Robert W Neel
email: robert.neel@lehigh.edu

Ludovic Sacchelli
email: ludovic.sacchelli@inria.fr

LOCALIZED BOUNDS ON LOG-DERIVATIVES OF THE HEAT KERNEL ON INCOMPLETE RIEMANNIAN MANIFOLDS

Keywords: 2010 Mathematics Subject Classification. Primary 58J65; Secondary 58J35 58K55 35K08 heat kernel bounds, logarithmic derivatives, small-time asymptotics, incomplete manifolds

Bounds on the logarithmic derivatives of the heat kernel on a compact Riemannian manifolds have been long known, and were recently extended, for the log-gradient and log-Hessian, to general complete Riemannian manifolds. Here, we further extend these bounds to incomplete Riemannan manifolds under the least restrictive condition on the distance to infinity available, for derivatives of all orders. Moreover, we consider not only the usual heat kernel associated to the Laplace-Beltrami operator, but we also allow the addition of a conservative vector field. We show that these bounds are sharp in general, even for compact manifolds, and we discuss the difficulties that arise when the operator incorporates non-conservative vector fields or when the Riemannian structure is weakened to a sub-Riemannian structure.

Introduction

Uniform bounds for the logarithmic derivatives of the heat kernel on a Riemannian manifold, in small time, have been studied since the 90s. In particular, improving on earlier work of Sheu [START_REF] Shuenn | Some estimates of the transition density of a nondegenerate diffusion Markov process[END_REF], Hsu [START_REF]Estimates of derivatives of the heat kernel on a compact Riemannian manifold[END_REF] and Stroock-Turetsky [START_REF] Stroock | Upper bounds on derivatives of the logarithm of the heat kernel[END_REF] independently proved that, on a compact (smooth) Riemannian manifold M , there exist two families of positive constants C N and D N such that ( both holding for (t, x, y) ∈ (0, 1] × M × M . (These two families of bounds are widely-known to be equivalent, though, for completeness, we provide the details below.) Here p t (x, y) is the usual heat kernel (corresponding to the Laplace-Beltrami operator), ∇ N represents N th-order covariant differentiation with | • | the (pointwise) norm on N -tensors induced by the Riemannian mertric, and the subscript x indicates that the covariant derivative acts on the first spatial variable (that is, the x-variable). In this case, p t is symmetric, so the result also holds for y-derivatives, but the difference in technique for studying the two spatial variables will play a prominent role in what follows, so we emphasize where our derivatives fall throughout. The work of Sheu, Hsu, and Stroock-Turetsky naturally applies to x-derivatives because it considers perturbations of the entire (random) path induced by perturbing the starting point of the diffusion. However, a perturbation of the entire path is in principle a global object, making localization, and thus the extension of these results to non-compact manifolds, non-trivial (at least those without bounded geometry). Indeed, see [START_REF] Hsu | Bismut's gradient formula for vector bundles[END_REF] for later work in this spirit, looking at more general maps between vector bundles, but still over a compact Riemannian manifold. Recently, Chen-Li-Wu [START_REF] Chen | Logarithmic heat kernels: estimates without curvature restrictions[END_REF] extended [START_REF] Bailleul | Diffusion in small time in incomplete sub-Riemannian manifolds[END_REF] for N = 1, 2 to general complete Riemannian manifolds (that is, complete manifolds with no additional assumptions on the geometry).

Interest in such estimates comes from a few sources. For example, control of the logarithmic gradient (with respect to x) of p t (x, y) is central to the question of whether the Brownian bridge process is a semi-martingale up to its terminal time (which is essentially a question of the integrability of the drift term, given in terms of the logarithmic derivative). While it has long been known that the Brownian bridge is a semi-martingale on a compact Riemannian manifold (see [START_REF]Stochastic analysis on manifolds[END_REF]), extending this result to general complete manifolds was only accomplished a few years ago by Güneysu [START_REF] Güneysu | On the semimartingale property of Brownian bridges on complete manifolds[END_REF], which required sufficient localization of the bound on the logarithmic gradient of the heal kernel. More generally, control of the logarithmic gradient and logarithmic Hessian of the heat kernel, on compact manifolds (or complete manifolds with bounded geometry), has been used in various aspects of stochastic analysis on loop space; we refer to the introduction of [START_REF] Chen | Logarithmic heat kernels: estimates without curvature restrictions[END_REF] for an overview of the literature.

