (X_t)dt= _∞

0

r(t )P_t^Gf (x)dt, μ(dx)-almost surely,

which follows from Fubini’s theorem and the fact that μ(dx)-almost surely, (t, ω)→ r(t )f (Xt(ω))1t <τG(ω)∈ L¹(Px(dω)⊗dt ), since

E_x _∞

0

1t <τGr(t )f (X_t)dt≤ f_∞ExR(τ_G) <∞, μ(dx)-almost surely.

A second application of Fubini’s theorem implies that

f, _∞

0

r(t )P_t^Gf (·)dt

= _∞

0

r(t )

f, P_t^Gf dt,

which follows from the product integrability of(t, x)→f (x)P_t^Gf (x)r(t )with respect to dt μ(dx)which in turn is granted by the following upper bound.

|f|, _∞

0

r(t )P_t^Gf (·)dt

≤

|f|, _∞

0

r(t )P_t^G|f|(·)dt

≤ f²_∞EμR(τ_G).

This shows that 2 implies 3 and 4 as well as the equalities of (2.3).

Remark 1. Note that the equivalence between points3and4of Theorem2.2remains true for the non-killed process, after removing the spectral projection on the0-eigenspace.More precisely,we have for all bounded measurable functionsf such thatμ(f )=0,

]0,∞[Λr(ξ )d(Eξf, f )= _∞

0

r(t )Pt /2f²dt.

Example 2.4. Consider the caser(t )=t^l, l≥0.We have forξ ≥0Λ_r(ξ )=Γ (l+1)ξ⁻^(l⁺¹⁾.Hence Eμτ_G^l⁺¹<∞ ⇐⇒

[0,∞[ξ⁻^(l⁺¹⁾d

E_ξ^Gf, f

<∞, (2.4)

for allf non-negative and bounded.In the next section we will explain how to use the spectral condition to obtain functional inequalities forX^Gand then forX.

Example 2.5. Consider the caser(t )=e^λt, λ >0.We have Λ_r(ξ )= 1

ξ−λ, ifξ > λ, Λ_r(ξ )= +∞ otherwise.

Put

λ₀=sup

λ >0,Eμe^λτ^G<∞ . We obtain that

]0,λ₀[dE^G_ξ =0,i.e.spectral measure does not charge]0, λ0[.

(7)

2.2. Polynomial spectral condition and Nash inequality for the killed process

In [9], Liggett introduced the following Nash inequality for a Dirichlet formE(f, f )associated to a linear operator generating a strongly continuous Markovian semigroup with invariant probability measureμ.

μ

f −μ(f )2

≤CE^1/p(f, f )Φ^1/q(f ), f ∈D(E). (2.5)

Here 1< p, q <∞with 1/p+1/q=1,Cis a positive constant, andΦ:L²(μ)→ [0,∞]satisfiesΦ(cf )=c²Φ(f ), for anyc∈Randf ∈L²(μ).It is shown in [9] that if in additionΦ(P_tf )≤Φ(f )∀f ∈L²(μ),∀t >0, then the inequality (2.5) is equivalent to

∃C >0, P_t(f )−μ(f )²

2≤CΦ(f )

t^q⁻¹ (2.6)

for allf∈L²(μ)andt >0. If the semi-group ofXis conservative, symmetric and ergodic,μ(f )=E0f. Hence we will consider the following form of the Nash inequality:

f −E₀(f )²≤CE^1/p(f, f )Φ^1/q(f ), f ∈D(E). (2.7)

Let us point out again that for the killed process the semi-group is not conservative, transient, and E₀=0. The following proposition shows that the conditionEIGμτ_G^l⁺¹<∞implies the Nash inequality in the form (2.7) for the killed process.

Proposition 2.6. Letl≥0and suppose thatEμτ_G^l⁺¹<∞.Let Φ(f )=

[0,∞[ξ⁻^(l⁺¹⁾d

E_ξ^Gf, f

, f ∈L²G(μ). (2.8)

Then the Nash inequality (2.7)holds for the killed process withp=^l_l⁺₊²₁ andq =l+2.Furthermore,Φ satisfies Φ(f )≤_{Γ (l}¹₊₂₎f²_∞Eμτ_G^l⁺¹.

