Diffusion in small time in incomplete sub-Riemannian manifolds

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manifolds

Ismaël Bailleul, J Norris

To cite this version:

Ismaël Bailleul, J Norris. Diffusion in small time in incomplete sub-Riemannian manifolds. 2019.

�hal-02047605�

sub-Riemannian manifolds

I. BAILLEUL and J. NORRIS¹

Abstract. For incomplete sub-Riemannian manifolds, and for an associated second-order hypoelliptic operator, which need not be symmetric, we identify two alternative conditions for the validity of Gaussian-type upper bounds on heat kernels and transition probabilities, with optimal constant in the exponent. Under similar conditions, we obtain the small- time logarithmic asymptotics of the heat kernel, and show concentration of diffusion bridge measures near a path of minimal energy. The first condition requires that we consider points whose distance apart is no greater than the sum of their distances to infinity. The second condition requires only that the operator not be too asymmetric.

1 – Introduction

LetM be a possibly unbounded connected smooth manifold, which is equipped with a smooth sub-Riemannian structure X₁, . . . , X_m and a positive smooth measure ν. Thus, X1, . . . , Xm are smooth vector fields on M which, taken along with their commutator brackets of all orders, span the tangent space at every point, and ν has a positive smooth density with respect to Lebesgue measure in each coordinate chart. Consider the symmet- ric bilinear form aonT^˚M given by

apxq:“

m

ÿ

`“1

X_{`</sub>pxq bX<sub>`}pxq.

Let L be a second order differential operator on M with smooth coefficients, such that L1“0 and Lhas principal symbol a{2. In each coordinate chart,L takes the form

L“ 1 2

d

ÿ

i,j“1

a^ijpxq B² BxⁱBx^j `

d

ÿ

i“1

bⁱpxq B

Bxⁱ (1.1)

for some smooth functionsbⁱon the coordinate chart. Writepfor the Dirichlet heat kernel of L inM with respect to ν, and write B “ `

Bt :tP r0, ζq˘

for the associated diffusion process. For x, yof M and tP p0,8q, set

Ω^t,x,y:“

! ω PC`

r0, ts, M˘

:ω₀ “x and ω_t“y )

.

Consider the case where B₀ “x. While the explosion time ζ of B may be finite, we can still disintegrate the sub-probability law µ^t,x of B restricted to the event tζ ą tu by a unique family of probability measures `

µ^t,x,y :yPM˘

, weakly continuous in y, such that µ^t,x,y`

Ω^t,x,y˘

“1 and

µ^t,xpdωq “ ˆ

M

µ^t,x,ypdωqppt, x, yqνpdyq.

1Research supported by EPSRC grant EP/103372X/1 1

The probability measuresµ^t,x,y are calledbridge measures.It will be convenient to consider these measures all on the same space Ω^x,y :“ Ω^1,x,y. So define σ_t : Ω^t,x,y Ñ Ω^x,y by σtpωqs:“ωst and defineµ^x,y_t onΩ^x,y by

µ^x,y_t :“µ^t,x,y˝σ_t^´1.

We focus mainly on two problems, each associated with a choice of the endpointsxand y, and with the limit tÑ0. The first is to give conditions for the validity of Varadhan’s asymptotics for the heat kernel

tlogppt, x, yq Ñ ´dpx, yq²

2 , (1.2)

where d is the sub-Riemannian distance. The second is to give conditions for the weak limit

µ^x,y_t Ñδγ (1.3)

where γ is a path of minimal energy in Ω^x,y. We wish to understand, in particular, what can be said without symmetry or ellipticity of the operatorL, andwithout compactness or even completeness of the underlying space M. The heat kernel and the bridge measures have a global dependence onL, while the limit objects have a more local character, so the limits depend on some localization of diffusion in small time. We will give two sufficient conditions for this localization, the first generalizing from the Riemannian case a criterion of Hsu [8] and the second requiring a ‘sector condition’ which ensures that the asymmetry inLis not too strong. We will thus give new conditions for the validity of (1.2) and (1.3), which do not require completeness, symmetry or ellipticity; no control on the diffusivity a or the symmetrizing measure ν is needed either. In a companion paper [3], we have further investigated the small-time fluctuations of the diffusion bridge around the minimal pathγ, which reveal a Gaussian limit process.

We define from the quadratic form athe energy Ipγq of an M-valued path γ and the distance functions on M in the classical way; see the Notations paragraph at the end of this section.

LetKbe a closed set inM. WritepK^c for the Dirichlet heat kernel ofLinK^c, extended by 0outside K^cˆK^c. Define

ppt, x, K, yq:“ppt, x, yq ´pK^cpt, x, yq. (1.4) Then

ppt, x, K, yq “ppt, x, yqµ^x,y_t

´

ω PΩ^x,y:ωsPK for somesP r0,1s(¯ .

We callppt, x, K, yqtheheat kernel throughK. In the case whereK^cis relatively compact, we write ppt, x, Kq for the hitting probability for K, given by

ppt, x, Kq:“PxpT ďtq “1´ ˆ

K^c

pK^cpt, x, yqνpdyq (1.5) whereT :“infttP r0, ζq:B_tPKu.² Define also

dpx, Kq:“inf dpx, zq:zPK( dpx, K, yq:“inf

!

dpx, zq `dpz, yq:zPK )

2Note that

ppt, x, Kq ě ˆ

M

ppt, x, K, yqνpdyq ě ˆ

K

ppt, x, yqνpdyq

and the first inequality is strict if the process explodes, while the second inequality is always strict because the process returns toU with positive probability after hittingK.

and note that

dpx, Kq `dpy, Kq ďdpx, K, yq.

Define

dpx,8q:“sup

!

dpx, Kq:K closed andMzK relatively compact )

.

It is clear thatdp¨,8qis either finite or identically infinite. By the sub-Riemannian version of the Hopf-Rinow theorem, the second case occurs if and only if M is complete for the sub-Riemannian metric. Note that the triangle inequality does not apply ‘atK’ or ‘at8’, and dpx, Kq may exceed dpx,8q ifMzK is not relatively compact.

1. Theorem – Suppose that there is a smooth 1-formβ onM such that

Lf “ ¹₂divpa∇fq `apβ,∇fq (1.6) where the divergence is understood with respect toν. Then, for allx, yPM and any closed set K in M with MzK relatively compact, we have

lim sup

tÑ0

tlogppt, x, Kq ď ´dpx, Kq²{2 (1.7) and

lim sup

tÑ0

tlogppt, x, K, yq ď ´pdpx, Kq `dpy, Kqq²{2. (1.8) Moreover, if there is a finite positive constant λsuch that

sup

xPM

apβ, βqpxq ďλ² (1.9)

then, for any closed set K in M, lim sup

tÑ0

tlogppt, x, K, yq ď ´dpx, K, yq²{2. (1.10) Moreover, all the above upper limits hold uniformly in x and y in compact subsets of MzBK.