The primary goal of the present paper is to further extend (1) to possibly-incomplete Riemannian manifolds under the weakest condition on the distance to infinity available, for all values of N . (This obviously includes all complete manifolds, with no additional conditions.) Moreover, we consider not only the usual heat kernel associated to the Laplace-Beltrami operator, but we allow the addition of a (smooth) conservative vector field. That is, if we let ∆ LB be the Laplace-Beltrami operator, we consider the diffusion operator ∆ = ∆ LB + Z where Z is the gradient vector field of some smooth function on M . The point is that Z being conservative is exactly the condition that there exists a (smooth) symmetrizing reference measure, namely a volume measure ν on M such that the heat kernel is symmetric when written with respect to ν. (See Section 2 for more details on the heat kernel with respect to an arbitrary volume.) More precisely, our main result is the following. Theorem 1. Let M be a (possible incomplete) Riemannian manifold, let ∆ = ∆ LB + Z, where ∆ LB is the Lapace-Beltrami operator and Z is a smooth vector field, and let p t (x, y) be the heat kernel for ∆ (with respect to the Riemannian volume measure). Suppose that Z is conservative (of course, this includes the case Z ≡ 0), and let K ⊂ M × M be compact such that there exists ε > 0 so that, for all (x, y) ∈ K, {z : d(x, z) + d(z, y) < d(x, y) + ε} has compact closure. Then there exist sequences of constants C N and D N such that, for any N ≥ 1,

∇ N x log p t (x, y) ≤ C N d(x, y) t + 1 √ t N and ∇ N x p t (x, y) ≤ D N d(x, y) t + 1 √ t N p t (x, y)
for all t ∈ (0, 1] and (x, y) ∈ K.

The underlying logic of our proof is that y-derivatives are fundamentally easier to localize, because the Markov property applies forward in time, meaning perturbing the endpoint of a (random) path only requires perturbing the path in a neighborhood of the endpoint, as opposed to perturbing the entire path. The existence of a symmetrizing measure then allows us to transfer bounds on y-derivatives to x-derivatives, which, as indicated above, are generally of more interest.

We complement this bound with a natural improvement away from the cut locus. Let Cut(M ) ⊂ M × M be the set of (x, y) such that y is in the cut locus of x, or equivalently, x is in the cut locus of y (the relationship is symmetric).

Theorem 2. Let everything be as in Theorem 1, with the additional condition that K ∩ Cut(M ) = ∅. Then, for any N ≥ 1,

-4t∇ N x log p t (x, y) -∇ N x d 2 (x, y) = O(t) as t 0,
uniformly over (x, y) ∈ K.

A version of this result (with the O(t) replaced by o(1)) was proven on a compact Riemannian manifold in the case when Z ≡ 0 in [START_REF] Malliavin | Short time behavior of the heat kernel and its logarithmic derivatives[END_REF] (see also [START_REF] Gong | The log-Sobolev inequality on loop space over a compact Riemannian manifold[END_REF] for a variant when N = 2) and then extended to general complete manifolds in [START_REF] Chen | Logarithmic heat kernels: estimates without curvature restrictions[END_REF]. We are able to localize it and thus extend it to incomplete M just as above.

Of course, these two results can be combined in various ways. To give one example, we have the following.

for d(x, y) > c for all (t, x, y) ∈ (0, 1] × K.

Remark 4. We have stated Theorems 1 and 2 and Corollary 3 for x-derivatives, as emphasized, but they hold for y-derivatives as well, which is a central feature of the proofs.