Proof. In virtue of Theorem2.2, the conditionEμτ_G^l⁺¹<∞is equivalent to

[0,∞[ξ⁻^(l⁺¹⁾d

E_ξ^Gf, f

<∞, (2.9)

for all boundedf∈Bb(G). Letf∈D(EG). Suppose thatp⁻¹+q⁻¹=1 and write, using Hölder’s inequality:

f−E₀f²= f²=

[0,∞[d

E_ξ^Gf, f

=

[0,∞[ξ^1/pξ⁻^1/pd

E_ξ^Gf, f

≤

[0,∞[ξd

E_ξ^Gf, f^1/p

×

[0,∞[ξ⁻^q/pd

E_ξ^Gf, f^1/q

=EG^1/p(f )Φ^1/q(f ),

where Φ(f )=

[0,∞[ξ⁻^q/pd

E^G_ξf, f .

Now we choosepandq in such a way that Φ(f )=

[0,∞[ξ⁻^(l⁺¹⁾d

E_ξ^Gf, f .

(8)

This choice is given byp=^l_l⁺₊²₁andq=l+2. Finally we obtain for allf∈D(EG) f −E0f²≤EG(f )^(l⁺^1)/(l⁺²⁾×Φ^1/(l⁺²⁾(f ),

whereΦ satisfiesΦ(cf )=c²Φ(f )for anyc∈RandΦ(f ) <∞for all boundedf. Also Φ

P_t^Gf

=

[0,∞[ξ⁻^(l⁺¹⁾e⁻^{2ξ t}d

E_ξ^Gf, f

≤Φ(f ).

Remark 2. Since the proof of the above Proposition2.6uses only the condition(2.9),we deduce from this the follow- ing:If for the non-killed process the spectral condition

]0,∞[ξ⁻^(l⁺¹⁾d(Eξf, f ) <∞ (2.10)

holds for allf ∈Bbsuch thatμ(f )=0,then the Nash inequality(2.7)holds,withΦ(f )=

]0,∞[ξ⁻^(l⁺¹⁾d(Eξf, f ) andq=l+2.

However,in general,it is difficult to find sufficient conditions,expressed in terms of the generator of the process, ensuring condition(2.10)for the non-killed process,whereas conditions ensuringEμτGl+1<∞are by now classical, for suitable choices of the setG,see Section3.3below.

3. Polynomial moments and Nash inequality for the non-killed process

In this section we show how polynomial modulated moments are related to the Nash inequality for the non-killed process. The result can be summarized as follows. For alll >0, for allε >0 we have: “integrability of moments of orderl+1 ⇒Nash inequality giving rise toL²convergence of the semigroup with speedt⁻^(l⁺¹⁾ ⇒existence of moments of orderl+1−ε.”

For the second implication we work under the general conditions of Section2. For the first implication “moments imply Nash” we work in the diffusion case only. In dimension 1, no hypothesis on the diffusion is imposed. In higher dimension, however, we need a non-degeneracy condition which is a local Poincaré inequality (see the comments in Remark5).

3.1. Polynomial moments ⇒Nash inequality.One-dimensional diffusion case

In this subsection we show that the Nash inequality for a killed diffusion process onRimplies the Nash inequality for the non-killed process. Fix somea∈Rand letG⁻= ]−∞, a[,G⁺= ]a,∞[. We use some well-known techniques which are specific to the one-dimensional case.

SinceX is a diffusion, it possesses a scale functionSand a corresponding speed measurem, see e.g. Revuz and Yor [12], Chapter VII. We suppose thatXis recurrent. This implies that limx→±∞S(x)= ±∞. Note moreover that mcharges any non-empty open set. Denote by dSthe measure induced byS(x). LetF (x)be a real function onR. We shall write dF dS, if there exists a functionf (x)inL¹_loc(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)

, F]a,∞[=

F ∈F: F (x)=0, x≤a

, (3.1)

F]−∞,a[=

F ∈F: F (x)=0, x≥a .

(9)

We do not assume that dSandmare absolutely continuous with respect to the Lebesgue measure. We cite the following theorem from [10].

Theorem 3.1 ([10]). The diffusionXism-symmetric.The Dirichlet space associated withXis the function spaceF given by(3.1),and the Dirichlet form has the expression

E(F, F )= _∞

−∞

dF dS

2

(x)dS(x), F ∈F.

