Short-time asymptotics in Dirichlet spaces

Short-Time Asymptotics in Dirichlet Spaces

JOSÉ A. RAMÍREZCourant Institute

Abstract

This work is involved with the short-time asymptotics of the heat semigroup ina general setting. A generalization of Fang’s version of Varadhan’s formula isproved. A spectral gap or the possibility of obtaining one by an appropriatechange of measure is required.c© 2001 John Wiley & Sons, Inc.

1 Introduction

Given an elliptic differential operator of the formL f (x) = 12∇(a(·)∇ f )(x)

on Rn, it is possible to associate a metric to it that, in some sense, is the right one

for this operator. For this, take|γs|2a−1 = ∑i, j a−1

i, j (γs)γis γ

js and define the intrinsic

distance by

(1.1) d2a(x, y) = inf

γ

∫ 1

0|γs|2a−1 ds.

Here the infimum is taken over Lipschitz pathsγ : [0,1] → Rn such thatγ0 = x

andγ1 = y. Equivalently, this intrinsic distance can be defined as

da(x, y) = sup〈∇ f,a∇ f 〉≤1

f (x)− f (y) .

In 1967 Varadhan [27] proved that the heat kernelpt(x, y) on a Riemannianmanifold has the following asymptotic behavior ast gets small:

limt↓0

t log pt(x, y) = −1

2d2

a(x, y) .

This gives one argument for the above metric being “right” for the operator. Theproof requires smoothness conditions on the operator coefficients and curvaturebounds on the manifold. Since then, the question of whether it is possible to recoverVaradhan’s formula in a more general setting has been investigated in many papers,notably by Davies [4], who obtained the sharp upper bound in a fairly generalcase (no smoothness or curvature bounds). Later, Norris and Stroock [18] provedthe sharp lower bound for general operators inR

n. This was further generalizedby Norris [17] to Lipschitz manifolds (a more complete historical account can befound there).

Communications on Pure and Applied Mathematics, Vol. LIV, 0259–0293 (2001)c© 2001 John Wiley & Sons, Inc.

260 J. A. RAMÍREZ

Then came the question of what happens in infinite dimensions. The workstarted with Fang [9], who proved a version of this result for the Ornstein-Uhlen-beck process on Wiener space. Whether this result also holds for the Ornstein-Uhlenbeck process on the path space over a Riemannian manifold was the originof our research on the topic. Recently this was further studied by Zhang [28] andAida and Kawabi [1], who proved the result for more general processes on Wienerspace, and Aida and Zhang [2], who dealt with the Ornstein-Uhlenbeck process onpath groups.

Similar results to the ones stated below were proved almost simultaneously byM. Hino [12]. Although the conclusions are the same, the two hypotheses are notcomparable. Moreover, the methods are very different. Below we have includedremarks indicating where the differences are. At this point we only mention thatthere is at least one important case that is covered by our theorem but not by Hino’sresult. This example is the path space over a Riemannian manifold that is not agroup.

In these nonlocally compact spaces, the heat kernel is usually not well defined.Thus, instead we try to estimate the following:

(1.2) limt↓0

t log Pt(A, B) .

Here our replacementPt(A, B) for the heat kernel is given by

Pt(A, B) =∫A

Tt1B(x)µ(dx) =∫B

Tt1A(x)µ(dx)

whereTt is the evolution semigroup and 1A is the indicator function of the setA.This is the probability that the associated process starts inA and ends up inB attime t . The second equality expresses the symmetry assumption that we are goingto make for the rest of this work.

1.1 Background and Assumptions

Let (�,B, µ) be a probability space. On the Hilbert spaceL2(µ) we considera closed Dirichlet formE : D(E)× D(E) → R with domainD(E) dense inL2(µ),that is, a nonnegative definite bilinear form that satisfies

f ∈ D(E) ⇒ f ∧ 1 ∈ D(E) and E( f ∧ 1) ≤ E( f )

and such that the norm

‖ f ‖D = [| f |2L2 + E( f )]1/2 = [

( f, f )+ E( f )]1/2

determines a Hilbert space. This Hilbert space will be denoted byD and is assumedto be separable. Also, assume that 1∈ D andE(1) = 0.

SHORT-TIME ASYMPTOTICS IN DIRICHLET SPACES 261

Notation.Whenever convenient, we use( f, g) = ∫f (x)g(x)dµ(x) to denote

the scalar product inL2(�,µ). E( f )meansE( f, f ). The operatorL is the infini-tesimal generator, andTt = exp(tL) is the corresponding semigroup. Also,a ∨ bmeans maximum anda ∧ b minimum.

An equivalent of the squared gradient is necessary. The corresponding object inthis setting is usually calledcarré du champ(or just “squared gradient”). This is anonnegative, definite, symmetric, continuous bilinear form0 : D×D → L1(X, µ)such that

E( f, g) = 1

2

∫�

0( f, g)dµ .

It can be defined from∫φ0( f, g)dµ = E( f, φg)+ E(g, φ f )− E( f g, φ) ,

whenever the right-hand side defines a functional (inφ) that is bounded inL∞ andcan be identified to some function0(u,u) ∈ L1 (see [3] for more details). Weassume its existence.

It is easy to check that the result we pursue is not true for nonlocal generators(a Poisson process, for example); therefore our generator should be local in somesense. In this generality, the local character of the generator could be alternativelyexpressed in terms of thecarré du champoperator. Assume that, for everyF ∈C1(R) bounded andf ∈ D, we haveF( f ) ∈ D and

(1.3) 0(F( f ), F( f ))(x) = [F ′( f (x))]20( f, f )(x) .

Of course, there are other ways of describing locality; see [3]. From property (1.3),it can be deduced that, forF ∈ C2(R), we have

(1.4) LF( f ) = F ′( f )L f + 1

2F ′′( f )0( f, f ) ,

which certainly implies the local property forL in spaces where the topology iswell behaved with respect toE .

It will be explicitly stated when a spectral gap is needed for the operatorL.Such a property is equivalent to the Poincaré inequality, i.e.,

(1.5)∫ (

f −∫

f dµ

)2

dµ ≤ csE( f, f )

for some finite constantcs. This, in turn, implies the following inequality:

(1.6)∫ f − 1

µ(A)

∫A

f dµ

2

dµ ≤ cs(A)E( f, f )

for any positive measure setA and a constantcs(A).

262 J. A. RAMÍREZ

1.2 Main ResultSet, as before,Pt(A, B) = ∫

B Tt1A(x)dµ(x). With thecarré du champat handwe can define the distance that will be used to identify the asymptotic behavior ofPt(A, B). That definition is

(1.7) d(A, B) = sup0( f, f )≤1

infx∈A,y∈B

f (x)− f (y) .

Note.From definition (1.7) and for the rest of this work except for Section 4(Examples), when we write “inf” or “sup,” we really mean them in an essentialsense. Otherwise it makes no sense in our context. All statements below are trueup to a set of measure zero (except in Section 4).

The following is our main theorem, which we prove in Section 3:

THEOREM 1.1 Assume a Dirichlet space as described in Section1.1. If A and Bare measurable, withµ(A) > 0 andµ(B) > 0, we have

lim supt↓0

2t log Pt(A, B) ≤ −d2(A, B) (the upper bound)

If the spectral gap holds, we have the following:

lim inft↓0

2t log Pt(A, B) ≥ −d2(A, B) (the lower bound)

with d2(A, B) defined as in(1.7).

Notice that the definition of the distance depends only on the measure class ofµ. However, the spectral gap property does depend on the particular measureµ.This could mean that the gap property is not really required. In fact, the spectral gapassumption can be relaxed a bit to be valid with an extra potential in the measure.This is proved in Theorem 5.1.

We now proceed to state the result of M. Hino and compare both results. Define

02( f, g) = 1

2{L0( f, g)− 0(L f, g)− 0( f,Lg)} .

We say that02 has a lower bound ofK > −∞ if, for any f ∈ D, we have that

(1.8) 02( f, f ) ≥ K0( f, f ) .

In the paper of M. Hino the basic assumption used to establish the lower boundis a lower boundK on02 rather than the existence of a spectral gap. The bound on02 is some kind of a curvature bound. Take, for example, a Riemannian manifoldwith L = 1

21 the Laplacian. One has, from Bochner’s formula,

02( f, f ) = |∇∇ f |2 + Ric(∇ f,∇ f ) .

This says that the bound in (1.8) largely depends on a similar bound for the Riccicurvature.

As with curvature, whenK > 0 the inequality (1.8) implies a spectral gap (thisis the Bakry-Emery argument). But whenK ≤ 0, nothing can be said. On theother hand, one cannot deduce (1.8) from the spectral gap either.


An acknowledgment is in order at this point. Originally the second part of ourproof for the lower bound (Section 3.3) required an extra condition onA or B (anda probabilistic method) in order to apply the Tauberian theorem (cf. Lemma 3.11).This condition seemed in accordance with previous results [1, 2, 9] where one ofthe sets was required to be open. The result by Hino made the breakthrough ofdropping this condition for the first time and prompted us to find a better methodthat worked for general positive measure sets. The section on the final form of thelower bound (Section 3.3) was rewritten in light of this.