In Section 2, we describe the geometric context and notation in more detail, including the stochastic aspect. We also recall previous estimates of Léandre [START_REF] Léandre | Majoration en temps petit de la densité d'une diffusion dégénérée[END_REF] and Bailleul-Norris [START_REF] Bailleul | Diffusion in small time in incomplete sub-Riemannian manifolds[END_REF], proven in the generality of sub-Riemannian manifolds, which underlie our localization procedure. In Section 3, we prove the basic localization results for heat kernel derivatives. In this section, we work in greater generality than just Riemannian manifolds, since the method applies to any diffusion satisfying certain smalltime asymptotics, and thus is of independent interest. In particular, the results of this section apply on sub-Riemannian manifolds as well. In Section 4, we apply the general localization results to prove Theorems 1 and 2. Finally, in Section 5, we discuss a few complementary issues. First, we note that the the upper bound given in Theorem 1 is sharp in the power of t for all N , which follows from a recently proven result in [START_REF] Neel | Uniform, localized asymptotics for sub-riemannian heat kernels and diffusions[END_REF] giving the coefficient of 1/t N in terms of the joint cumulant of geometrically relevant random variables. We illustrate this in the most important case, that of N = 2, by treating the explicit example of the circle "by hand." Second, we discuss the general (Riemannian) non-compact case, in which no symmetry assumptions are placed on ∆ + Z. In this case, while one can still localize y-derivatives, it is unclear whether or not the corresponding result for x-derivatives holds. This is further related to allowing ∆ to also include a (smooth) potential. Finally, we observe that in the sub-Riemannian situation, even for a symmetric operator on a compact manifold, no estimate of the type (1) is known, because uniform control of the heat kernel, at the level of precision of (1), near the diagonal is currently unavailable. This is essentially because the diagonal, on a true sub-Riemannian manifold (that is, one which is not Riemannian) is always non-strictly abnormal, and behavior of the heat kernel at abnormals is poorly understood.

As alluded to above, many of the localization ideas we employ also apply to sub-Riemannian manifolds, and this direction is pursued in [START_REF] Neel | Uniform, localized asymptotics for sub-riemannian heat kernels and diffusions[END_REF]. Moreover, we make use of existing estimates on sub-Riemannian manifolds, though often only because they accommodate the case when Z = 0 without additional justification. For example, (1) on a compact M for non-zero Z is established in [START_REF] Neel | Uniform, localized asymptotics for sub-riemannian heat kernels and diffusions[END_REF]. Nonetheless, the eventual results in the Riemannian case are significantly stronger, admit shorter proofs, and draw from different literature, making it best to treat them separately from the sub-Riemannian case. For those wishing to avoid any reference to sub-Riemannian results, we point out that the necessary estimates to prove a weaker version of Theorem 1, following the approach in this paper, were essentially already present in the 90s literature on Riemannian manifolds, as discussed in Remark 10.

Riemannian background

For a (possibly incomplete) Riemannian manifold M , we let ∆ LB be the Laplace-Beltrami operator, and we consider the diffusion operator ∆ = ∆ LB + Z where Z is a smooth vector field. We let X t be the diffusion generated by ∆; in particular, X t is a Brownian motion (run at √ 2 times the normal speed) with drift. Of course, it is possible that X t explodes in finite time, and we understand that the process is killed upon explosion.

If we let µ be the Riemannian volume measure, then X t has a smooth transition function p t (x, y) ∈ C ∞ ((0, ∞) × M × M ) with respect to µ, that is, for any Borel set A ⊂ M P (X t ∈ A|X 0 = x) = y∈A p t (x, y) dµ(y).

(Note that the integral over all of M is 1 if and only if the process does not explode, otherwise the integral will be strictly less than 1.) This transition function is also the associated (minimal) heat kernel for ∆ (or fundamental solution to the heat equation), meaning that ∂ t p t (x, y) = ∆ x p t (x, y) and lim t 0

p t (x, y) = δ x (y)
where ∆ x denotes ∆ acting on the x-variable (that is, on the first spatial variable) and δ x is the usual point mass at x. One consequence is that, for any smooth, compactly supported function φ on M , the corresponding solution to the heat equation with initial values given by φ is φ t (x) = y∈M p t (x, y)φ(y) dµ(y). (See [START_REF] Mckean | Stochastic integrals[END_REF] and [START_REF] Dodziuk | Maximum principle for parabolic inequalities and the heat flow on open manifolds[END_REF] for classical treatments of the existence and basic properties of the heat kernel on incomplete Riemannian manifolds, and [START_REF] Grigor | Heat kernels on weighted manifolds and applications, The ubiquitous heat kernel[END_REF] for a modern survey in the context of weighted Riemannian manifolds.)

Note that the right-hand side of ( 1) is bounded from below by a positive constant for (t, x, y) in any compact subset of (0, ∞) × M × M , so, for any t 0 ∈ (0, 1) the estimate holds on any compact subset of M × M for all t ∈ [t 0 , 1] simply by smoothness of the heat kernel. Thus establishing (1) for t ∈ (0, 1] is really a matter of the small-time asymptotics of the heat kernel. Said differently, once we have shown (1) holds uniformly for t ∈ (0, t 0 ), we can extend it to t ∈ (0, 1] at the cost of possibly adjusting the constants C N . Our basic approach is to combine a localization result for the heat kernel itself with a global result on y-derivatives of the heat kernel in order to localize these derivatives. To provide context for the argument, we first introduce the necessary notation and describe the known estimates in the Riemannian case.