The restriction of the Dirichlet form E on F]a,∞[ is the Dirichlet form E]a,∞[ associated with the semigroup (P_t^]^a,^∞[)_t_≥₀ of the process X killed when it exits ]a,∞[.The killed process X^]^a,^∞[ is symmetric with respect to the measureI_]a,∞[·m(dx).

The same is true(with obvious modifications)forE_]−∞,a[. The proof of this theorem is given in [10].

We can now state the Nash inequality for the non-killed processX. Fora∈Rintroduce the hitting timeT_a= inf{t≥0, Xt=a}. We writeμ(·)=_m(¹_R₎m(·)for the renormalized speed measure.

Theorem 3.2. Letl >0.Suppose that for somea∈R _+∞

−∞ ExT_a^l⁺¹m(dx) <∞. (3.2)

Then there exists a functionalΦ:L²(μ)→ [0,+∞]such that the Nash inequality μ

F−μ(F )2

≤E^1/p(F, F )Φ^1/q(F ), F∈F, (3.3)

holds with p= ^l_l⁺₊²₁ and q =l+2. The functional Φ satisfies Φ(cF )=c²Φ(F ) for all c∈R,F ∈L²(μ),and Φ(P_tF )≤Φ(F )for allt >0.Moreover,there exists a finite constantC >0such that

Φ(F )≤CF−μ(F )²_∞ ∀F∈L². (3.4)

Remark 3. Note also that Φ(F )≤C

supR F −inf

R F 2

=COsc(F )². (3.5)

Proof. Fix a pointa∈R. Then the variational formula for the variance gives for allF ∈F, _+∞

−∞

F (x)−μ(F )2

μ(dx)≤ _+∞

−∞

F (x)−F (a)2

μ(x)

= _a

−∞

F (x)−F (a)2

μ(dx)+ _+∞

a

F (x)−F (a)2

μ(dx).

Write

F₋(x)=

F (x)−F (a)

1_{x<a}, F₊(x)=

F (x)−F (a) 1_{x>a}.

ThenF₋∈F]−∞,a[ andF₊∈F]a,∞[. Hence we can apply Proposition2.6for bothG⁻= ]−∞, a[,G⁺= ]a,∞[. Denote

E]−∞,a[=E− and E]a,+∞[=E+

(10)

and

Φ₋(u)=

[0,∞[ξ⁻^(l⁺¹⁾d

E_ξ^G⁻u, u

, and Φ₊(u)=

[0,∞[ξ⁻^(l⁺¹⁾d

E_ξ^G⁺u, u .

Then, withp=^l_l⁺₊²₁ andq=l+2, _a

−∞

F (x)−F (a)2

μ(dx)+ _+∞

a

F (x)−F (a)2

μ(dx)

≤E₋^1/p(F₋)Φ₋^1/q(F₋)+E₊^1/p(F₊)Φ₊^1/q(F₊)

=Φ₋^1/q(F₋) a

−∞

dF dS

2

(t)dS(t) 1/p

+Φ₊^1/q(F₊) _+∞

a

dF dS

2

(t)dS(t) 1/p

≤

Φ₋^1/q(F₋)+Φ₊^1/q(F₊) ^+∞

−∞

dF dS

2

(t)dS(t) 1/p

=E^1/p(F )Φ_a^1/q(F ), where

Φ_a(F )=

Φ₋^1/q(F₋)+Φ₊^1/q(F₊)q

.

The above result holds for anya∈R. Hence we can put Φ(F )=sup

t≥0

ainf∈RΦ_a(P_tF ). (3.6)

ThenΦ(cF )=c²Φ(F )andΦ(PtF )≤Φ(F )are trivially satisfied. It remains to show that under the conditions of the theorem,Φ satisfies (3.4). In virtue of Theorem2.2,

Φ₋(F₋)= 1 Γ (l+1)

R1_]−∞,a[(x)

F (x)−F (a)

×Ex

_T_a

0

s^l×

F (X_s)−F (a)

dsμ(dx)

≤ 4

Γ (l+2)F−μ(F )²

∞

R1_]−∞,a[(x)ExT_a^l⁺¹μ(dx). (3.7)

In the same way, Φ₊(F₊)≤ 4

Γ (l+2)F −μ(F )²

∞

R1_]a,∞[(x)ExT_a^l⁺¹μ(dx), and thus, withq=l+2,

Φ_a(F )≤2^q 4

Γ (l+2)F −μ(F )²_∞EμT_a^l⁺¹.