2 Preliminaries: The Intrinsic Metric

First we present some technical tools that are going to be useful when dealingwith the carré du champ. Then we study some properties of the set distance in-troduced in (1.7). We also defined(·, A), the distance function to the setA. Asbefore, all statements are true up toµ-negligible sets.

2.1 Properties of0(·, ·)Being nonnegative, thecarré du champsatisfies the following convexity in-

equality:

(2.1) 0

(∑i

αi fi ,∑

i

αi fi

)≤∑

i

αi0( fi , fi )

where∑

i αi = 1. We want to extend this to continuous averages. But first wepresent a lower semicontinuity type lemma for0(·, ·).LEMMA 2.1 Assume that fk ⇀ f weakly inD and0( fk, fk) ⇀ 0∞ weakly in L1;then f ∈ D(0) and, moreover,

0( f, f ) ≤ 0∞ a.s.

PROOF: Takeφ ∈ L∞ ∩ D positive and consider the Dirichlet form

Eφ( f, g) =∫φ0( f, g)dµ .

Call Dφ the corresponding Dirichlet space. Suppose that we have strong conver-

gence inD : fk → f . Therefore we have that

Eφ( fk − f, fk − f ) =∫φ0( fk − f, fk − f )dµ ≤ cE( fk − f, fk − f ) → 0 ,

and thereforefk → f in Dφ. Thus, we find∫

φ0( f, f )dµ = Eφ( f, f ) = limk↑∞

Eφ( fk, fk)

= limk↑∞

∫φ0( fk, fk)dµ =

∫φ0∞ dµ .

264 J. A. RAMÍREZ

If we only have fk ⇀ f weakly in D, then we choose a convex combinationfk = ∑

i αi fi that converges strongly tof . Because of inequality (2.1), the limitof 0( fk, fk) is less than0∞. This proves the lemma. �

Now we extend the convexity inequality to continuous averages.

LEMMA 2.2 Suppose ft ∈ D is a family of functions, jointly measurable for(t, x) ∈ [0, T] ×� and satisfying that t7→ ft is bounded as a map from[0, T] toD. Denote

ft = 1

T

∫ T

0ft dt .

We have

0( fT , fT ) ≤ 1

T

∫ T

00( ft , ft)dt .

PROOF: Suppose thatt 7→ ft ∈ D is continuous. Denote the Riemann sums offt by

Rn( f ) = 1

T

∑i

f

(iT

n

)T

n.

For finite convex combinations we have the inequality

0(Rn( f ), Rn( f )) ≤ 1

T

∑i

0

(f

(iT

n

), f

(iT

n

))T

n.

Multiply by a test functionφ ∈ L∞ ∩ D and integrate to obtain

(2.2)∫φ0(Rn( f ), Rn( f )

)dµ ≤ 1

T

∑i

∫φ0

(f

(iT

n

), f

(iT

n

))dµ

T

n.

Sincet 7→ ft ∈ D is continuous,t 7→ ft ∈ Dφ is also. Thus, the right-hand side of

(2.2) converges to the right integral. This gives

limn↑∞

∫φ0(Rn( f ), Rn( f )

)dµ ≤ 1

T

∫ T

0

∫φ0( ft , ft)dµ .

Therefore, it is enough to prove that

Rn( f ) ⇀1

T

∫ T

0ft dt whenn ↑ ∞ ,

because then we can use the previous lemma to conclude. The convergence occursbecause, forg ∈ D arbitrary,

E(g, Rn( f )) = 1

T

∑i

E(g, f (iT/n))T

n→∫ T

0E(g, ft)dt .

Here, again, we used the continuity assumption onft .


Finally, if we only have boundedness ofE( ft , ft), we can approximateft byaverages over small intervals[t − ε, t]:

f εt = 1

ε

∫ t

t−εfs ds.

Here we setfs = 0 for s < 0. These approximations satisfy thatt 7→ f εt ∈ D iscontinuous because of the original boundedness. The approximationsf εt convergeweakly to ft . This is so since

E( f εT , g) = 1

T

∫ T

0

1

ε

∫ t

t−εE( fs, g)ds dt→ 1

T

∫ T

0E( ft , g)dt

by the Lebesgue lemma. Therefore, using the previous lemma,∫φ0( fT , fT )dµ ≤

∫φ lim inf

ε↓00( f εT , f εT )dµ

≤∫φ

1

T

∫ T

00( f εt , f εt )dt dµ

≤ 1

T

∫ T

0

1

ε2

∫ t

t−ε

∫ t

t−ε

∫φ0( fs, fr )dµ ds dr dt

→ 1

T

∫ T

0

∫φ0( ft , ft)dµ dt

where the last line also comes from the Lebesgue lemma. We finally obtain theconclusion in the generality required. �

Finally, let us remark that if0( f, f ) ≤ C and0(g, g) ≤ C, the followinginequalities hold true:

0( f ∨ g, f ∨ g) ≤ C and 0( f ∧ g, f ∧ g) ≤ C .

This can be seen by approximating the functionals(x, y) 7→ x ∨ y and(x, y) 7→x ∧ y by differentiable functions and using (1.3).

2.2 Weak Solutions

A certain equation (3.3) that will turn out to be very useful later is shown to bevalid here. As in most of this work, the notationL p (without a specific space) isreserved forL p(�,µ).

LEMMA 2.3 Suppose f∈ L2. Letwt(x) = logTt f (x), and assume that|wt |L2 ≤Cw for t ∈ [t0, t1] where t1, t0 ∈ (0,∞). This function satisfies the following:

(2.3) (φt , wt)− (φt0, wt0) =∫ t

t0

{(∂rφr , wr )− E(φr , wr )+ 1

2

∫φr0(wr , wr )dµ

}dr

266 J. A. RAMÍREZ

for anyφt(x) ∈ H1([t0, t1]; L2)∩ L2([t0, t1]; D∩ L∞). In other words, the functiont 7→ (φt , wt) is absolutely continuous and satisfies

∂t(φt , wt) = (∂tφt , wt)− E(φt , wt)+ 1

2

∫φt0(wt , wt)dµ

for almost every t∈ [t0, t1].

PROOF: Assume first thatf ≥ c > 0. Notice that because of this lower boundon f , we can change tow(x, t) = logc Tt f (x) where logc x = logx if x > c andsomeC2 continuation otherwise. The statement will not change. Then the equalityfor f ∈ D(L) is just a simple computation using formula (1.4) (see also equation(3.3) below).

It now only remains to approximatef by

fn = 1

n

∫ 1/n

0Ts f (x)ds,

so that fn ∈ D(L) and fn → f strongly in D, since this implies thatwn =logc Tt fn → w strongly inD (this is standard since logc ∈ C2).

Back to the general case, we setf η = f +η andwη = log f η for η > 0 so thatthe conclusion is valid for this new function. Takingφt = 1 in equation (2.3) gives

(2.4)1

2

∫ t1

t0

E(wηr , wηr )dr =

∫wηt1 dµ−

∫wηt0 dµ ≤ 2Cw ,

and thereforewη ∈ L2([t0, t1]; D)with a uniform bound. Therefore one can extracta weakly convergent subsequence (inL2([t0, t1]; D)) from (wη)η and take limits inequation (2.3). If we had a linear equation, the proof would be finished at thispoint. The difficulty here is that the last term in that equation is nonlinear. All wecan get at this point is an inequality (by using Lemma 2.1), which is enough forthe applications that we have in mind. Nevertheless, we will proceed and prove theequality.

We will prove thatwη → w strongly in L2([t0, t1]; D) asη ↓ 0. This willmean that we can take limits in equation (2.3) when applied towη and obtain thecorresponding equation forw. It is clear thatwη → w strongly inL2([t0, t1]; L2)

(and more) since the sequence is pointwise increasing and|wt |L2 ≤ Cw.It remains to control the Dirichlet form ofwη − w. Consider

0(wη − w,wη − w) = 1

Tt f η0(Tt f η, wη − w)− 1

Tt f0(Tt f, wη−w)

= 1

Tt f η0(Tt f η − Tt f, wη − w)

+(

1

Tt f η− 1

Tt f

)0(Tt f, wη − w)


= 1

Tt f η0(η,wη − w)+ Ttη

Tt f η Tt f0(Tt f, wη − w)

= η

Tt f η0(w,wη − w) .(2.5)

The expressionη(Tt f η)−1 = η(Tt f + η)−1 is bounded by 1 and decreases to 0 inthe set whereTt f > 0. Since|w|L2 ≤ Cw, we know thatTt f > 0 a.s. Henceη(Tt f η)−1 ↓ 0 a.e., so that, by using dominated convergence, we conclude

limη↓0

∫ t1

t0

E(wηr − wr , wηr − wr )dr = lim

η↓0

∫ t1

t0

∫η

Tt f η0(wr , w

ηr − wr )dµ dr = 0 .

�

Even with less regularity onφt , the result is still true. This is seen, for example,in the next two lemmas. We skip the proofs since they follow the same pattern asLemma 2.3.

LEMMA 2.4 If in Lemma2.3φt = wt , the equality

|wt1|L2 − |wt0|L2 =∫ t1

t0

{−E(wr , wr )+ 1

2

∫φr0(wr , wr )dµ

}dr

is true for any t0, t1 ∈ (0,∞).