The sector condition (1.9) limits the strength of the asymmetry of Lwith respect to ν and plays the role of a uniform bound on the drift. We will deduce from Theorem 1 the small-time logarithmic asymptotics of the heat kernel.

2. Theorem – Suppose thatL has the form (1.6). Define S:“

!

px, yq PMˆM :dpx, yq ďdpx,8q `dpy,8q )

. Then, as tÑ0, uniformly on compacts in S,

tlogppt, x, yq Ñ ´dpx, yq²{2. (1.11) Moreover, if L satisfies (1.9), then (1.11) holds uniformly on compacts inM ˆM.

We will deduce from Theorem 1 also the following concentration estimate for the bridge measuresµ^x,y_t onΩ^x,y. A path γ PΩ^x,y is minimal ifIpγq ă 8and

Ipγq ďIpωqfor all ω PΩ^x,y.

We will say that γ is strongly minimal if, in addition, there exist δ ą 0 and a relatively compact open set U inM such that

Ipγq `δ ďIpωqfor all ωPΩ^x,y which leaveU .³ (1.12)

3WhenM is complete for the sub-Riemannian distance, all metric balls are relatively compact, so every minimal path is strongly minimal. Also, if there is a unique minimal path γ P Ω^x,y, which is strongly minimal, then, by a weak compactness argument, for all relatively compact domainsU containingγ, there is aδą0such that (1.12) holds.

3. Theorem – Suppose that L has the form (1.6). Let x, yPM and suppose that there is a unique minimal path γ PΩ^x,y. Suppose either that

dpx, yq ădpx,8q `dpy,8q,

or that the drift inLsatisfies the boundedness assumption (1.9)andγ is strongly minimal.

Write δγ for the unit mass at γ. Then

µ^x,y_t Ñδ_γ weakly on Ω^x,y as tÑ0.

Theorems 1, 2 and 3 are our main results; they are proved in Section 5 as a consequence of Proposition 6 and Proposition 7 that give Gaussian-type upper bounds, for heat kernels and hitting probabilities respectively. The proof of the latter builds on the extension to the incomplete case of the dual characterization for complete sub-Riemannian metrics, proved by Jerison and Sanchez-Calle [11], given here in Section 3.

Notations. For an absolutely continuous path γ :r0,1s ÑM, the energyIpγq is given by Ipγq “inf

ˆ ₁

0

@ξt, apγtqξt

Ddt

where the infimum is taken over all measurable pathsξ :r0,1s ÑT^˚M such thatξ_tPT_γ^˚_tM for all tand, for almost all t,

γ9_t“apγ_tqξ_t.

If γ is not absolutely continuous or there is no such path ξ, then we set Ipγq “ 8. The sub-Riemannian distance is given by

dpx, yq “inf!a

Ipγq:γ PΩ^x,y )

.

It is known thatddefines a metric onM which is compatible with the topology ofM. Acknowledgements. The authors would like to thank Michel Ledoux and Laurent Saloff- Coste for helpful discussions. J.N. would like to acknowledge the hospitality of Universit´e Paul Sabatier, Toulouse, where this work was completed.

2 – Discussion and review of related works

‚The small-time logarithmic asymptotics for the heat kernel (1.2) or (1.11) were proved by Varadhan [21] in the case when M “ R^d and a is uniformly bounded and uniformly positive-definite. Azencott [1] considered the case where a is positive-definite but M is possibly incomplete for the associated metric d. He showed [1, Chapter 8, Proposition 4.4], that the condition

dpx, yq ămax dpx,8q, dpy,8q(

(2.1) is sufficient for a Gaussian-type upper bound which implies the small time logarithmic asymptotics (1.2). In particular, completeness is sufficient. He showed also [1, Chapter 8, Proposition 4.10], that such an upper bound holds for p_Upt, x, yq, without further condi- tions, whenever U is a relatively compact open set in M. Azencott also gave an example [1, Chapter 8, Section 2], which shows that (1.2) can fail without a suitable global condi- tion on the operator L. Hsu [8] showed that Azencott’s condition (2.1) for (1.2) could be relaxed to

dpx, yq ďdpx,8q `dpy,8q (2.2)

and gave an example to show that (1.2) can fail without this condition.

Azencott and Hsu’s methods in [1] and [8] work ‘outwards’ from relatively compact subdomains U in M and make essential use of the following identity, which allows to control p in terms of pU. See [1, Chapter 2, Theorem 4.2]. Let U, V be open sets in M withV compactly contained inU. Then, forxPM andyPV, we have the decomposition

ppt, x, yq “1_Upxqp_Upt, x, yq ` ˆ

r0,tqˆBV

p_Upt´s, z, yqµ_xpds, dzq (2.3) where

µ_x “

8

ÿ

n“1

µⁿ_x, µⁿ_x`

r0, ts ˆA˘

“P_x`

B_Tn PA, T_nďt˘ where we setS0 “0 and define recursively forně1

T_n:“inf

!

těS_n´1:B_tPV )

, S_n:“inf

!

těT_n:B_tRU )

. This can be combined with the estimate

µ_x`

r0, ts ˆ BV˘

ďCpU, Vqt, CpU, Vq ă 8

to obtain estimates onppt, x, yq from estimates on p_Upt, x, yq. The same identity (2.3) is also used elsewhere to deduce estimates under local hypotheses from estimates requiring global hypotheses. See for example [12] on hypoelliptic heat kernels, and [6] on Hunt processes.

‚ Varadhan’s asymptotics (1.2) were extended to the sub-Riemannian case by L´eandre [13, 14] under the hypothesis

M “R^d and X₀, X₁, . . . , X_m are bounded with bounded derivatives of all orders.

(2.4) Here,X₀ is the vector field on M which appears when we write Lin H¨ormander’s form

L“ 1 2

m

ÿ

`“1

X<sub>`</sub><sup>2</sup>`X₀.

Our Theorem 2 extends (1.2) to a general sub-Riemannian manifold for operators L of the form (1.6), subject either to Hsu’s condition (2.2), understood for the sub-Riemannian metric, or to the sector condition (1.9).