We deduce, usingP_tF∞≤ F∞andμ(P_tF )=μ(F )that infa Φ_a(P_tF )≤2^q 4

Γ (l+2)P_tF −μ(P_tF )²

∞inf

a∈REμT_a^l⁺¹

≤CF −μ(F )²

∞,

(11)

where

C=2^l⁺² 4 Γ (l+2) inf

a∈REμT_a^l⁺¹,

and this implies (3.4).

Remark 4. If the Nash inequality(3.3)holds with a functionalΦsatisfying the properties of Theorem3.2,then Liggett [9],Theorem2.2,shows that

P_tF−μ(F )²

2≤CΦ(F )

t^l⁺¹ , F ∈F.

Hence under the assumption of integrability ofl+1-moments of hitting times we obtain a polynomial decay of the transition semigroupP_t ofXat the same ratet⁻^(l⁺¹⁾.

3.2. Polynomial moments ⇒Nash inequality.General diffusion case

In this section we come back to the general conditions of Section2and consider theμ-symmetric Hunt processXon the LCCB spaceEsuch thatμ(E)=1, with semigroup(P_t)_t_≥₀and associated Dirichlet form(E,D(E))onL²(μ).

Assumption 3.3. Assume that the Dirichlet form(E,D(E))is regular and admits a carré du champΓ.

Following Bouleau and Hirsch [1], Proposition 4.1.3, this means that there exists a unique positive symmetric and continuous bilinear form fromD(E)×D(E)intoL¹(μ),denoted byΓ and calledthe carré du champoperator, such that∀f, g, h∈D(E)∩L^∞,

E(f h, g)+E(gh, f )−E(h, f g)=

hΓ (f, g)dμ. (3.8)

Assumption 3.4. Assume that the Dirichlet form(E,D(E))is local.

In this case, by [1], Proposition 6.1.1,

∀f∈D(E), E(f, f )=1 2

E

Γ (f, f )dμ.

Note that the locality of the form is equivalent to assume that the processXis a diffusion process, in the sense thatX has a.s. continuous trajectories, see Theorem 4.5.1 of [8].

Assumption 3.5. Assume thatEis recurrent,i.e. 1∈D(E)and E(1,1)=0.

Recall the definition of the spacesDp,p≥2, similar to the definition of Sobolev spaces, ([1], Definition 6.2.1):

Dp=

f ∈D(E)∩L^p;Γ (f, f )^1/2∈L^p and, forf∈Dp,

f_Dp= f_L^p+ Γ (f, f )^1/2_L^P.

The following proposition is proved in [1] (Proposition 6.2.3).

Proposition 3.6. Letp, q, r≥2with_p¹+¹_q=¹_r.Then

f ∈Dp and g∈Dq ⇒ f g∈Dr and f g_Dr ≤ f_Dpg_Dq.

(12)

For any setGand anyr >0 we setG_r= {x∈E: dist(x,G) < r}. Under Assumptions3.3,3.4and3.5the following theorem holds:

Theorem 3.7. Letl >0.Suppose there exists an open subsetG⊂Eandr >0such thatμ(G_r\ ¯G) >0and such that the following conditions are satisfied.

1. ForA∈ {Gr,G¯^c}, Eμτ_A^l⁺¹<∞.

2. μsatisfies a local Poincaré inequality in restriction toG_r\ ¯G,i.e.

G_r\ ¯G

f²dμ≤CP(G, r)

G_r\ ¯G

Γ (f, f )dμ for allf ∈D(E)such that

G_r\ ¯Gfdμ=0.

3. There exists a regularized indicator functionu∈D(E)associated to the setsGandG_r such that0≤u≤1, u≡1 onG,¯ u≡0onG^c_r,which verifies

C(u, r):=Γ (u, u)

∞<∞. (3.9)

Then there exists a functionalΦ:L²(μ)→ [0,∞]such that the following Nash inequality holds.