We will frequently restrict or extend the domain of the heat kernel, and for any open set U , we let p U t (x, y) be the heat kernel on U (which should be understood with Dirichlet boundary conditionsthat is, it is the transition density for the process killed upon explosion, or equivalently, upon the first exit from U ). Note that since we are not assuming that our Riemannian manifolds are complete, there is no real distinction between a Riemannian manifold and an open subset of a Riemannian manifold. For a closed set A, we let p t (x, A, y) = p t (x, y) -p A c t (x, y), where p A c t (x, y) is extended to be zero for either x or y inside of A (that is, outside of A c ). In particular, p t (x, A, y) gives the contribution to p t (x, y) from paths that pass through A. Further, we define d(x, A, y) = inf{d(x, z) + d(z, y) : z ∈ A}, which gives the distance from x to y via paths that pass through A. We will also consider the heat kernel with respect to different reference measures, and we will write p ν t (x, y) for the heat kernel with respect to a smooth volume measure ν (meaning a measure with a smooth, positive density in any system of local cooridnates) when there is a chance of confusion. This means that, for any Borel set

A ⊂ M P (X t ∈ A|X 0 = x) = y∈A p ν t (x, y) dν(y),
and thus dν = dν dµ dµ for a smooth, positive Radon-Nikodym derivative dν dµ . Assume that M a possibly incomplete Riemannian manifold and ∆ is symmetric, with respect to some smooth reference volume ν. Then Theorem 1.1 of Bailleul-Norris [START_REF] Bailleul | Diffusion in small time in incomplete sub-Riemannian manifolds[END_REF] gives that (here the heat kernel is understood with respect to ν)

(2) lim sup t 0 4t log p t (x, A, y) ≤ -d 2 (x, A, y)
uniformly for x and y in any compact subset of M \∂A. Further, under the same assumptions, Theorem 1.2 of [START_REF] Bailleul | Diffusion in small time in incomplete sub-Riemannian manifolds[END_REF] says that

(3) lim t 0 4t log p t (x, y) = -d 2 (x, y)
uniformly for (x, y) in any compact subset of M × M . Next, consider a Riemannian metric on R n given by a global orthonormal frame Z 1 , . . . , Z n , where the Z i are smooth, bounded vector fields with bounded derivatives of all orders. Further, suppose that if we write ∆ as n i=1 Z 2 i + Z 0 , then Z 0 is also a smooth, bounded vector field with bounded derivatives of all orders. (We don't pursue the question of when a metric admits such a representation, since eventually we localize the following estimate to a small enough neighborhood of a point.) In this case, Léandre [START_REF] Léandre | Majoration en temps petit de la densité d'une diffusion dégénérée[END_REF] proves that, for any multi-index α, (4) lim sup

t 0 4t log ∂ α y p t (x, y) ≤ -d 2 (x, y),
uniformly for (x, y) in any compact subset of R n × R n . Here d(x, y) is the Riemannian distance, the partial derivatives are understood with respect to the standard Cartesian coordinates, and the result holds for any smooth reference volume on R n (which is clear via the same logic as in Lemma 6 below).

Note that this result doesn't require any symmetry of ∆.

We continue with a simple consequence of stochastic calculus for exit times of balls on M .

Lemma 5. Let A be a compact subset of a (possibly incomplete) Riemannian manifold M , let a be a positive constant, and let σ a be the first exit time of the diffusion X t (determined by having infinitesimal generator ∆) from the open ball of radius a around X 0 = x (that is, the first time the diffusion moves a distance a from its starting point, which never happens if the diffusion explodes first). Then there exists T > 0 such that

P (σ a < T |X 0 = x) < 1 2 for any x ∈ A.
Proof. Since σ a is non-decreasing in a, without loss of generality we can assume that a is small enough so that an a-neighborhood of A, briefly A a , is relatively compact, and that a is less than the injectivity radius at all x ∈ A. This follows from the fact that the injectivity radius is continuous and everywhere positive, and thus is bounded from below on a compact. (The continuity of the injectivity radius appears widely known; for example, it is asserted without proof in [START_REF] Chavel | Riemannian geometry[END_REF]. For a recent proof, one can see Section 10.8 of [START_REF] Boumal | An introduction to optimization on smooth manifolds[END_REF].) Thus the squared distance function is smooth on A × A a .