LEMMA 2.5 In the same setting as in Lemma2.3, but with the exception thatφt =Tτ−t g for some g∈ L∞, we have the equality

(φt1, wt1)− (φt0, wt0) = 1

2

∫ t1

t0

∫φr0(wr , wr )dµ dr

for any t0, t1 ∈ (0, τ ).2.3 The Distance Function

Now we come to the problem of defining the distance function to a set. Herewe have to be careful in giving the right meaning to the naive definition

d(x, A) “=” sup0(u,u)≤1

infy∈A

u(x)− u(y) ,

since we do not have a topology to play with. We use the hypothesis that the spaceD is separable here.

For A a set with positive measure, letAA = {u ∈ D : 0(u,u) ≤ 1, u ≥ 0, u =0 a.e. onA}. We have the following:

THEOREM 2.6 Suppose that inequality(1.6) holds true. Then there exists a func-tion dA(·) = d(·, A) ∈ AA such that∫

C

d(x, A)dµ = supu∈AA

∫C

u(x)dµ

for any measurable set C; i.e.,dA ≥ u for every u∈ AA.

268 J. A. RAMÍREZ

PROOF: SinceD is separable, there exists a countable dense subsetAA ⊂ AA

in the topology of the Dirichlet space. Let us say

AA = { f1, f2, . . . } .From the last section, it is clear thatgk = f1∨ f2∨· · ·∨ fk is an element ofAA.

We can see thatAA is uniformly bounded inL2 from the spectral gap inequality,i.e.,

|u|2L2 =∫ u − µ(A)−1

∫A

u

2

dµ ≤ cE(u,u) ≤ c .

Therefore it is a bounded sequence inD. Clearly,0(gk, gk) is also bounded inL2.We can extract, then, a weakly convergent subsequencegkl so that0(gk, gk) alsoconverges weakly. We setd(·, A) (= dA) as its limit. Moreover, since thegk forma pointwise increasing sequence, the whole sequence converges todA.

Because of Lemma 2.1, we see that0(dA,dA) ≤ 1, and it is also clear that∫A dA dµ = 0, implying thatdA ∈ AA. Finally, anyu ∈ AA can be expressed as

u = s-lim fni where fni ∈ AA, giving∫C

u dµ = lim∫C

fni dµ ≤ lim∫C

gni dµ =∫C

dA dµ ,

and that proves the lemma. �

Remark.A spectral gap (or inequality (1.6)) is not really needed to have a distancefunction. It is sufficient if one wantsdA ∈ L2. In any other case, one can replaceAA by its intersection with{ f : f ≤ M} for someM > 0. Then the argument ofTheorem 2.6 gives “dA ∧ M .” Now let M ↑ ∞.

Concerning the relation between the distance function and the distance betweentwo sets, we have the following lemma:

LEMMA 2.7 It is true thatdA achieves the set distance, i.e.,

(2.6) infx∈B

d(x, A) = d(A, B) .

In particular, infx∈B d(x, A) = infx∈A d(x, B).

PROOF: It is clear that, by definition,

infx∈B

d(x, A) ≤ d(A, B) .

On the other hand, for a givenε > 0, take a functionf with 0( f, f ) ≤ 1 andsuch that

(2.7) infx∈B,y∈A

f (x)− f (y) ≥ d(A, B)− ε .


The function f can be “cut” and “brought down” (f = ( f ∨ a) − a wherea =supy∈A f (y)) in order to replace it with a function inf ∈ AA still satisfying (2.7).We have that

d(A, B)− ε = infx∈B

f (x) ≤ infx∈B

d(x, A)

with ε arbitrarily small. The proof is finished. �

Finally, here is a lemma that will be useful when approximating sets from in-side.

LEMMA 2.8 Let A and B be two positive measure sets. Suppose that we have asequence of nested measurable sets Bk ⊂ Bk+1 ⊂ B such that

µ(B \ Bk) ↓ 0 as k↑ ∞andd(A, Bk) ≤ M for all k; then, we have that

limk↑∞ d(A, Bk) = d(A, B) .

PROOF: It is clear that limk↑∞ d(A, Bk) ≥ d(A, B) by definition. On theother hand,d(·, Bk) ∧ d(A, Bk) is a decreasing sequence. Its limitd∞ satis-fies 0(d∞,d∞) ≤ 1 by Lemma 2.1. We also have thatd∞(x) = 0 for x ∈B. Henced∞ ≤ dB. This, in conjunction with equation (2.6), implies thatlimk↑∞ d(A, Bk) ≥ d(A, B). �

3 Large Deviations

We prove Theorem 1.1 here. We work on general measure spaces subject to theassumptions made before. Therefore all statements are true up to a set of measurezero.

3.1 Upper Bound

THEOREM 3.1 Let A and B be two measurable sets with positive measure; then

(3.1) lim supt↓0

t log Pt(A, B) ≤ −d2(A, B)

2.

PROOF: This can be easily proved by a well-known trick that uses semigroups.This is usually known as the “method of Davies.” However, the method was orig-inally introduced by Gaffney (see [11] and also [4]). For completeness, we repro-duce the argument here.

Let ut = Tt1A. Forα ∈ R and a functionw(x) that is constant inA and inBand satisfies0(w,w) ≤ 1, consider the function

f (t) =∫(eαwut)

2 dµ

270 J. A. RAMÍREZ

which is finite sincew is bounded. Differentiating int we obtain

f ′(t) =∫

2e2αwutLut dµ

= −2E(e2αwut ,ut)

= −2∫0(ut ,ut)e

2αw dµ− 4α∫

ut0(w,ut)e2αw dµ

≤ 2α2∫

u2t 0(w,w)e

2αw dµ

≤ 2α2 f (t) .

Thereforef (t) ≤ f (0)e2α2t . Hence, we find

|eαwut |L2 ≤ √µ(A)eαw(A)+α

2t .

In a similar way, forvt = Tt1B and−α, we can obtain∣∣e−αwvt

∣∣L2 ≤ √

µ(B)e−αw(B)+α2t .

Now setw(·) = d(·, A) ∧ d(A, B). We see thatw(A) = 0 andw(B) = d(A, B).With this, we can estimate

Pt(A, B) =∫

Tt/21ATt/21B dµ =∫

Tt/21A eαw Tt/21B e−αw dµ

≤ |ut/2|2L |vt/2|2L ≤ √µ(A)

√µ(B)e−αd(A,B)+α2t .

Optimizing inα we get

Pt(A, B) ≤ √µ(A)µ(B) e− d2(A,B)

2t ,

and this proves the theorem. �

3.2 Lower Bound: Averaged Form

In this section we prove an averaged version of the lower bound. In fact, weprove the following stronger theorem:

THEOREM3.2 Assume a spectral gap. Given a positive measure set A, let ut(x) =−t log(Tt1A)(x). Consider its time averages, i.e.,

uT = 1

T

∫ T

0ut dt .

We have thatuT ⇀ d2A/2 weakly inD.

Remark.It is not hard to see that this implies that

lim infT↓0

1

T

∫ T

0t log Pt(A, B)dt ≥ −d2(A, B)

2,


but we will not prove it here since the same procedure will be applied in the nextsection to deduce the final form of the lower bound.

We start somewhat informally. Let’s look at the equation thatut satisfies.(Tt1A)(x) is the (weak) solution to the heat equation(∂t − L)(Tt1A)(x) = 0, so itslogarithmw(x, t) = log(Tt1A)(x) satisfies

(3.2) ∂twt = Lwt + 1

20(wt , wt) .

Here we used property (1.4) to deduce the formula

L log f = L f

f− 1

2

1

f 20( f, f ) = L f

f− 1

20(log f, log f ) .

Therefore, forvt = −wt we have

(3.3) ∂tvt = Lvt − 1

20(vt , vt) .

For ut = −twt we find

(3.4) ∂tut = Lut + 1

t

(ut − 1

20(ut ,ut)

),

or

(3.5) t (∂tut − Lut) = ut − 1

20(ut ,ut) .

From equation (3.5) one expects that, in the limit ast ↓ 0, it will be true that0(u0,u0) = 2u0 (hereu0 = limt↓0 ut ). This is the point where, in order to provesomething like this, we first do some averaging int . This produces the neededprecompactness.

In fact, we are only interested in the inequality0(u0,u0) ≤ 2u0 because itimplies for the square root ofu0 (using (1.3)):

(3.6) 0(√

u0,√

u0) = 1

4u00(u0,u0) ≤ 2

4= 1

2.

Therefore, by definition of the distance,

infx∈B,y∈A

(√u0(x)− √

u0(y)) ≤ 2−1/2 d(A, B) .

If, in addition, one has thatu0(y) = 0 for y ∈ A (as is natural to expect), the aboveinequality would imply

infx∈B

(limt↓0

−t logTt1A

)= inf

x∈Bu0(x) = inf

x∈B,y∈A

(√u0(x)− √

u0(y))2

≤ 1

2d2(A, B)

which is almost the result we want.We start the rigorous proof by assuming that a spectral gap holds true for the

generator under consideration. We will use the inequality that was stated in (1.6). It

272 J. A. RAMÍREZ

will be the main tool in proving boundedness properties of the solutions of equation(3.4). For this, we will use a method that closely resembles those used for theexample in [5] and [22] to get nonsharp Gaussian bounds for the heat kernel.