‚A powerful approach to analysis of the heat equation emerged in the work of Grigor’yan [5] and Saloff-Coste [18, 19]. They showed that a local volume-doubling inequality, com- bined with a local Poincar´e inequality, implies a local Sobolev inequality, which then allows to prove regularity properties for solutions of the heat equation by Moser’s procedure, and then heat kernel upper bounds by the Davies–Gaffney argument. This was taken up in the general context of Dirichlet forms by Sturm who proved a Gaussian upper bound [20, Theorem 2.4] under such local conditions, without completeness and for non-symmetric operators. Moreover, in this bound, the intrinsic metric appears with the correct con- stant in the exponent, which allows to deduce the correct logarithmic asymptotic upper bound (1.2). This intrinsic metric corresponds in our context to the dual formulation of the sub-Riemannian metric. Our Gaussian upper bounds can be seen as applications of Sturm’s result. For greater transparency, we will re-run part of the argument in our context, rather than embed in the general framework and check the necessary hypothe- ses. The approach thus adopted no longer relies on working outwards from well-behaved heat kernels using (2.3), but reduces the global aspect to a certain sort ofL²-estimate for solutions of the heat equation, which requires no completeness in the underlying space.

One finds that the sector condition (1.9) on the drift is enough to prevent pathologies in theL²-estimate, thus dispensing with the need for condition (2.2) on the end-points x, y.

This is a significant extension: for example, (1.9) is satisfied trivially by all symmetric operatorsLf “ ¹₂divpa∇fq, without any control on the diffusivitya or the symmetrizing measure ν near infinity.

‚ The small-time convergence of bridge measures is known in the case of Brownian motion in a complete Riemannian manifold by a result of Hsu [7] and in the case of H¨ormander’s type operators L “ ¹₂ ř_m

`“1X` `X0, on a closed manifold, with no hori- zontality assumption on the drift X₀, by a result of Bailleul [2]. It is also known under the assumption (2.4) and subject to the condition thatapxq is positive-definite by work of Inahama [9] on R^d. While the limit is the expected one, given the well-known small-time large deviations behaviour of diffusions, a statement such as Theorem 3 appears new, both for incomplete manifolds and in the unbounded sub-Riemannian case.

Remark. We have not attempted to minimize regularity assumptions for coefficients but note that their use for upper bounds is limited to certain basic tools. The analysis [16]

of metric balls, in particular the volume-doubling inequality (3.2), is done for the case where X1, . . . , Xm are smooth. Also the Poincar´e inequality (3.6)is proved in [10] in this framework. These points aside, for upper bounds, the smooth assumptions on a, ν and β are used only to imply local boundedness. While the dual characterization of the distance function is unaffected by modification of a on a Lebesgue null set, the definition as an infimum over paths is more fragile, and current proofs that these give the same quantity rely on the continuity of a. In contrast to the Riemannian case [17], for lower bounds in the sub-Riemannian case, in particular for L´eandre’s argument using Malliavin calculus, current methods demand more regularity.

3 – Dual formulation of the sub-Riemannian distance

We review some basic analytic facts for sub-Riemannian manifolds before extending Jerison and Sanchez-Calle’s dual formulation of the sub-Riemannian distance [11] to pos- sibly incomplete settings.

3.1 – Analytic ingredients

The set-up of Section 1 is assumed. Nagel, Stein & Wainger’s analysis [16] of the sub-Riemannian distance and of the volume of sub-Riemannian metric balls implies the following statements. There is a covering ofM by charts φ:U ÑR^d such that, for some constantsαpUq P p0,1sand CpUq P r1,8q, for all x, yPU,

C^´1ˇ

ˇφpxq ´φpyqˇ

ˇďdpx, yq ďCˇ

ˇφpxq ´φpyqˇ ˇ

α. (3.1)

Moreover, there is a covering of M by open sets U such that, for some constant CpUq P p1,8q, for allxPU and allr P p0,8qsuch thatBpx,2rq ĎU, we have the volume-doubling inequality

ν`

Bpx,2rq˘

ďCν`

Bpx, rq˘

. (3.2)

Moreover, in [16, Theorem 1], a uniform local equivalent forνpBpx, rqqis obtained, which implies that, for allxPM,

rÑ0lim logν`

Bpx, rq˘

logr “Npxq. (3.3)

Here,Npxq is given by

Npxq “N₁pxq `2N<sub>2</sub>pxq `3N₃pxq ` ¨ ¨ ¨ (3.4)

whereN₁pxq ` ¨ ¨ ¨ `N_kpxqis the dimension of the space spanned at x by brackets of the vector fieldsX₁, . . . , X_m of length at mostk. While the limit (3.3) is in general not locally uniform, there is also the following uniform asymptotic lower bound on the volume of small balls, for any compact set F inM,

lim sup

rÑ0

sup

xPF

logν`

Bpx, rq˘

logr ďNpFq (3.5)

where

NpFq:“sup

xPF

Npxq ă 8.

We recall also the local Poincar´e inequality proved by Jerison [10]. There is a covering of M by open setsU such that, for some constantCpUq ă 8, for allxPU and allrP p0,8q such thatBpx,2rq ĎU, for all f PC_c⁸pMq, we have

ˆ

Bpx,rq

ˇ

ˇf ´ xfy_Bpx,rqˇ ˇ

2dν ďCr² ˆ

Bpx,2rq

a`

∇f,∇f˘

dν (3.6)

where xfyB :“ _νpBq¹ ´

Bf dν “: ffl

Bf dν is the average value of f on B. As Saloff-Coste claimed [19, Theorem 7.1], the validity of Moser’s argument, given (3.2) and (3.6), extends with minor modifications to suitable non-symmetric operators. This leads to the following parabolic mean-value inequality.

4. Proposition – Let L be given as in equation (1.6) and let U be a relatively compact open set in M. Then there is a constant CpUq ă 8 with the following property. For any non- negative weak solution u of the equation pB{Btqu_t “Lu_t onp0,8q ˆU, for all x PU, all tP p0,8q and all rP p0,8q such thatBpx,2rq ĎU andr² ďt{2, we have

utpxq²ďC

t

t´r² Bpx,rq

u²_sdν ds. (3.7)

Moreover, the same estimate holds if L is replaced by its adjoint Lˆunderν.

For a detailed proof, the reader may check the applicability of the more general results [4, Theorem 1.2] or [15, Theorem 4.6].

3.2 – Dual formulation of the sub-Riemannian distance

In Riemannian geometry, the distance function has a well known dual formulation in terms of functions of unit gradient. Jerison & Sanchez-Calle [11] showed that this dual formulation extends to complete sub-Riemannian manifolds. We now show that such a dual formulation holds without completeness, and for the distances to and through a given closed set.

5. Proposition – For allx, yPM and any closed subset K of M, we have dpx, K, yq “sup

!

w^{`</sup>pyq ´w<sup>´</sup>pxq:w<sup>´</sup>, w<sup>`} PF with w^`“w^´ onK )

(3.8) and

dpx, Kq “sup

!

wpxq:wPF with w“0 onK )

, (3.9)

where F is the set of all locally Lipschitz functions w on M such that a`

∇w,∇w˘ ď 1 almost everywhere.

