Diffusion in small time in incomplete sub-Riemannian manifolds
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.
Diffusion in small time in incomplete sub-Riemannian manifolds
October 23, 2018
1 Introduction and summary of results
Let be a connected manifold, which is equipped with a sub-Riemannian structure and a positive measure . Thus, are vector fields on which, taken along with their commutator brackets of all orders, span the tangent space at every point, and has a positive density with respect to Lebesgue measure in each coordinate chart. Consider the symmetric bilinear form on given by
Let be a second order differential operator on with coefficients, such that and has principal symbol . In each coordinate chart, takes the form
for some functions . Write for the Dirichlet heat kernel of in with respect to , and write for the associated diffusion process. For and , set
Consider the case where . While the explosion time of may be finite, we can still disintegrate the sub-probability law of restricted to the event by a unique family of probability measures , weakly continuous in , such that
Then is the law of the -diffusion bridge from to in time . It will be convenient to consider these bridge measures all on the same space . So define by and define on by
We focus mainly on two problems, each associated with a choice of the endpoints and , and with the limit . The first is to give conditions for the validity of Varadhan’s asymptotics for the heat kernel
where is the sub-Riemannian distance. The second is to give conditions for the weak limit
where is a path of minimal energy in . We wish to understand, in particular, what can be said without symmetry or ellipticity of the operator , and without compactness or even completeness of the underlying space . The heat kernel and the bridge measures have a global dependence on , 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  and the second requiring a ‘sector condition’ which ensures that the asymmetry in is not too strong. We will thus give new conditions for the validity of (2) and (3), which do not require completeness, symmetry or ellipticity, nor do they require any condition on the measure . In a companion paper , we have further investigated the small-time fluctuations of the diffusion bridge around the minimal path , which reveal a Gaussian limit process.
In this section, we state our three asymptotic results. In the next, we discuss related prior work. Later in the paper, we state three further results. The first of these, Proposition 4.1, shows that the dual characterization for complete sub-Riemannian metrics, proved by Jerison and Sanchez-Calle , extends to the incomplete case. Then Propositions 5.1 and 5.2 give Gaussian-type upper bounds, for heat kernels and hitting probabilities respectively, from which the asymptotic results are deduced.
Let be a closed set in and set . Write for the Dirichlet heat kernel of in , extended by outside . Define
We call the heat kernel through .
In the case where is relatively compact, we write for the hitting probability for , given by
Recall that the sub-Riemannian distance is given by
where denotes the energy
Suppose that there is a -form on such that
where the divergence is understood with respect to . Then, for all and any closed set in with relatively compact, we have
Moreover, if there is a constant such that
then, for any closed set in ,
Moreover, all the above upper limits hold uniformly in and in compact subsets of .
We will deduce from Theorem 1.1 also the following concentration estimate for the bridge measures on . A path is minimal if and
We will say that is strongly minimal if, in addition, there exist and a relatively compact open set in such that
The authors would like to thank Michel Ledoux and Laurent Saloff-Coste for helpful discussions. JN would like to acknowledge the hospitality of Université Paul Sabatier, Toulouse, where this work was completed.
2 Discussion and review of related work
The small-time logarithmic asymptotics for the heat kernel (2) were proved by Varadhan  in the case when and is uniformly bounded and uniformly positive-definite. Azencott  considered the case where is positive-definite but is possibly incomplete for the associated metric . He showed [1, Chapter 8, Proposition 4.4], that the condition
is sufficient for a Gaussian-type upper bound which then implies (2). In particular, completeness is sufficient. He showed also [1, Chapter 8, Proposition 4.10], that such an upper bound holds for , without further conditions, whenever is a relatively compact open set in . Azencott also gave an example [1, Chapter 8, Section 2], which shows that (2) can fail without a suitable global condition on the operator . Hsu  showed that Azencott’s condition (11) for (2) could be relaxed to
and gave an example to show that (2) can fail without this condition.
The methods in  and  work ‘outwards’ from relatively compact subdomains in and make essential use of the following identity, which allows to control in terms of . See [1, Chapter 2, Theorem 4.2]. Let be open sets in with compactly contained in . Then, for and , we have the decomposition
where we set and define recursively for
This can be combined with the estimate
to obtain estimates on from estimates on . The same identity (13) is also used elsewhere to deduce estimates under local hypotheses from estimates requiring global hypotheses. See for example  on hypoelliptic heat kernels, and  on Hunt processes.
Here, is the vector field on which appears when we write in Hörmander’s form
A powerful approach to analysis of the heat equation emerged in the work of Grigor’yan  and Saloff-Coste [18, 19]. They showed that a local volume-doubling inequality, combined with a local Poincaré 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 constant in the exponent, which allows to deduce the correct logarithmic asymptotic upper bound (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 hypotheses. The approach thus adopted no longer relies on working outwards from well-behaved heat kernels using (13), but reduces the global aspect to a certain sort of -estimate for solutions of the heat equation, which requires no completeness in the underlying space. One finds that the sector condition (7) is enough to prevent pathologies in the -estimate, thus dispensing with the need for condition (12). This is a significant extension: for example, (7) is satisfied trivially by all symmetric operators , without any control on the diffusivity or the symmetrizing measure 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 . For a compact sub-Riemannian manifold, it was shown by Bailleul . It is also known under the assumption (14) and subject to the condition that is positive-definite by work of Inahama . While the limit is the expected one, given the well-known small-time large deviations behaviour of diffusions, a statement such as Theorem 1.3 appears new, both for incomplete manifolds and in the non-compact sub-Riemannian case.