LEMMA 3.3 ut is bounded in L2.

PROOF: Setvηt (x) = − logTt(1A ∨ η)(x) whereη > 0 is a small number. Thisnew function satisfies the same equation asvt . In addition, it is trivial to check thatit belongs toL2, and it is easy to see that it satisfies (3.3) (see Lemmas 2.3 and 2.4).

Multiply by vηt equation (3.3), integrate the space variable, and use inequality(1.6) to obtain

1

2∂t

∣∣vηt ∣∣2L2 = −E(vηt , vηt )− 1

2

∫vηt 0(v

ηt , v

ηt )dµ

= −E(vηt , vηt )− 2

9E((vηt )

3/2, (vηt )

3/2)

≤ −cs(A)

∣∣∣∣∣∣(vηt )3/2 − 1

µ(A)

∫A

(vηt )

3/2 dµ

∣∣∣∣∣∣2

L2

≤ −cs(A)(1 − ε)∣∣(vηt )3/2∣∣2L2 − c

(1 − 1

ε

) 1

µ(A)

∫A

(vηt )

3/2dµ

2

≤ −c∣∣vηt ∣∣3L2 + M ,(3.7)

wherec andM are constants independent ofη. We used the inequality(a − b)2 ≥(1 − ε)a2 + (1 − ε−1)b2. The estimate on the last integral follows easily from thefollowing lemma:

LEMMA 3.4 Let f (say∈ L2) be a nonnegative function that is not identically zero(a.s.). If we set

g(x) ={

f (x)(Tt f (x))−1 if f (x) 6= 0

0 if f (x) = 0 ,

then g∈ L1 (in particular, finite a.e.) and∫g(x)dµ(x) ≤ 1 for all t .

In particular, the set{x : f (x) 6= 0} ∩ {x : Tt f (x) = 0} has measure zero.


PROOF: Let f ε = f ∨ ε for ε > 0 small. First write, using Jensen’s inequality,∫f ε(x)

1

T2t f ε(x)dµ(x) ≤

∫f ε(x)Tt

(1

Tt f ε(x)

)(x)dµ(x)

=∫

Tt f ε(x)1

Tt f ε(x)dµ = 1 ,

and afterwards use Fatou’s lemma asε ↓ 0 to conclude. �

Now apply the lemma together with the fact that(− logx)3/2 ≤ kx−1 if x ≤ 1(for some constantk > 0) to obtain∫

A

(vηt )

3/2 dµ ≤∫A

(− log(Tt(1A ∨ η)))3/2 dµ

≤ k∫A

1

Tt(1A ∨ η) dµ < k∫(1A ∨ η) 1

Tt(1A ∨ η) dµ < k .

Back to (3.7), if we were able to deducef (t) = ∣∣vηt ∣∣2L2 ≤ ct−2 (for someconstantc independent ofη), Lemma 3.3 would follow. In order to prove that, wework out the differential inequalityf ′ ≤ −c f 3/2 + M whereM is constant. Byintegrating the inequality, given thatf is positive, we see that

(3.8) f (t) ≤ c

(t −

∫ t

0

M

f 3/2(t)dt

)−2

.

If f (t) ≤ (M/(εc))2/3 for all t < t0, there is nothing to prove. If not, sayf (t1) >(M/(εc))2/3 for somet1 < t0, from the inequality we see that

f ′(t1) ≤ −cM

cε+ M ≤

(−1

ε+ 1

)M < 0

if ε is, say, less than 1. Consequently, we havef ′(t) < 0 and f (t) > (M/(εc))2/3

for all t < t1. Therefore,∫ t

0

M

f 3/2(t)dt ≤ cεt ⇒ t −

∫ t

0

M

f 3/2(t)dt ≥ (1 − cε)t

for t < t1, and from (3.8) we getf (t) ≤ c((1 − cε)t)−2 ≤ ct−2.

Therefore we have∣∣vηt ∣∣2L2 ≤ ct−2 wherec is independent ofη. Let η ↓ 0 and

use Fatou’s lemma to obtain the same bound for|vt |2L2. This implies thatut = tvt

is bounded inL2. �

In particular, from Lemma 3.3 we already get a (nonsharp) lower bound:

COROLLARY 3.5 We have

lim inft↓0

t log Pt(A, B) ≥ −C .

274 J. A. RAMÍREZ

PROOF: By using Jensen’s inequality and Lemma 3.3,

t log Pt(A, B) = t log

1

µ(B)

∫B

Tt1A dµ

+ t logµ(B)

≥ 1

µ(B)

∫B

t logTt1A dµ+ O(t)

≥ 1

µ(B)

∫t logTt1A dµ+ O(t)

= − 1

µ(B)|ut |L1 + O(t) ≥ −C .

�

Remark.We also conclude from Lemma 3.3 that equation (3.2) (and therefore alsoequations (3.3) and (3.4)) is satisfied in a weak sense (see Lemma 2.3).

We now go for the sharp lower bound. From the boundedness inL2 we want todeduce some kind of boundedness inD, but we can only do so for time averages.

LEMMA 3.6 ut is bounded in the Dirichlet spaceD.

PROOF: Like before, letuηt (x) = −t logTt(1A ∨ η)(x). This time integrateequation (3.4) to deduce

t∂t |uηt |L1 = |uηt |L1 − 1

2E(uηt ,u

ηt

).

Rearranging terms, integrating from 0 tot , and integrating by parts, we get

1

2t

∫ t

0E(uηs,u

ηs

)ds = 1

t

∫ t

0

(|uηs|L1 − s∂s|uηs|L1

)ds

= |uηt |L1 − 1

t

[s|uηs|L1

]t0 − 1

t

∫ t

0|uηs|L1 ds

= 2|uηt |L1 − |uηt |L1 .(3.9)

Therefore, sinceut andut are both bounded inL2 (⇒ boundedness inL1), all thatis needed is that

1

t

∫ t

0E(uηs,u

ηs

)ds ≥ E

(uηt , u

ηt

),

and this follows easily from the positivity ofE (Lemma 2.2). All this tells us that

E(uηt , u

ηt

) ≤ 2|uηt |L1 − |uηt |L1 .

Finally, letη ↓ 0 to conclude (use Lemma 2.1 on the left and monotonic con-vergence on the right). �


At this point we have weak precompactness ofut because it is bounded inD,which is a Hilbert space. Pick any subsequence ofut . Now, from this subsequenceextract further a subsequence(utk) that converges weakly inD (utk ⇀ u0 ∈ D) andsuch thatutk (no average here) converges weakly inL2 (utk ⇀ u0 ∈ L2).

Before proceeding further with the analysis, we want to prove the followinglemma. This will be needed to compareu0 with the distance to the setA as wasexplained earlier.

LEMMA 3.7 u0 = 0 a.s. in A.

PROOF: It is enough to see the following:∫A

u2t dµ = t2

∫A

(logTt1A)2 dµ ≤ ct2

∫A

1

Tt1Adµ → 0 .

We used the elementary inequality(logx)2 ≤ cx−1 for some constantc > 0 ifx ≤ 1 and Lemma 3.4. �

Now we go back to relation (3.4). Take any functionφ ∈ D, multiply throughby it in (3.5), integrate inx andt , and proceed as in (3.9) to obtain

(3.10)1

2(φ, 0t) = 2(ut , φ)− (ut , φ)− 1

t

∫ t

0sE(φ,us)ds,

where we have set0t = 1t

∫ t0 0(us,us)ds. We will treat the last term on the right

first. Integrating by parts, we find

1

t

∫ t

0sE(φ,us)ds = s

t

∫ s

0E(φ,ur )dr

∣∣∣t0− 1

t

∫ t

0

∫ s

0E(φ,ur )dr ds

= sE(φ, us)

∣∣∣t0− 1

t

∫ t

0sE(φ, us)ds

→ 0

ast ↓ 0. Hence, taking the limit alongtk, we deduce that0tk converges weakly tothe limit 2u0 − u0. This limit will be denoted by00. We have00/2 = 2u0 − u0 ≤2u0. Here we just dropped theu0 term. However, we will see below that this doesnot give the sharp bound.

By using Lemma 2.1, we have proved that

(3.11) 0(u0, u0) ≤ 4u0 .

First, this says that0(u0, u0) = 0 in the set{u0 = 0}. In addition, we have fromLemma 3.7 thatu0 = 0 in A. Therefore that implies, in the same way as in (3.6),0(u1/2

0 , u1/20 ) ≤ 1. Hence, we obtain infx∈S u0(x) ≤ d2(A, S) for any setS or,

alternatively,

(3.12) u0(x) ≤ d2(x, A)

276 J. A. RAMÍREZ

(the functiond(x, A) is defined in Lemma 2.6). This is not quite the expected resultbecause of the absence of the factor of1

2 on the right (compare to (3.6)). Noticethat this is so because we had to drop the termu0 from 00 since we do not knowif u0 = u0. But having this preliminary result at hand, we can repeat the aboveprocedure starting from (3.10). However, this time we try to bound the term withno average instead of just dropping it.