Proof – Denote the right hand sides of (3.8) and (3.9) byδpx, K, yq andδpx, Kq for now.

First we will show that δpx, K, yq ď dpx, K, yq. Let ω P Ω^x,y and suppose that ω is absolutely continuous with driving path ξ and that ωt P K. Let w^´, w^{`</sup> P F, with w<sup>`}“w^´on K. It will suffice to consider the case whereω|_r0,ts andω|_rt,1s are simple, and then to choose relatively compact chartsU₀ and U₁ forM containingω|_r0,ts and ω|_rt,1s respectively. Then, given εą0, since ais continuous, for i“1,2, we can find smooth functionsf_i^´, f_i^` onU_i such thatˇ

ˇf_i^˘pzq ´w^˘pzqˇ

ˇďεanda`

∇f_i^˘,∇f_i^˘˘ pzq ď 1`εfor all zPUi. Then

w^{`</sup>pyq´w<sup>´</sup>pxq “w<sup>`}pyq´w<sup>`</sup>pωtq`w^´pωtq´w^´pxq ďf₁^{`</sup>pyq´f<sub>1</sub><sup>`}pωtq`f<sub>0</sub><sup>´</sup>pωtq´f<sub>0</sub><sup>´</sup>pxq`4ε and

f₁^{`</sup>pyq ´f<sub>1</sub><sup>`}pω_tq `f₀^´pω_tq ´f₀^´pxq

“ ˆ _t

0

@∇f₀^´pωsq,ω9s

Dds` ˆ ₁

t

@∇f₁^`pωsq,ω9s

Dds

“ ˆ _t

0

@∇f₀^´pωsq, apωsqξs

Dds` ˆ ₁

t

@∇f₁^`pωsq, apωsqξs

Dds

ď ˆˆ _t

0

a`

∇f₀^´,∇f₀^´˘

pωsqds` ˆ ₁

t

a`

∇f₁^{`</sup>,∇f<sub>1</sub><sup>`}˘ pωsqds

˙1{2ˆˆ ₁

0

apξs, ξsqds

˙1{2

ďa

p1`εqIpωq.

Hence w^`pyq ´w^´pxq ďa

Ipωq. On taking the supremum over w^˘ and the infimum overω, we deduce that

δpx, K, yq ďdpx, K, yq. (3.10)

For w P F with w “ 0 on K and for y P K, we can take w^´ “ ´w and w^` “ 0 in (3.8) to see that δpx, Kq ď δpx, K, yq. Hence, on taking the infimum over y P K in (3.10), we obtain

δpx, Kq ďdpx, Kq.

Now we prove the reverse inequalities. Consider a smooth symmetric bilinear form

¯

a on T^˚M such that ¯a ě a and ¯a is everywhere positive-definite. Write I¯for the associated energy function and write d¯and δ¯for the distance functions obtained by replacingaby ¯ain the definitions ofdand δ. Set

w^`pzq “dpx, K, zq,¯ w^´pzq “dpx, zq,¯ wpxq “dpx, Kq.¯

Note that w^{`</sup> “w<sup>´</sup> and w “0 on K. Since ¯ais positive-definite, the functions w<sup>´</sup>, w<sup>`}andware locally Lipschitz, and their weak gradients∇w^˘ and∇wsatisfy, almost everywhere,

¯ a`

∇w^˘,∇w^˘˘

ď1, ¯a`

∇w,∇w˘ ď1.

Hence

dpx, K, yq “¯ w^`pyq ´w^´pxq ď¯δpx, K, yq ďδpx, K, yq, dpx, Kq “¯ wpxq ďδpx, Kq ď¯ δpx, Kq.

We will show that, for all ε ą 0 and all c P r1,8q, we can choose ¯a so that, for all x, yPM withdpx, yq ďc,

dpx, yq ďdpx, yq `¯ ε.

Then, for this choice ofa, we have also, for all closed sets¯ K with dpx, Kq, dpy, Kq ď c´1,

dpx, K, yq ďdpx, K, yq `¯ 2ε, dpx, Kq ďdpx, Kq `¯ ε.

Since εą0 and cě1are arbitrary, this completes the proof. The idea in choosing ¯a is as follows. While we have no control over the behaviour ofanear 8, neither do we have any constraint on how small we can choose ¯a´anear8. Givenεą0, this will allow us to choosea¯ so that, for any path ¯γ PΩ^x,y withI¯p¯γq ă 8, we can construct another pathγ PΩ^x,y withIpγq ďI¯p¯γq `ε.

It will be convenient to fix smooth vector fieldsY1, . . . , YponMwhich span the tangent space at every point, so that

a0pxq “

p

ÿ

i“1

Yipxq bYipxq

is a positive-definite symmetric bilinear form onT^˚M. There exists an exhaustion of M by open sets pUn:nPNq, such thatUn is compactly contained in Un`1 for alln.

SetU₀ “ H. Letpδ_n:nPNqbe a sequence of constants, such thatδ_nP p0,1sfor alln, to be determined. There exists a positive smooth functionf on M such that f ďδn

on MzU_n´2 for all n. We take ¯a“ a`f<sup>2</sup>a<sub>0</sub>. Writed<sub>0</sub> and I<sub>0</sub> for the distance and energy functions associated witha`a0. Recall that we writed¯and I¯for the distance and energy functions associated with ¯a. Thend₀ ďd¯ďd. Set ε_n“d₀`

BU_n,BU_{n`1</sub>˘ . By the sub-Riemannian distance estimate (3.1), there are constants αn P p0,1s and C<sub>n</sub>ă 8, depending only on nand on the open sets pU<sub>n</sub>:nPNq and the vector fields X<sub>1</sub>, . . . , X<sub>m</sub> and Y<sub>1</sub>, . . . , Y<sub>p</sub>, such that, for all x, yPU<sub>n`2},

dpx, yq ďC_nd₀px, yq^αn.

Fix a constant c P r1,8q. Fix x, y P M with dpx, yq ďc and suppose that ω P Ω^x,y satisfies I¯pωq ď c². There exist absolutely continuous paths h : r0,1s Ñ R^m and k:r0,1s ÑR^p such that, for almost allt,

ω9t“

m

ÿ

`“1

X<sub>`</sub>pωtqh9<sup>`</sup>_t`

p

ÿ

i“1

fpωtqYipωtqk9ⁱ_t

and ˆ ₁

0

|h9_t|²dt` ˆ ₁

0

|k9_t|²dt“Ipωq.¯

By reparametrizingω if necessary, we may assume that|h9_t|²` |k9<sub>t</sub>|<sup>2</sup> “Ipωq¯ for almost all t. Consider for now the case where ω<sub>t</sub>PU<sub>n`1</sub>zU_n´1 for all tfor some nand define a new pathγ by

γ9t“

m

ÿ

`“1

X`pγtqh9<sup>`</sup>_t, γ0 “x.