For anyf∈D(E)withμ(f )=0, μ

f²

≤E^1/p(f, f )Φ^1/q(f ), (3.10)

wherep=(l+2)/(l+1)andq=l+2. Φ satisfiesΦ(af )=a²Φ(f )for alla∈Rand Φ(P_tf )≤Φ(f )for all t≥0, f ∈L²(μ).Moreover,

Φ(f )≤2^q/p 1+C(u, r)C_P(G, r)q/p 1

Γ (l+2)×Osc(f )² Eμτ_G^l⁺¹

r

1/q

+ Eμτ^l_¯⁺¹

Gc

1/qq

. (3.11)

Proof. Letf ∈D(E)withμ(f )=0. Let

c= 1

μ(G_r\ ¯G)

G_r\ ¯G

fdμ.

The use of this constant will become clear in formula (3.16) later. By the variational definition of the variance, we have that

f²(x)μ(dx)≤ f (x)−c2

μ(dx).

Denotef˜=f−cand letube the regularized indicator ofG. Writef˜= ˜f u+ ˜f (1−u). Using Proposition3.6, since u∈D_∞andf˜∈D2=D(E), we havef u˜ ∈D(E). Hence bothf u˜ andf (1˜ −u)belong toD(E). Now we can write

E

f (x)˜ 2

μ(dx)=

E

f u˜ + ˜f (1−u)2

(x)μ(dx)

≤2

E

(f u)˜ ²(x)μ(dx)+

E

f (1˜ −u)2

(x)μ(dx)

=2

Gr

(f u)˜ ²(x)μ(dx)+

¯ G^c

f (1˜ −u)2

(x)μ(dx)

.

(13)

In a first step, we want to apply the Nash inequality onG_r. For that sake, note thatf u˜ ∈D(E)and its quasicontinuous modification is zero onG^c_r. (For the definition of quasicontinuity, we refer the reader to Chapter 2.1 of [8].) Hence by (4.3.1) of [8],f u˜ ∈D(EG_r). Therefore, we introduce

Φ^G^r(f )˜ =

[0,∞[ξ⁻^(l⁺¹⁾d

E_ξ^G^rf u,˜ f u˜ and obtain, applying Proposition2.6,

G_r

(f u)˜ ²(x)μ(dx)≤ EGr(f u,˜ f u)˜ 1/p

Φ^G^r(f )˜1/q

≤E^1/p(f u,˜ f u)˜ Φ^G^r(f )˜1/q

, (3.12)

where the first inequality follows from Proposition2.6, and the second sinceEG_r is just the restriction of the Dirichlet formEtoFGr.

In the same way,f (1˜ −u)∈D(EG_¯^c). Introducing Φ^G^¯^c(f )˜ =

[0,∞[ξ⁻^(l⁺¹⁾d

E_ξ^G^¯^cf (1˜ −u),f (1˜ −u) ,

we obtain

¯ G^c

f (1˜ −u)2

(x)μ(dx)≤E^1/pf (1˜ −u),f (1˜ −u) Φ^G^¯^c(f )˜1/q

. (3.13)

In order to controlE(f u,˜ f u)˜ andE(f (1˜ −u),f (1˜ −u)), we use (Proposition 6.2.3 of [1] and Cauchy–Schwarz) that Γ (f u,˜ f u)˜ ≤2

Γ (f ,˜ f )˜ + ˜f²Γ (u, u)

. (3.14)

By the locality of the form, it is classical to show thatΓ (f ,˜ f )˜ =Γ (f, f )andΓ (u, u)=0 onG¯andG^c_r. Hence, E(f u,˜ f u)˜ =1

2

G_r

Γ (f u,˜ f u)˜ dμ

≤

E

Γ (f, f )dμ+

G_r\ ¯G

f˜²(x)Γ (u, u)(x)μ(dx)

≤

E

Γ (f, f )μ(dx)+C(u, r)

Gr\ ¯G

f˜²(x)μ(dx),

which implies that

E(f u,˜ f u)˜ ≤2E(f, f )+C(u, r)

G_r\ ¯G

f˜²(x)μ(dx).

The role ofuand 1−ubeing symmetric, we get in the same way Ef (1˜ −u),f (1˜ −u)

≤2E(f, f )+C(u, r)

Gr\ ¯G

f˜²(x)μ(dx), with the same constantC(u, r). Putting things together, we conclude that

E

f²(x)μ(dx)≤

2E(f, f )+C(u, r)

G_r\ ¯G

f˜²(x)μ(dx) 1/p

Ψ^1/q(f ), (3.15)