LEMMA 3.8 If we have

u0(x) ≤ Cd2(x, A)

2for some C> 1, this can be improved to

u0(x) ≤(

2 − 1

C

)d2(x, A)

2.

PROOF: From now onφ ∈ D is positive and bounded. That is enough to obtainpointwise (a.e.) bounds likef ≤ g from a relation like( f, φ) ≤ (g, φ) for all suchφ. We start by estimating

(ut , φ) =∫

−t logTt1Aφ dµ

≥ − |φ|L1 t log1

|φ|L1

∫Tt1Aφ dµ

= − |φ|L1 t log∫

Tt1Aφ dµ+ t |φ|L1 log |φ|L1

≥ − |φ|L1 t log Pt(A, Sφ)− |φ|L1 t log supφ + o(1)

whereSφ = {x : φ(x) > 0}. Taking the limit astk ↓ 0 along our subsequenceand using the hypothesis and the known upper bound for the large deviations, weobtain

(u0, φ) ≥ |φ|L11

2d2(A, Sφ) ≥ 1

C|φ|L1 inf

x∈Sφu0(x) .

Plugging that back into the limit of (3.10), we get

(3.13)1

2(0(u0, u0), φ) ≤ 2(u0, φ)− 1

C|φ|L1 inf

x∈Sφu0(x) .

It should be intuitively clear that with the above inequality we improve the estimateto

(3.14)1

20(u0, u0) ≤

(2 − 1

C

)u0 ,

and therefore we finally get

u0(x) ≤(

2 − 1

C

)d2(x, A)

2.


We postpone the easy but technical proof of (3.14) until the end of the section.�

The lemma implies that we find a recurrence relation of the form

Ck+1 = 2 − 1

Ck, C1 = 2 ,

for the bound ofu0 in terms ofdA. Notice that 2− 1C ≤ C (sinceC2 ≥ 2C−1) with

equality only whenC = 1. That says that our sequence of iterations will finallyconverge to give the desired result, that is,

(3.15) u0(x) ≤ 1

2d2(x, A) ,

which is true for any limit pointu0 of ut .The other inequality comes easily from the upper bound proved in the previous

section. Write, for any weakly convergent subsequenceuTk ⇀ u0,

d2(A,C)

2≤ lim inf

T↓0

1

T

∫ T

0−t log Pt(A,C)dt

≤ limTk↓0

1

Tk

∫ Tk

0−t log Pt(A,C)dt

= lim supTk↓0

1

Tk

∫ Tk

0

−t log

1

µ(C)

∫C

Tt1A dµ

+ t logµ(C)

dt

≤ lim supTk↓0

1

Tk

∫ Tk

0

1

µ(C)

∫C

ut dµ dt

= lim supTk↓0

1

µ(C)

∫C

uTk dµ = 1

µ(C)

∫C

u0 dµ .

Assume thatDε = {x : 2u0(x) ≤ d2A(x) − 2ε} has positive measure. TakeCε =

Dε ∩ {x : d2A(x) ≤ d2(A, Dε)+ ε} and notice that it also has positive measure and

d2(A, Dε) = d2(A,Cε). Therefore, by using the previous inequality, we have that

1

µ(Cε)

∫Cε

d2A(x)

2dµ− ε

2≤ d2(A,Cε)

2≤ 1

µ(Cε)

∫Cε

u0 dµ

≤ 1

µ(Cε)

∫Cε

d2A(x)

2− ε ,(3.16)

from which we obtain a contradiction. Therefore it is true thatµ(Dε) = 0 for anyε > 0. This proves thatu0 = d2

A/2 for any limit point u0 and therefore showsconvergence and concludes the proof.

278 J. A. RAMÍREZ

We now go back and prove that (3.13) and (3.11) imply (3.14).

LEMMA 3.9 Suppose we have

1

2

(0(u0, u0), φ

) ≤ 2(u0, φ)− 1

C|φ|L1 inf

x∈Sφu0(x)

for everyφ ∈ D ∩ L∞ positive, and also that0(u0, u0) ≤ 4 u0 almost everywhere.We can deduce

1

20(u0, u0) ≤

(2 − 1

C

)u0 .

PROOF: Let Ank = {k/n ≤ u0 < (k + 1)/n}. Now pick Fi : R → R smooth

functions withFi (x) = 0 for x ≥ (k + 1)/n andx ≤ k/n and such that

Fi → 1(k/n,(k+1)/n) asi ↑ ∞ .

We see that the functionsFi (u0) satisfy0(Fi (u0), Fi (u0)) ≤ ci andFi (u0) = 0outsideAn

k. In addition, ifk > 0 we have thatd(A, Ank) > 0. Therefore we can

apply (3.13) toFi (u0)φ in order to get

1

2(0(u0, u0), Fi (u0)φ) ≤ 2(u0, Fi (u0)φ)− 1

C

(inf

x∈Sφ∩Ank

u0(x)

)∫Fi (u0)φ dµ

≤ 2(u0, Fi (u0)φ)− k

Cn

∫Fi (u0)φ dµ

and take limits asi ↑ ∞ (bounded convergence theorem):

1

2

∫An

k

φ0(u0, u0)dµ ≤ 2∫An

k

φu0 dµ− k

Cn

∫An

k

φ dµ .

This inequality, which is true for everyφ, gives

1

2

∫E∩An

k

0(u0, u0)dµ ≤ 2∫

E∩Ank

u0 dµ− k

Cnµ(E ∩ An

k)

for any measurable setE andk > 0. By using the fact thatu0 < (k + 1)/n in Ank,

we see that

(3.17)1

2

∫E∩An

k

0(u0, u0) <

(2 − 1

C

) ∫E∩An

k

u0 + 1

2nµ(E ∩ An

k) .

If k = 0 we can use the coarse inequality (3.11) to obtain

(3.18)1

2

∫E∩An

0

0(u0, u0)dµ ≤ 2∫

E∩An0

u0 dµ < 2µ(E ∩ An0)

1

n.


Adding inequalities (3.18) and (3.17) fromk = 1 up to certain finitek = nM:

1

2

∫E∩{u0<M}

0(u0, u0)dµ ≤(

2 − 1

C

) ∫E∩{u0<M}

u0 dµ+ 2

nµ(E ∩ {u0 < M})

≤(

2 − 1

C

) ∫E∩{u0<M}

u0 dµ+ 2

n.

Now letn ↑ ∞ andM ↑ ∞ in that order to obtain the desired bound. �

3.3 Lower Bound: Final Form

We want to get rid of the average we had in the last section. For this, we clearlyhave to give up weak convergence inD and work with L2. We will prove thefollowing:

THEOREM 3.10 Assume a spectral gap. Given A, a set of positive measure, letut(x) = −t log(Tt1A)(x). We have that ut ⇀ d2

A/2 weakly in L2 as t ↓ 0.

The lower bound for Theorem 1.1 will be easily deduced from here. We beginby stating the main tool in dispensing with the average. This will be given by thefollowing Tauberian theorem:

LEMMA 3.11 Let f be a function defined on some interval[0, τ ] for which it istrue that

1

t

∫ t

0f (s)ds → L as t ↓ 0 ∈ R

and also that f(t)− f (s) ≤ Cs (t − s) (without absolute value) for all t > s small

enough. Then one can deduce that f(t) → L as t ↓ 0.

This lemma can be deduced from an application of a slightly different form ofPitt’s addition to Wiener’s Tauberian theorem (in the form of [19]) or from elemen-tary considerations.

We proceed to eliminate the time average. But first we present a little digressionintended to clarify the idea. Ideally one would like to apply the Tauberian lemmato

f (t) = −t log Pt(A, B)

since it is easy to obtain convergence in average forf (t) from Theorem 3.2. Butdifferentiating gives

f ′(t) = − log Pt(A, B)− td

dtPt(A, B) ≤ C

t− t

d

dtPt(A, B) .

The bound on the first summand comes right out of Lemma 3.5 and it is good. Theproblem is that we lack control on the second term, although one thinks of it asnegative (and therefore irrelevant). It seems negative from the fact thatPt(A, B)is, in spirit, increasing witht . The problem is that we cannot substantiate this

280 J. A. RAMÍREZ

claim (not even in some asymptotic sense). The following lemma is, however,proven along these lines:

LEMMA 3.12 We have, for t≤ τ andτ > 0,

limt↓0

−t∫

Tτ−t1B log(Tt1A)dµ = limt↓0

(Tτ−t1B,ut

) =∫

Tτ1Bd2

A

2dµ .

PROOF: Set f (t) = (Tτ−t1B,ut). We check the first condition of Lemma 3.11by proving that the averaged limit is true:

limT↓0

1

T

∫ T

0f (t)dt = lim

T↓0

1

T

∫ T

0

∫Tτ−t1But dµ dt =

∫Tτ1B

d2A

2dµ .