Then Ipγq ďIpωq¯ . By Gronwall’s lemma, there is a constant A_nP r1,8q, depending only on n and on the open sets pUn : n P Nq and the vector fields X1, . . . , Xm and Y₁, . . . , Y_p, such that

d0pγ1, yq ďDAnδn

provided that

cA_nδ_nďε_n´2^ε_n`1. (3.11)

We will ensure that (3.11) holds, and hence that γ1 PUn`2. Then dpx, yq ďdpx, γ1q `dpγ1, yq ď

b

I¯pωq `Cnd0pγ1, yq^αn ď b

I¯pωq `CncAnδ_n^αn. We return to the general case. Then there is an integerkě1 and there is a sequence of timest₀ďt₁ ď. . .ďt_k and there is a sequence of positive integersn₁, . . . , n_k such

thatt₀ “0,t_k“1, and |n_{j`1</sub>´n<sub>j</sub>| “1and ω<sub>tj</sub> P BU<sub>nj`1} forj“1, . . . , k´1, and ω_tPU¯_nj`1zU_nj´1

for allt P rtj´1, tjs and all j “1, . . . , k, and, ifk ě2,ωt P BUn1 for some tP rt0, t1s.

Set

S_n“ tt_j :j P t1, . . . , k´1u andn_j`1 “nu, χ_n“ |S_n|.

Sinceω must hit eitherBU_n`1 orBU_n´1 immediately prior to any time inS_n, we have pε_n´1^ε_nqχ_nďc.

We have shown that

dpω_tj´1, ω_tjq ď pt_j´t_j´1q

bIpωq `¯ C_njcA_njδ_n^α_j^nj so

dpx, yq ď

k

ÿ

j“1

dpωtj´1, ωtjq ď b

I¯pωq `Cn1cAn1δn<sup>α</sup>1<sup>n1</sup> `

8

ÿ

n“1

CncAnχnδ_n^αn.

Now we can choose the sequencepδn:nPNq so that (3.11) holds and 2

8

ÿ

n“1

Cnc²Anδ_n^αn εn´1^εn ďε.

Then, on optimizing overω, we see thatdpx, yq ďdpx, yq `¯ εwhenever dpx, yq ďc, as

required.

4 – Gaussian-type upper bounds

Recall the definitionppt, x, K, yqof the heat kernel viaKgiven in (1.4) and the definition of the hitting probabilityppt, x, Kqgiven in (1.5).

6. Proposition – Let L be given as in equation (1.6)and suppose that the drift in Lsatisfies the sector/boundedness condition (1.9). Then there is a continuous functionC:MˆM Ñ p0,8qsuch that, for all x, yPM and alltP p0,8q, for

r“min

# t dpx, yq,

ct

4,dpx,8q

4 ,dpy,8q 4

+

we have

ppt, x, yq ď Cpx, yq aνpBpx, rqqa

νpBpy, rqqexp

"

´dpx, yq² 2t `λ²t

2

*

. (4.1)

Moreover, for any closed set K “ MzU in M, there is a continuous function Cp¨,¨, Kq: U ˆU Ñ p0,8q such that, for allx, yPU and all tP p0,8q, for

r“min

# t dpx, K, yq,

ct

4,rpx, Kq

4 ,rpy, Kq 4

+

, rpx, Kq “mintdpx,8q, dpx, Kqu we have

ppt, x, K, yq ď Cpx, y, Kq aνpBpx, rqqa

νpBpy, rqqexp

"

´dpx, K, yq² 2t `λ²t

2

*

. (4.2)

The statements above remain true with the constant 4 replaced by 2, by the local volume-doubling inequality. The value4 will be convenient for the proof.

Proof – We omit the proof of (4.1), which is a simpler version of the proof of (4.2). For (4.2), we will show that the argument used in [17, Theorem 1.2], for the case where ais positive-definite andβ “0, generalizes to the present context⁴. Consider the set MĂ:“M^´YM^{`</sup>, whereM<sup>˘</sup>“KYU<sup>˘</sup>and U<sup>´</sup>, U<sup>`} are disjoint copies ofU “MzK.

Write π for the obvious projection MĂÑ M. For functions f defined on M, we will write f also for the function f ˝π on MĂ. Thus we will sometimes consider a as a symmetric bilinear form on T^˚U^˘ and β as a 1-form on U^˘. Define a measure rν on MĂby

νpAqr :“νpAXKq `<sup>1</sup><sub>2</sub>ν`

πpAXU^´q˘

`<sup>1</sup><sub>2</sub>ν`

πpAXU^`q˘ . Note thatν “rν˝π^´1. Now define

ppt, x, yq “r

$

’&

’%

ppt, x, yq `p_Upt, x, yq, ifx, yPU^˘,

ppt, x, yq ´p_Upt, x, yq, ifxPU^˘ and yPU^¯, ppt, x, yq, ifxPK oryPK.

Given bounded measurable functions f^´, f^{`</sup> on M with f<sup>´</sup> “f<sup>`} on K, write f for the function on MĂ such that f “ f^˘˝ π on M^˘, and set f¯ “ pf^´`f<sup>`</sup>q{2 and f^U “ pf^{`</sup>´f<sup>´</sup>q{2. Letφ<sup>´</sup> and φ<sup>`} be smooth functions on M, of compact support, withφ^´“φ^` onK and defineφon MĂandφ¯and φ^U onM similarly. FortP p0,8q, define functions ut on M,Ă u¯ton M and u^U_t onU by

utpxq “ ˆ

MĂ

ppt, x, yqfpyqr νpdyqr and

¯ u_tpxq “

ˆ

M

ppt, x, yqf¯pyqνpdyq, u^U_tpxq “ ˆ

M

p_Upt, x, yqf^Upyqνpdyq.