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  of metric balls, in particular the volume-doubling inequality (16), is done for the case where are . Also the Poincaré inequality (20) is proved in  in this framework. These points aside, for upper bounds, the assumptions on , and are used only to imply local boundedness. While the dual characterization of the distance function is unaffected by modification of 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 . In contrast to the Riemannian case , for lower bounds in the sub-Riemannian case, in particular for Léandre’s argument using Malliavin calculus, current methods demand more regularity.
3 Review of some analytic prerequisites
The set-up of Section 1 is assumed. Nagel, Stein & Wainger’s analysis  of the sub-Riemannian distance and of the volume of sub-Riemannian metric balls implies the following statements. There is a covering of by charts such that, for some constants and , for all ,
Moreover, there is a covering of by open sets such that, for some constant , for all and all such that , we have the volume-doubling inequality
Moreover, in [16, Theorem 1], a uniform local equivalent for is obtained, which implies that, for all ,
Here, is given by
where is the dimension of the space spanned at by brackets of the vector fields of length at most . While the limit (17) 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 in ,
We recall also the local Poincaré inequality proved by Jerison . There is a covering of by open sets such that, for some constant , for all and all such that , for all , we have
where is the average value of on .
As Saloff-Coste claimed [19, Theorem 7.1], the validity of Moser’s argument, given (16) and (20), extends with minor modifications to suitable non-symmetric operators. This leads to the following parabolic mean-value inequality.
Let be given as in equation (4) and let be a relatively compact open set in . Then there is a constant with the following property. For any non-negative weak solution of the equation on , for all , all and all such that and , we have
Moreover, the same estimate holds if is replaced by its adjoint under .
4 Dual formulation of the sub-Riemannian distance
In Riemannian geometry, the distance function has a well known dual formulation in terms of functions of sub-unit gradient. Jerison & Sanchez-Calle  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.
For all and any closed subset of , we have
where is the set of all locally Lipschitz functions on such that almost everywhere.
Denote the right hand sides of (22) and (23) by and for now. First we will show that . Let and suppose that is absolutely continuous with driving path and that . Let , with on . It will suffice to consider the case where and are simple, and then to choose relatively compact charts and for containing and respectively. Then, given , since is continuous, for , we can find functions on such that and for all . Then
Hence . On taking the supremum over and the infimum over , we deduce that
Now we prove the reverse inequalities. Consider a symmetric bilinear form on such that and is everywhere positive-definite. Write for the associated energy function and write and for the distance functions obtained by replacing by in the definitions of and . Set
Note that and on . Since is positive-definite, the functions , and are locally Lipschitz, and their weak gradients and satisfy, almost everywhere,
We will show that, for all and all , we can choose so that, for all with ,
Then, for this choice of , we have also, for all closed sets with ,
Since and are arbitrary, this completes the proof. The idea in choosing is as follows. While we have no control over the behaviour of near , neither do we have any constraint on how small we can choose near . Given , this will allow us to choose so that, for any path with , we can construct another path with .
It will be convenient to fix vector fields on which span the tangent space at every point, so that
is a positive-definite symmetric bilinear form on . There exists an exhaustion of by open sets , such that is compactly contained in for all . Set . Let be a sequence of constants, such that for all , to be determined. There exists a positive function on such that on for all . We take . Write and for the distance and energy functions associated with . Recall that we write and for the distance and energy functions associated with . Then . Set . By the sub-Riemannian distance estimate (15), there are constants and , depending only on and on the open sets and the vector fields and , such that, for all ,
Fix a constant . Fix with and suppose that satisfies . There exist absolutely continuous paths and such that, for almost all ,
By reparametrizing if necessary, we may assume that for almost all . Consider for now the case where for all for some and define a new path by
Then . By Gronwall’s lemma, there is a constant , depending only on and on the open sets and the vector fields and , such that
We will ensure that (25) holds, and hence that . Then
We return to the general case. Then there is an integer and there is a sequence of times and there is a sequence of positive integers such that , , and and for , and
for all and all , and, if , for some . Set
Since must hit either or immediately prior to any time in , we have
We have shown that
Now we can choose the sequence so that (25) holds and
Then, on optimizing over , we see that whenever , as required. ∎
5 Gaussian-type upper bounds
Recall from Section 1 the notions of distance and heat kernel through a given closed set .
The statements above remain true with the constant replaced by , by the local volume-doubling inequality. The value will be convenient for the proof.
We omit the proof of (26), which is a simpler version of the proof of (27).
For (27), we will show that the argument used in
[17, Theorem 1.2],
for the case where is positive-definite and , generalizes to the present context
Note that . Now define
Given bounded measurable functions on with on , write for the function on such that on , and set and . Let and be functions on , of compact support, with on and define on and and on similarly. For , define functions on , on and on by
Then and solve the heat equation with Dirichlet boundary conditions in and respectively. It is straightforward to check that on , where and we extend by on . Hence