With Lemma 3.2, this is standard. To see it, note that∣∣∣∣ 1

T

∫ T

0

∫Tτ−t1But dµ dt −

∫Tτ1B

d2A

2dµ

∣∣∣∣ ≤1

T

∫ T

0|Tτ−t1B − Tτ1B|L2 |ut |L2 dt +

∣∣∣∣∫

Tτ1B

(uT − d2

A

2

)dµ

∣∣∣∣ → 0 ,

sinceuT ⇀ d2A/2 weakly from Theorem 3.2 andTτ−t1B → Tτ1B strongly inL2.

For the second hypothesis in Lemma 3.11, we proceed formally to differentiatef (t) in order to obtain

f ′(t) =∫∂t Tτ−t1B(x)ut(x)dµ(x)+

∫Tτ−t1B(x)∂tut(x)dµ(x)

= −∫

LTτ−t1But dµ+∫

Tτ−t1B

(Lut + 1

t

(ut − 1

20(ut ,ut)

))dµ

= 1

t

∫Tτ−t1But dµ− 1

2t

∫Tτ−t1B0(ut ,ut)dµ

≤ C

t,

since the second term in the third line is negative andut is bounded inL2. Thisargument can be seen to be rigorous by using Lemma 2.5. Namely, by multiplyingequation (2.4) by−t and integrating by parts, one can deduce(

Tτ−t1B,uηt

)− (Tτ−s1B,u

ηs

) =∫ t

s

1

r

{(Tτ−r 1B,u

ηr

)− (Tτ−r 1B, 0(u

ηr ,u

ηr ))}

dr

≤ C

s(t − s)

uniformly in η. Let η ↓ 0 to complete the proof. �

Now we finish the proof of Theorem 3.10. We have

(3.19) −t log Pt(A, B) = −t log∫B

Tt1A dµ ≤ 1

µ(B)

∫B

ut dµ+ t logµ(B) .


Therefore, lettingt ↓ 0, we obtain

1

2d2(A, B) ≤ lim inf

t↓0

1

µ(B)

∫B

ut dµ = 1

µ(B)

∫B

u0 dµ ,

whereu0 is a weak limit ofut that achieves the lim-inf. As in the previous section(see equation (3.16) and the paragraph before), this impliesu0 ≥ d2

A/2 for anyweak limit ofut .

For the other inequality, consider∫B

ut dµ ≤∫

Tτ−t1But dµ+∫B

|1 − Tτ−t1B| ut dµ

≤ f (t)+ |1B − Tτ−t1B|L2 |ut |L2

≤ f (t)+ C(|1B − Tτ−t1B|L2 + |Tτ1B − Tτ−t1B|L2

)where f (t) is the one from Lemma 3.12. Lettingt ↓ 0 we obtain∫

B

ut dµ ≤∫

Tτ1Bd2

A

2dµ+ C |1B − Tτ1B|L2 .

Finally, lettingτ ↓ 0, we get thatu0 ≤ d2A/2 for any weak limitu0 of ut . This

finishes the proof of Theorem 3.10. Now we can prove the lower bound.

PROOF OFTHEOREM 1.1: Recall equation (3.19). This clearly implies

(3.20) lim supt↓0

−t log Pt(A, B) ≤ 1

µ(B)

∫B

d2A

2dµ .

This is almost our final lower bound. In order to obtain the result we want, we picka smaller setCε instead ofB. For example, set

Cε = B ∩ {x : d2(x, A) < d2(A, B)+ ε}

so that the last integral in equation (3.20) (withCε instead ofB) can be boundedby ∫

Cε

d2(x, A)dµ ≤ µ(Cε)(d2(A, B)+ ε) .

Notice that because infx∈B d(x, A) = d(A, B), we haveµ(Cε) > 0 (as a re-minder, “inf” means “essinf”). Finally,

lim supt↓0

−t log Pt(A, B) ≤ lim supt↓0

−t log Pt(A,Cε) ≤ d2(A, B)+ ε

2(3.21)

whereε > 0 is arbitrary. By lettingε ↓ 0 we obtain what was the main purpose ofthis work. �

282 J. A. RAMÍREZ

Remark.The result holds even if the distance between the two sets is zero; i.e.,there is no functionf satisfying0( f, f ) ≤ 1 and “separating” the two sets (that is,infx∈A f (x)−supy∈B f (y) > 0). The limit is zero in that case, as it should be. Thiscan be traced back to the limitu0 found in the last section, which cannot “separate”the two sets because itscarré du champis bounded. Notice that this includes thecase where there are no functions for which0( f, f ) is bounded (except constants).

4 Examples

Here we discuss some applications of the previous results. We do not work withgeneral measure spaces here, so there is a distinction between “inf” and “essinf.”

4.1 Dirichlet Forms on Locally Compact Spaces

We intend to complement the results in a series of papers by K. T. Sturm [25,26] by proving the lower bound in the asymptotics of the heat kernel. This is ageneral framework that certainly includes many finite-dimensional examples. Wefirst describe the additional assumptions that are needed in this section.

The space� is assumed locally compact, separable, and Hausdorff. The mea-sureµ is a Radon measure with support�. Define the intrinsic metric by

d(x, y) = sup{

f (x)− f (y) : f ∈ D ∩ C(�), 0( f, f ) ≤ 1}.

Additional assumptions needed are:

(1) Strong regularity. The topology induced by the intrinsic metric is the sameas the original one and balls are relatively compact in�.

(2) Doubling property. There is a constantN such that

µ(B2r (x)) ≤ 2Nµ(Br (x))

uniformly for x ∈ �.(3) Poincaré inequality. There is a constantCP such that for all ballsBr (x) ⊂

� we have∫Br (x)

∣∣ f − fx,r

∣∣2 dµ ≤ CPr 2∫

Br (x)

0( f, f )dµ

where fx,r = µ(Br (x))−1∫

Br (x)f dµ.

If one defines distance between sets in the natural way, namely,

d(A, B) = infx∈A,y∈B

d(x, y) ,

then from results in [24] one hasd(A, B) = d(A, B). Moreover, due to continuityof the metric,

(4.1) d(x, y) = limr ↓0

d(Br (x), Br (y)) .

In addition, from the properties described above, a nonsharp bound on the heatkernel is deduced in [26]. We state it in the form that is relevant here. To be


precise, under the above assumptions it is true that, for any ballBR(x), if t > 0 issufficiently small, there is aC2 = C2(R, x) and ac ≥ 1 such that

(4.2) pt(x, y) ≥ C2 e− cd2(x,y)2t for everyy ∈ BR(x) .

Although, in general, we do not have a spectral gap, the Poincaré inequalitygives one for finite balls. That is, if� ⊂ BR(x) for someR > 0, then we obtaina spectral gap. We see that the conditions of Theorem 1.1 are valid for the spaceunder consideration.

In general, if the space is not contained in a ball of finite radius, for givenη > 0,fix x ∈ A andy ∈ B such that = d(x, y) ≤ d(A, B)+ (1

2)η. Change the setsAandB to A = Bη(x) ∩ A andB = Bη(x) ∩ B. Therefored(A, B) ≥ d(A, B)− η.Consider the Dirichlet space onB = B2`(x) with Neumann boundary conditionsat ∂B. The Dirichlet form is given by

EB( f, g) =∫B

0( f, g)dµ .

As discussed above, the asymptotic result (specifically, the lower bound) is validhere. Denote byPB the associated stochastic process. We proceed to compare thisto the original one.

Pt(A, B) ≥ Pt(A, B)

≥ P[X0 ∈ A,Xt ∈ B, τ∂B > t

]= PB

[X0 ∈ A,Xt ∈ B, τ∂B > t

]= PB

t (A, B)− PB[X0 ∈ A,Xt ∈ B, τ∂B ≤ t

].

The third line comes from the fact that the two processes are the same up to thetime when they hit the boundary (∂B). The uncomfortable last term is too small tobother the asymptotics,

PB[X0 ∈ A,Xt ∈ B, τ∂B ≤ t

] ≤ PB[X0 ∈ A, τ∂B ≤ t

]≤ Ce− (2`−η)2

2t ≤ Ce− 2d2(A,B)2t ;

the other one gives almost the right rate:

lim inft↓0

t log PBt (A, B) ≥ −d2(A, B)

2≥ (d(A, B)+ (1/2)η)2

2.

Therefore, we obtain

lim inft↓0

2t log Pt(A, B) ≥ −(

d(A, B)+ 1

2η

)2

.

Finally, letη ↓ 0, thus obtaining the lower bound in the asymptotics ofPt(A, B).Therefore we have that for anyε > 0 there is aT = T(ε, A, B) > 0 and a

284 J. A. RAMÍREZ

C1 = C1(ε, A, B) > 0 such that for anyt < T the estimate

Pt(A, B) ≥ C1e− d2(A,B)

2t (1+ε)

holds true. By using this, (4.1), and (4.2), we can get pointwise estimates.

THEOREM 4.1 Under the above assumptions, we have

lim inft↓0

2t log pt(x, y) ≥ −d2(x, y) .