Then u¯_t and u^U_t solve the heat equation with Dirichlet boundary conditions in M and U respectively. It is straightforward to check that ut “ u^˘_t ˝π on M^˘, where u^˘_t “u¯t˘u^U_t and we extendu^U_t by 0 onK. Hence

ˆ

MĂ

φu_tdνr“ ˆ

M

φ¯¯u_tdν` ˆ

U

φ^Uu^U_t dν and so

d dt

ˆ

MĂ

φutdrν“ d dt

ˆ

M

φ¯¯utdν` d dt

ˆ

U

φ^Uu^U_t dν

“ ´1 2

ˆ

M

ap∇φ,¯ ∇¯u_tqdν` ˆ

M

apφβ,¯ ∇¯u_tqdν

´1 2

ˆ

U

a`

∇φ^U,∇u^U_t˘ dν`

ˆ

U

a`

φ^Uβ,∇u^U_t ˘ dν

“ ´1 2

ˆ

MĂ

ap∇φ,∇u_tqdrν` ˆ

MĂ

apφβ,∇u_tqdν.r (4.3) Letpw^´, w^{`</sup>qbe a pair of bounded locally Lipschitz functions onM such thatw<sup>´</sup>“w<sup>`} onKanda`

∇w^˘,∇w^˘˘

ď1almost everywhere. Define a functionwonMĂby setting

4The idea is to combine a standard argument for heat kernel upper bounds with a reflection trick. In terms of Markov processes, we give a random sign to each excursion of the diffusion process intoU, viewing it as taking values inU^´orU^`. Then a generalization of the classical reflection principle for Brownian motion allows to express the density for paths fromxto yviaK in terms of this enhanced process. In fact the heat kernelprfor this process may be written in terms ofpandpU, and we find it technically simpler to defineprin those terms, rather than set up the enhanced process.

w“w^˘˝πonM^˘. FixθP p0,8qand setψ“θw. We deduce from (4.3) by a standard argument that

d dt

ˆ

MĂ

pe^´ψu_tq²drν“ ´ ˆ

MĂ

a`

∇pe^´2ψu_tq,∇u_t˘ drν`2

ˆ

MĂ

a`

βe^´2ψu_t,∇u_tqdrν

“ ´ ˆ

MĂ

a`

∇u_t,∇u_t˘

e^´2ψdrν`2 ˆ

MĂ

a`

pβ`∇ψqu_t,∇u_t˘

e^´2ψdrν

ď ˆ

MĂ

a`

β`∇ψ, β`∇ψ˘

pe^´ψu_tq²drν ďρ ˆ

MĂ

pe^´ψu_tq²dνr where

ρ:“

›

›

›a`

β`∇ψ, β`∇ψ˘›

›

›8ď pλ`θq². Then, by Gronwall’s inequality,

ˆ

MĂ

pe^´ψu_tq²dνrďe^ρt ˆ

MĂ

pe^´ψfq²drν. (4.4)

There exists a locally finite cover U of U by sets of the form Bpx, rpx, Kq{4q, where we recall that rpx, Kq “ mintdpx,8q, dpx, Kqu. For V :“ Bpx, rpx, Kq{4q P U, set Vr :“ Bpx,7rpx, Kq{8q. Then Vr is a relatively compact open subset of U. By the triangle inequality, for all V P U and all x P V, we have Bpx, rpx, Kq{2q Ď Vr. Fix V₁, V₂ P U and write CpV_iq for the constants appearing in the parabolic mean-value inequality for L on Vr₁ and for Lˆon Vr₂. Fix xPV₁,y PV₂ and t P p0,8q, and recall that we set

r“min

# t dpx, K, yq,

ct

4,rpx, Kq

4 ,rpy, Kq 4

+ .

Writex^´andy^{`</sup>for the unique points inU<sup>´</sup>andU<sup>`}respectively such thatπpx^´q “x and πpy^`q “y. Set

B^´:“

!

zPU^´:πpzq PBpx, rq )

, B^`:“

!

zPU^`:πpzq PBpy, rq )

.

Takef^´“0and choose f^`ě0 supported onBpy, rqand such that´

Mpf^`q²dν“2.

Then ´

MĂf²drν “ 1. Note that w ď w^´pxq `r on B<sup>´</sup> and w ě w<sup>`</sup>pyq ´r on B^`. Hence we obtain from (4.4), for all sě0,

e^{´2θpw´pxq`rq} ˆ

B^´

u²_sdνrďe^ρse^{´2θpw`pyq´rq}. (4.5) Since u^´_t ě 0 and pB{Btqu^´_t “ Lu^´_t on p0,8q ˆ U, by the parabolic mean-value inequality, for all τ P p0,8qsuch that r²ďτ{2,

u_τpx^´q²ďCpV₁q

τ τ´r² B^´

u²_sdνdsr ďCpV₁qνpBpx, rqq^´1e^{´2θpw`pyq´w´pxq´2rq`ρτ}. (4.6) Set v_spzq “ pps, x, K, zq, then v_s ě 0 and pB{Bsqv_s “ Lvˆ _s on p0,8q ˆ D. By the parabolic mean-value inequality again,

ppt, x, K, yq²ďCpV2q

t

t´r² Bpy,rq

pps, x, K, zq²νpdzqds

ďCpV2q

t t´r² B^`

pps, xr ^´, zq²νpdzqds.r

Recall that r² ďt{4. For each sP rt´r², ts, we can take f^` “c_‹pps, x, K, .q1_Bpy,rq, wherec‹ is chosen so that´

MĂf²drν “1. For this choice of f^`, we have u_spx^´q²“

ˆ

B^`

pps, xr ^´, zq²rνpdzq.

Hence

ppt, x, K, yq² ď CpV₂q νpBpy, rqq

t t´r²

u_spx^´q²ds

ď CpV1qCpV2q

νpBpx, rqqνpBpy, rqqe^{´2θpw`pyq´w´pxq´2rq`ρt}.

Here, we applied (4.6) with τ “ s, noting that s ě 3t{4, so r² ď t{4 ď s{2. We optimize overpw^´, w^`q and take θ“dpx, K, yq{t to obtain

ppt, x, K, yq ď CpV1, V2, x, yq aνpBpx, rqqa

νpBpy, rqq exp

"

´dpx, K, yq² 2t `λ²t

2

*

where

CpV1, V2, x, yq:“e2`λdpx,K,yq{2a

CpV1qCpV2q.

Finally, since U is locally finite, there is a continuous function Cp¨,¨, Kq : U ˆU Ñ p0,8q such that CpV1, V2, x, yq ď Cpx, y, Kq for all V1, V2 P U and all x P V1 and

yPV₂.

7. Proposition – Let Lbe given as in equation (1.6). LetU be a relatively compact open set in M and set K:“MzU. There is a constantCpUq ă 8with the following property. For all xPU and alltP p0,8q, and forr “t{dpx, Kq,

ppt, x, Kq ď C

aνpBpx, rqqexp

"

´dpx, Kq² 2t

*

. (4.7)

Proof – We adapt the argument of the proof of Proposition 6. Since νpUq ă 8 and ppt, x, Kq ď1, it will suffice to consider the case wheredpx, Kq² ě2t. We modify the measure ν and the1-formβ on K, if necessary, by multiplication by suitable smooth functions, so that νpKq ď1 and apβ, βqpxq ďλ² for allxPM, for someλă 8. This does not affect the value ofppt, x, Kq for x PU. Set f “ 1`1<sub>U</sub><sup>`</sup> ´1_U^´ and define, forxPMĂ,

u_tpxq “ ˆ

MĂ

ppt, x, yqfr pyqνpdyq.r

Then ppt, x, Kq “ u_tpx^´q for x P U. Fix a locally Lipschitz function w on MĂ such that w“0 on KYU^` and ap∇w,∇wq ď1 almost everywhere. Then, as we showed at (4.5), for allθP r0,8q,

ˆ

MĂ

pe^θwu_tq²dνrďe^ρt ˆ

MĂ

f²dνr“e^ρt`

2νpUq `νpKq˘ where

ρ:“sup

xPM

a`

β´θ∇w, β´θ∇w˘

pxq ď pλ`θq².

By the same argument as that leading to (4.6), there is a constant CpUq ă 8 with the following property. For all x PU and all t P p0,8q, for all r P p0,8q such that Bpx,2rq ĎU and r² ďt{2, we have

utpx^´q² ďC

t t´r² B^´

u²_sdrνds

where B^´ :“ π^´1pBpx, rqq XU^´. Since dpx, Kq² ě 2t, we can take r “ t{dpx, Kq. Note thatwěwpx^´q ´r on B. Then

ppt, x, Kq² “utpx^´q² ďC

t t´r² B

u²_sdrνdsďCν`

Bpx, rq˘_´1 exp

!

´2θpwpx^´q ´rq `ρt )

and, by optimizing overε,θand w, using Proposition 5, we obtain ppt, x, Kq ď C

aνpBpx, rqqexp

"

´dpx, Kq² 2t

* .

5 – Proofs of Theorems 1, 2, 3

Proof of Theorem 1 – The asymptotic upper bound (1.10) for the heat kernel through K, under condition (1.9), follows directly from the Gaussian upper bound (4.2) and the asymptotic lower bound (3.5) for the volume of small balls, on letting t Ñ 0.

Similarly, the asymptotic upper bound (1.7) for the hitting probability for K, when MzK is relatively compact, follows from (4.7) and (3.5). It remains to show (1.8).

For this, we adapt an argument of Hsu [8] for the Riemannian case. Consider the L-diffusion process `

B_t:tP r0, ζq˘ . Set

T “infttP r0, ζq:BtPKu.

We use the identity

ppt, x, K, yq “Ex

”

ppt´T, BT, yq1_tTătu ı

. (5.1)

Note thatPxpT ďtq “ppt, x, Kqand the estimate (4.7) applies. We estimateppt, z, yq forzPKusing (2.3). ChooseV relatively compact containing the closure ofU. Then, forzP BU,

ppt, z, yq “1UpxqpVpt, z, yq ` ˆ

r0,tqˆBU

pVpt´s, z¹, yqµxpds, dz¹q where

µx

`r0, ts ˆ BU˘

ďCpU, Vqt, CpU, Vq ă 8.

For allzP BU,

p_Vpt, z, yq ď C_Vpz, yq aνpBpz, rpt, zqqqa

νpBpy, rpt, zqqqexp

"

´dpz, yq²

2t `λ²_Vt 2

*

where

rpt, zq “min

# t dpz, yq,

ct

4,dpz,BVq

4 ,dpy,BVq 4

+

, λ²_V “sup

zPV

apβ, βqpzq ă 8.

Now

zPBUinf dpz, yq “dpy, Kq, sup

zPBU

dpz, yq “CpU, yq ă 8

and, forr ą0 sufficiently small inf

zPU¯νpBpz, rqq ěr^NpUq`1¯ .

Fortą0 sufficiently small, we haverpt, zq “t{dpz, yqfor all zP BU, and then aνpBpz, rpt, zqqqa

νpBpy, rpt, zqqq ě ˆ t

CpU, yq

˙_NpUq`1¯

.

Hence, fortą0sufficiently small, and all zP BU, pVpt, z, yq ď CU,Vpyq

t^NpUq`1¯ exp

"

´dpy, Kq²

2t `λ²_Vt 2

*

where

C_U,Vpyq “ sup

zPBU

C_Vpz, yq ˆCpU, yq^NpUq`1¯ .

This estimate, along with (4.7), allows us to short-cut some steps in Hsu’s argument.

On substituting the estimates into (5.1) and using the elementary [8, Lemma 2.1], we conclude as claimed that

lim sup

tÑ0

tlogppt, x, K, yq ď ´`

dpx, Kq `dpy, Kq˘₂ {2.

Proof of Theorem 2 – First we will show the lower bound

lim inf

tÑ0 tlogppt, x, yq ě ´dpx, yq²{2 (5.2) locally uniformly in x and y. Given εą0, there exists a simple path γ PΩ^x,y, with driving path ξ say, such that a

Ipγq ď dpx, yq `ε. We can and do parametrize γ so thatapξt, ξtq “Ipγq for almost alltP r0,1s. Fixδ ą0and consider the open set

U :“

!

zPM :dpz, γ_tq ăδ for sometP r0,1s )

.

We can and do choose δ so thatU is compactly contained in the domain of a chart.

Choose ně1 such that

dpx, yq `ε

n ďδ

and fix η P p0, δ{4q. For k“0,1, . . . , n, sett_k “k{n and x_k “γ_tk and suppose that y_kPBpx_k, ηq. Then, fork“1, . . . , n,

dpy_k´1, y_kq ădpx_k´1, x_kq `2η“a

Ipγq{n`2ηď`

dpx, yq `ε˘

{n`2η, d`

y_k´1, MzU˘

`dpy<sub>k</sub>, MzUq ě2pδ´ηq ą`

dpx, yq `ε˘

{n`2η.

We can identify the chart with a subset ofR^d and choose extensionsXr₀,Xr₁, . . . ,Xr_m toR^d of the restrictions ofX0, X1, . . . , Xm toU such that the extended vector fields are all bounded with bounded derivatives of all orders, such that Xr1, . . . ,Xrm is a sub-Riemannian structure onR^d, and such thatXr₀“χX₀ for some smooth function χ vanishing outside the chart. Then, by L´eandre’s lower bound [14, Theorem II.3] in R^d, fork“1, . . . , n, uniformly in y_k´1 and y_k,

lim inf

tÑ0 tlogppt, yr _k´1, y_kq ě ´dpy_k´1, y_kq²{2.