PROOF: Supposey ∈ BR/2(x). Write the Chapman-Kolmogorov equation forpt(x, y), and estimate it as follows:

pt(x, y) =∫∫

ptδ/2(x, z)pt (1−δ)(z, w)ptδ/2(w, y)dµ(z)dµ(w)

≥∫

Br (y)

mu(w)∫

Br (x)

mu(z)ptδ/2(x, z)pt (1−δ)(z, w)ptδ/2(w, y)

≥ C2(r, x)C2(r, y)∫

Br (y)

mu(w)∫

Br (x)

dµ(z)e− 2cr2tδ pt (1−δ)(z, w)

≥ C2(r, x)C2(r, y)e− 2cr2tδ Pt (1−δ)(Br (y), Br (x))

≥ C1(x, y, r )C2(r, x)C2(r, y)e− 2cr2tδ e− d2(Br (y),Br (x))

2t (1+ε) .

Take logarithms, multiply by 2t , and take the limit ast ↓ 0 to obtain

lim inft↓0

2t log pt(x, y) ≥ −4cr2

δ− d2(Br (y), Br (x))(1 + ε) .

Finally, let r ↓ 0 andδ ↓ 0 in that order to obtain the conclusion of the theorem.�

The corresponding upper bound is proved in [25].

4.2 Ornstein-Uhlenbeck Process on Wiener Space

A spectral gap inequality is valid here. Therefore everything proved in the lasttwo chapters apply to this case.

This case was worked out by S. Fang in [9]. We present here how to recover hisresult from Theorem 1.1. We follow his notation more or less. All we have to dois compare the distances and sets used. Define

dH (x, A) = infa∈A

‖x − a‖H .

First we will see that the set distance used in [9] is the same as ours in the casesof interest.


LEMMA 4.2 The distance between sets, as defined in(1.7), is greater or equal tothe distance dH defined in[9],

d(A, B) ≥ dH (A, B) = sup(essinfx∈A dH (x, B),essinfy∈B dH (y, A)

).

We have equality if one of the sets is open. Here, as in the reference,A =supp(1Adµ) ∩ A. Also, theessinfare in the sense of the measure

µ(A) = supK

{µ(K )}where the supremum is taken over K⊂ A compact.

PROOF: Let us first show thatd(A, B) ≥ dH (A, B). If A is compact,dH (·, A)is known to be measurable (in fact, its level sets{dH (·, A) ≤ λ} are compact by aSobolev embedding). We will proved(x, A) ≥ dH (x, A).

To check that this is true, we notice that|dH (x + h, A) − dH (x, A)| ≤ ‖h‖H

as in proposition 3.1 of [9]. This implies0(dH (·, A)) ≤ 1 and therefored(·, A) ≥dH (·, A) a.s. (see [14] or [6]).

Without loss of generality, we can assume that bothA = A andB = B. Sup-poseessinfx∈AdH (x, B) > λ. Then we have that

µ ({dH (x, L) ≤ λ} ∩ A) = 0

for any compactL ⊂ B. But sinced(x, L) ≥ dH (x, L) (becauseL is compact) weobtain

µ ({d(x, L) ≤ λ} ∩ A) = 0

for any compactL ⊂ B with µ(L) > 0. Therefored(A, L) ≥ λ. Now take asequence(Ln) of compact sets that exhaustB (µ(B \ Ln) ↓ 0) and use Lemma 2.8to getd(A, B) ≥ λ. The same can be done interchangingA andB to obtain that

d(A, B) ≥ dH (A, B) .

For the reverse inequality, suppose thatA is open. SetdA = d(·, A). It satisfies0(dA,dA) ≤ 1 a.e. and therefore|dA(x + h) − dA(x)| ≤ ‖h‖H for all x ∈ X,h ∈ H , and a versiondA of dA (see [6]). Also,dA = 0 a.e. inA. We now showthatdA can be chosen to be equal to zeroeverywherein A.

SetA = {x ∈ A : dA(x) > 0} so thatµ(A) = 0. If a ∈ A and we can take asequencehn → 0 in H such thatdA(a + hn) = 0 (recall thatdA = 0 a.s. inA), wewill have

dA(a) ≤ dA(a + hn)+ ‖hn‖H → 0 ,

which impliesdA(a) = 0, ora 6∈ A. Therefore, ifa ∈ A, then there existsε > 0depending ona such that

BH (a, ε) = {x : ‖x − a‖H < ε} ⊂ A .

Take(gk), a dense set ofH (in the H -topology). It is not difficult to see thatA + H = A + {gk} from the above property forA. This says thatA + H is adenumerable sum of zero-measure sets (each translation bygk has measure zero by

286 J. A. RAMÍREZ

quasi invariance). Thenµ(A + H) = 0 anddA can be redefined there to be zerowithout affecting itsH -Lipschitz condition.

That dA = 0 everywhere inA together with theH -Lipschitz condition tells usthat for anya ∈ A,

dA(x) ≤ ‖x − a‖H .

Now take the infimum overa ∈ A to getd(·, A) ≤ dH (·, A) a.e. But again, sinceA is open,dH (·, A) = dH (·, A). �

The result of all this is the following reformulation of Theorem 1.1:

THEOREM 4.3 If A and B are measurable sets withµ(A) > 0 andµ(B) > 0, wehave

lim supt↓0

2t log Pt(A, B) = −d2H (A, B) .

4.3 Ornstein-Uhlenbeck Process on Path Space

The generalization of the result from the last section to path spaces over Rie-mannian manifolds is discussed here. For some background on path spaces, thearticles by Hsu [13] and Stroock [23] are very nice references.

Consider a (compact, connected,d-dimensional) Riemannian manifoldM . Fol-lowing standard notation, denote byO(M) its orthonormal frame bundle and byH1(u), . . . , Hd(u) the canonical horizontal vector fields at a pointu ∈ O(M).Also, letπ : O(M) → M be the projection map.

Let I = [0,1] ⊂ R, and WM the space of continuous pathsX· : I → Mendowed with a measureP that is the image under the rolling map of Wienermeasure. That is the map defined by the Stratonovich equation

dUt =d∑

i=1

Hi (Ut) ◦ dwit , U0 = u0 , Xt = π(Ut) ,

wherew : I → Rd is Brownian motion. The intermediate variableU : I → O(M)

is called horizontal Brownian motion.Again, the topology on the space is given by a supremum norm,

d∞(x, y) = supt∈I

d(Xt , yt) .

Thecarré du champis theH -norm squared of the gradient, i.e.,

0(F, F)(x) = ‖DF(x)‖2H ;

moreover, ifF(x) = f (Xt1, . . . ,Xtn) depends only on a finite number of coordi-nates, we can compute it as follows:

(4.3) ‖DF(x)‖2H =

n∑i=1

(ti − ti−1)

∣∣∣∣∣∣n∑

j =i

U (x)−1tj ∇ j f (Xt1, . . . ,Xtn)

∣∣∣∣∣∣2

.


We assume that the Ricci curvature is bounded below. With that assumption, thespectral gap was proved here by Fang [8].

We will content ourselves with this and not try to identify the distance as min-imization over paths. This requires a version of Rademacher’s theorem for thesespaces that is only known for path groups. For the path group case, the result belowis due to Aida and Zhang [2] if eitherA or B is an open set. That case can also bededuced from Hino’s work [12].

We have the following:

THEOREM4.4 If A and B are measurable withµ(A) > 0 andµ(B) > 0, we have

lim supt↓0

2t log Pt(A, B) ≤ −d2(A, B) .

If the Ricci curvature of M is bounded below,

limt↓0

2t log Pt(A, B) = −d2(A, B) .

4.4 More General Processes on Path Space

Let L∞(H, H) denote the Banach space of all bounded linear operators onH using the operator norm. Consider the following bilinear form on Dom(D) ⊂L2(WM):

E A( f, g) = 1

2

∫ (A(x)D f (x), Dg(x)

)H

dµ(x)

whereµ is the Wiener measure and(·, ·)H is the usual scalar product in Cameron-Martin space.A is a strongly measurable map fromWM into L∞(H, H) such thatA(x) : H → H is a symmetric linear bounded operator. Moreover, suppose that,uniformly for x ∈ WM ,

c‖h‖2H ≤ (

A(x)h, h)

H≤ C‖h‖2

H

where 0< c ≤ C < ∞.These conditions make the new Dirichlet formE A(·, ·) equivalent to the Orn-

stein-Uhlenbeck one (A = I ). Therefore, we have the following:

• Since the operatorD is itself closed (cf. [13]), the newly defined formis a Dirichlet form. Moreover, the space satisfies quasi regularity (this isequivalent to the existence of a process [16]). This is seen from the factthat path space satisfies it, and theE A-capacities are equivalent to the onesdefined byE from the last section (in addition, the topology is the same).

• Sincec > 0, the spectral gap for Ornstein-Uhlenbeck implies the spectralgap for the new process.

In conclusion, we obtain the same statement as in the previous example (i.e.,Theorem 4.4). This type of examples was first considered by Zhang [28] and Aidaand Kawabi [1]. However, these studies imposed extra regularity conditions on theoperatorA; the present result is more general.

288 J. A. RAMÍREZ

4.5 Fleming-Viot Process

Let M1 denote the set of probability distributions on a compact Polish spaceE.We endowM1 with the weak topology. In this section (and only here)µ ∈ M1 isan element of the space and not the invariant measure of the process.