On the other hand, by Theorem 1, for k“1, . . . , n, uniformly in y_k´1 and y_k, lim sup

tÑ0

tlogppt, yr _k´1,R^dzU, y_kq ď ´dpy_k´1,R^dzU, y_kq²{2

ď ´pdpy_k´1, MzUq `dpy_k, MzUqq²{2 Hence, by our choice ofnand η, uniformly in yk´1 and yk,

lim inf

tÑ0 tlogp_Upt, y_k´1, y_kq “lim inf

tÑ0 tlogpr_Upt, y_k´1, y_kq ě ´`

dpx_k´1, x_kq `2η˘₂ {2.

Now, by a standard chaining procedure, we obtain, uniformly iny₀ and y_n, lim inf

tÑ0 tlogpUpt, y0, ynq ě ´n 2

n

ÿ

k“1

pdpx_k´1, x_kq `2ηq<sup>2</sup> ě ´pdpx, yq `ε`2ηnq²{2.

This implies (5.2), since p_Upt, x, yq ďppt, x, yqand εand η may be chosen arbitrarily small.

It remains to show the upper bound lim sup

tÑ0

tlogppt, x, yq ď ´dpx, yq²{2 (5.3) locally uniformly in x and y. In the case where L satisfies (1.9), this follows from Theorem 1 by taking K “M. On the other hand, given εą0 and a compact set F inS, there is a relatively compact open set U in M such that, forK “MzU and all px, yq PF,

dpx, yq ´εďdpx, Kq `dpy, Kq.

Now the restriction ofL toU satisfies (1.9), so lim sup

tÑ0

tlogp_Upt, x, yq ď ´d_Upx, yq²{2ď ´dpx, yq²{2 (5.4) uniformly in px, yq PF, while, by Theorem 1,

lim sup

tÑ0

tlogppt, x, K, yq ď ´pdpx, Kq `dpy, Kqq<sup>2</sup>{2 (5.5) also uniformly in px, yq P F. Since ppt, x, yq “ p<sub>U</sub>pt, x, yq ` ppt, x, K, yq and ε is

arbitrary, (5.3) follows from (5.4) and (5.5).

In the following proof, we introduce an auxiliary real Brownian bridge, from 0 to 1 of speedε. This is known to converge weakly to a uniform drift as εÑ0. So this auxiliary process provides a new coordinate which acts as a surrogate for time, thereby allowing us to lift the small-time estimate for the heat kernel to a weak convergence result for the associated bridge.

Proof of Theorem 3 – Consider first the case where L satisfies (1.9) and γ is strongly minimal. We will show, for allδą0, for

Γ_tpδq:“

!

ω_t:ωPΩ^x,y, Ipωq ădpx, yq²`δ )

and for

r “δ^1{4`

dpx, yq²`δ˘1{2

that we have lim sup

εÑ0

εlogµ^x,y_ε

´

ωPΩ^x,y :d`

ωt,Γtpδq˘

ěr for sometP r0,1s(¯

ď ´δ{2. (5.6) Then, since γ is the unique minimal path in Ω^x,y and γ is strongly minimal, for all ρ ą 0, there exists δ ą 0 such that, for all ω P Ω^x,y, we have Ipωq ě dpx, yq² `δ whenever dpω_t, γ_tq ěρ for sometP r0,1s. Hencedpz, γ_tq ăρ for all zPΓ_tpδq and all tP r0,1s. Then it follows from (5.6) that, as εÑ0,

µ^x,y_ε

´

ω PΩ^x,y:dpωt, γtq ăr`ρ for alltP r0,1s(¯ Ñ1 showing that µ^x,yε Ñδγ weakly on Ω^x,y.

Consider the operatorLrand measure rν on MĂ“MˆR given by Lr“L`1

2 ˆ B

Bτ

˙₂

, rνpdx, dτq “νpdxqdτ

whereτ denotes the coordinate inR. Then

Lfr “ ¹₂divprĂ a∇fq `rapβ,r ∇fq wheredivĂ is the divergence associated to rν and where

rapx, τq “apxq ` B Bτ b B

Bτ, B

βpx, τr q, v˘ B Bτ

F

“ xβpxq, vy, vPT_xM.

Moreover,rahas a sub-Riemannian structure and rapβ,r βqpx, τr q “apβ, βqpxq ďλ²

Write Ω^0,1pRq for the set of continuous paths σ : r0,1s Ñ R such that σ₀ “ 0 and σ1 “1. For σPΩ^0,1pRq, define

Ipσq “

$

&

% ˆ ₁

0

|σ9_t|²dt, ifσ is absolutely continuous,

8, otherwise.

Setxr“ px,0q and yr“ py,1q, and define Kr :“MĂzU ,r Ur :“

!

pγ_t, σ_tq:pγ, σq PrΓpδq, tP r0,1s )

where

Γpδqr :“

!

pγ, σq PΩ^x,yˆΩ^0,1pRq:Ipγq `Ipσq ădpx, yq<sup>2</sup>`1`δ )

.

Then Kr is closed in MĂ. Writeβε^0,1 for the law on Ω^0,1pRq of a Brownian bridge from 0 to1of speed ε. Then, with obvious notation,

ppt,r x,r yq “r ppt, x, yq 1

?2πe^´1{p2tq, µr^rx,r_ε^ypdω, dτq “µ^x,y_ε pdωqβ^0,1_ε pdτq.

By Theorem 1, we have lim sup

tÑ0

tlogppt,r x,r K,r yq ď ´r dprx,r K,r yqr²{2“ ´pdpx, yq²`1`δq{2 so

lim sup

εÑ0

εlogµr^x,r_ε^ry`

pω, τq:pωt, τtq PKr for sometP r0,1s(¯ ďlim sup

εÑ0

εlogppε,r x,r K,r yq ´r lim inf

εÑ0 εlogppε,r rx,yq ď ´δ{2r (5.7) where we have used the lower bound from Theorem 2. By standard estimates, we also have

εÑ0limεlogβ_ε^0,1

´

τ :|τ_t´t| ě?

δ{2for sometP r0,1s(¯

“ ´δ{2. (5.8) Suppose then that ω PΩ^x,y and τ PΩ^0,1pRq satisfy pω_t, τ_tq PUr and |τ_t´t| ă ?

δ{2 for all t P r0,1s. Then, for each t P r0,1s, there exist s P r0,1s and γ P Ω^x,y and σPΩ^0,1pRq such that

ωt“γs, τt“σs, Ipγq ădpx, yq²`δ, Ipσq ă1`δ.

Then |σ_s´s| ď?

δ{2so|t´s| ď?

δ and so

dpω_t,Γ_tpδqq² ďdpω_t, γ_tq²“dpγ_s, γ_tq² ď |t´s|Ipγq ďδ^1{2pdpx, yq²`δq.

The estimates (5.7) and (5.8) thus imply (5.6).