A differentiable curve is a functionω : [0,1] → M1 that is continuous in varia-tion and differentiable with respect to the variational norm (‖·‖var). LetC1(M1) de-note the set of real-valued continuous functionsu ∈ C(M1) such thatt → u(ω(t))is differentiable for every differentiable curveω and whose derivative satisfies

d

dtu(ω(t)) = 〈Du(ω(t)), ω(t)〉

whereDu : M × E → R is bounded and measurable.Du is computed from

Du(µ, x) = d

dt

∣∣∣∣t=0

u((1 − t)µ+ tδx)

whereδx is the point mass atx ∈ E. With the derivative at hand, we define acarré du champoperator by0(u,u)(µ) = ∫

(Du(µ, x))2µ(dx). In particular, thelocality hypothesis (1.3) is satisfied.

The generator of the Fleming-Viot process with mutation operatorA is givenby

Lu(µ) = 1

2

∫D2u(µ, x)µ(dx)+

∫ADu(µ, x)µ(dx) .

Suppose that the mutation operator is of the form

A f (x) = θ

2

∫E

( f (y)− f (x))ν0(dy) ,

whereθ > 0 andν0 ∈ M1 is such that suppν0 = E. This process has a uniquestationary distribution that moreover makes the process reversible (for a survey,see [7]). In this case, a spectral gap of sizeθ/2 exists for the operatorL as provedby Stannat [21]. Therefore, we see that Theorem 1.1 applies to the Fleming-Viotprocess.

In this particular case, the quantityd(A, B) can be computed from a pointwisemetric in the natural way. The (pointwise) intrinsic metric is defined as

d(µ, ν) = sup{u(µ)− u(ν) : u ∈ C1(M1); 0(u,u) ≤ 1

}.

In [20], Schied proved that it can be computed as follows:

d(µ, ν) = 2 arccos∫ √

dν

dη

dµ

dηdη

for anyη ∈ M such thatµ andν are both absolutely continuous with respect toη.In particular, the functiondν0(·) = d(·, ν0) is bounded a.s.


5 Extension of the Main Result

In this section we present an extension of Theorem 1.1 to spaces that can bemade to have a spectral gap.

5.1 On the Associated Process

In the proof of Theorem 5.1 our methods are mainly probabilistic. Therefore wehave to assume also the existence of a Markov process associated to the Dirichletform. This is a collection((Xt)t≥0, (Px)x∈�) such that there is a filtrationFt forwhich (Xt)t≥0 is Ft -adapted and Markov. It associates to the Dirichlet form in thesense that

Px

[Xt ∈ A

] = Tt1A(x)

is true a.s. With that, a probability measureP on the space of pathsX· : R → �

can be defined by starting from the invariant measureµ. This process is reversible;i.e., if rT is defined byrTXt = XT−t (time reversal aroundT), we have

P[X ∈ C

] = P[rTX ∈ C

].

The existence of the process and the properties to be described below havebeen studied thoroughly for regular Dirichlet spaces on locally compact spaces(see [10]) and generalizations to quasi-regular ones [16]. In fact, given a Hausdorffspace, quasi regularity is equivalent to the existence of the associated process (cf.[16]).

For the rest of this section, quasi regularity will be assumed forD. However, themethod used to prove the large-deviation estimates depends only on the (basicallyalgebraic) formulae given below.

The Doob-Meyer decomposition for the processf (Xt) − f (X0) for a functionf in the domain of the generator takes the following form:

(5.1) f (Xt)− f (X0) = M ft +

∫ t

0L f (Xs)ds,

whereM ft is a continuous martingale with respect toFt . If it happens thatf ∈ D,

we have the quadratic variation (bracket) ofMt given by

〈M〉t =∫ t

00( f, f )(Xs)ds.

This identification for the bracket ofM ft means that we further have the following

Doob-Meyer decomposition:

(M ft )

2 = ( f (xt)− f (X0)−∫ t

0L f (Xs)ds)2

= Martingale+∫ t

00( f, f )(Xs)ds.(5.2)

290 J. A. RAMÍREZ

Since our process is reversible, we can apply (5.1) to the process reversedaroundT in order to obtain

(5.3) f (XT−t)− f (XT ) = 1

2

(M f

t − (M f

T − M fT−t

)).

This is the forward-backward martingale decomposition (see [15]). It can be ex-tended to functions inD by standard approximation procedures.

5.2 Changing the Invariant MeasureWe review here what happens to the Dirichlet space when we do an absolutely

continuous change of invariant measure. Take9 ∈ D satisfying∫

exp(−29)dµ =1. Define a new Dirichlet form by

E9( f, f ) =∫0( f, f )(x)e−29(x)dµ(x) =

∫0( f, f )(x)dµ9(x) ,

whenever the right-hand side is finite. With this we can take the completion withrespect to the norm

| f |D9 =

[∫| f |2 dµ9 + E9( f, f )

]1/2

in order to obtain a new Dirichlet spaceD9 that satisfies the same requirementsfrom Section 1.1 as the original.

Assume that9 ∈ D with a boundedcarré du champ(0(9,9) ≤ C9). Usinga straightforward computation, we see that the associated generator is given byL9 f = L f + 0(9, f ).

Given the generator and recalling the Cameron-Martin formula for changes ofmeasure, one can guess what shape the Radon-Nikodym factor will take for theprocess. Let us confirm those suspicions. For the new measure, it should be truethat

(5.4) E9[g(X0) f (XT )] =∫

f (x)E9x [g(XT )]e−29(x)µ(dx) .

On the other hand, by expanding the left-hand side,

E9[g(X0) f (XT )] =∫

g(x)E9x [ f (XT )]e−29(x)dµ(x)

= E[ f (X0) f (XT )ZTe−29(X0)]

= E[g(XT ) f (X0)ZTe−29(XT )]

=∫

f (x)Ex[g(XT )e2(9(XT )−9(X0)) ZT ]e−29(x)dµ(x) .(5.5)

Here we have called the martingale (with respect toFT ) ZT = d P9

d P

∣∣Ft

. Inaddition, ZT is the image ofZT under time reversal aroundT (t 7→ T − t), andtherefore it is a martingale adapted to the reversed filtrationFt .


Comparing (5.4) with (5.5), we see that we need to have

e9(XT )−9(X0) ZT = ZT

where ZT is a martingale. This can be achieved by letting9(XT ) − 9(X0) =M9

T + N9T be the Doob-Meyer decomposition and setting

(5.6) ZT = e−M9T − 1

2 〈M9 〉T .

That helps because9(XT )−9(X0) = 12(M

9T − M9

T ) by the Lyons-Zheng decom-position.

Since we have0(9,9) ≤ C9 , the expression for the bracket is given by

〈Mψ〉t =∫ t

00(9,9)(Xt)dt ≤ C9 t ,

and with that, it is easy to check thatZT is a martingale.Now we can go back and define a measureP9 in the same space in whichP is

defined by specifying the Radon-Nikodym derivative:

d P9

d P

∣∣∣∣Ft

= ZT = e−M9T − 1

2 〈M9 〉T .

It is standard to see that all the properties discussed in Section 5.1 also transfer tothis new Dirichlet space. Forf ∈ D, one just has to substitute

M ft 7→ M f

t −∫ t

00( f, 9)(Xs)ds,

N ft 7→ N f

t +∫ t

00( f, 9)(Xs)ds,

in order to o btain the corresponding martingale and zero-energy process associatedto f in this new space.

5.3 Extension of the Main ResultBased on the previous discussion, we can somehow relax the spectral gap con-

dition in Theorem 1.1. The following is the basis for such a possibility:

THEOREM 5.1 Assume that Theorem1.1 is true for P9 ; then it is also valid for P.

PROOF: Use the definition ofP9 and Holder inequality to obtain

P9t (A, B) = EP

[1A(X0)1B(Xt)Zt

]≤ Pt(A, B)1/q EP[Z p

t ]1/p

= Pt(A, B)1/q

× EP

[exp

{pM9

t − p2

2〈M9〉t

}exp−

{p − p2

2〈M9〉t

}]1/p

≤ Pt(A, B)1/q EP

[exp

{pM9

t − p2

2〈M9〉t

}]1/p

e−{ 1−p2 C9 t}

292 J. A. RAMÍREZ

whereC9 is such that0(9,9) ≤ C9 . Take logarithms, multiply by 2t , and takethe lim-inf ast ↓ 0 in order to obtain

−d2(A, B) = limt↓0

2t log P9[X0 ∈ A,Xt ∈ B] ≤ lim inft↓0

2t log Pt(A, B)× 1

q.

We used the fact that Theorem 1.1 is valid forP9 . Finally, letq ↓ 1 to concludethe same forP.

The other inequality follows by a similar procedure. Or just remember that theupper bound does not require a spectral gap. �

Acknowledgments. This work would not have been completed without thehelp and encouragement of my advisor Prof. S. R. S. Varadhan. I am also gratefulto Prof. P. Malliavin, whose suggestion was the seed of this work.

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JOSÉA. RAMÍREZ

Cornell UniversityDepartment of MathematicsMalott HallIthaca, NY 14853-4201E-mail: [email protected]

Received February 2000.

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Short-time asymptotics in Dirichlet spaces