In particular, for any x ∈ A, let a be fixed as above, and let r be the distance from x, and r t = r (X t ) be the usual radial process, where X 0 = x so that r 0 = 0. Then Itô's lemma implies there exists some (standard one-dimensional) Brownian motion W t such that r 2 t satisfies the SDE (written in integral form)

r 2 t∧σa = t∧σa 0 2r s dW s + t∧σa 0 1 2 ∆ r 2
s ds. Now by smoothness of the squared distance, there exists some α > 0 such that the last integral on the right-hand side is bounded from above by α (t ∧ σ a ), and thus also by αt, uniformly in x. (Indeed, it is well known that α can be given in terms of a lower bound on Ricci curvature and an upper bound on the length of Z over A a .) Moreover, the first integral on the right-hand side is a martingale started from 0, and its quadratic variation is bounded from above by 4a 2 t, uniformly in x. Thus, by Chebyshev's inequality, it converges to 0 in probability as t 0, uniformly in x. On the other hand, we have a 2 P x (σ a < t) ≤ r 2 t∧σa . Putting this together, we have

P x (σ a < t) ≤ 1 a 2 t∧σa 0 2r s dW s + αt ,
where the right-hand side goes to 0 in probability with t (for any fixed a > 0), uniformly over x ∈ A. The lemma follows.

Localizing derivatives of the heat kernel

For this section, we work in a more general situation than the rest of the paper. In particular, let M be a smooth manifold (of dimension n) with a smooth volume ν. We suppose that we have a diffusion X t on M (that may explode in finite time with positive probability) with a smooth transition density p t (x, y) with respect to ν; that is, started from any x ∈ M , X t almost surely has continuous paths, satisfies the strong Markov property (for stopping times with respect to the filtration it generates), and for any Borel A ⊂ M ,

P (X t ∈ A|X 0 = x) = y∈A p t (x, y) dν(y).
We also assume that d(•, •) is a metric on M which is compatible with the smooth manifold topology (so it is also continuous). Finally, we assume that the diffusion satisfies the conclusion of Lemma 5 (where balls are understood with respect to the metric d); that is, we assume that the probability of the diffusion from x leaving an open ball of radius a > 0 around x before time T can be made less than 1/2 by choosing T small enough, uniformly over all x in a compact. (Recall that the hitting time of a closed set is a stopping time for the filtration generated by X t .) The point is that the basic localization of y-derivatives can be established for such a p t (x, y) using only the type of small-time estimates described in the previous section, and no other aspects of the Riemannian structure.

We begin with some preliminary observations, essentially amounting to smooth calculus, which we collect in the following lemma. For any system of (smooth) coordinates u 1 , . . . , u n on an open set U ⊂ M with compact closure, we say that the coordinates are extendable if they can be extended to a neighborhood of the closure of U . Note that if we have any two such extendable coordinate systems on U , the transition function between them is smooth on a neighborhood of the closure of U . Lemma 6. Let M and p t (x, y) be as above. Let B ⊂ M be an open ball with compact closure that admits (smooth) coordinates u 1 , . . . , u n that extend to some neighborhood of B, and let K ⊂ M × M be compact, such that π 2 (K), the projection of K onto the second factor (respectively, π 1 (K), the projection onto the first factor) is contained in B. (Below we understand partial derivatives ∂ α on B to be with respect to the u i .)

Then there exists a sequence of positive constants C N such that, for any N ≥ 1,

(5) |∂ α log p ν t (x, y)| ≤ C N d(x, y) t + 1 √ t N
for any multi-index α of weight N , uniformly for (x, y) ∈ K, with the derivatives acting in the y-variable (respectively, the x-variable), if and only if there exists a sequence of positive constants D N such that, for any N ≥ 1,

(6) |∂ α p ν t (x, y)| ≤ D N d(x, y) t + 1 √ t N p ν t (
x, y) for any multi-index α of weight N , uniformly for (x, y) ∈ K, with the derivatives acting in the y-variable (respectively, the x-variable).

Further, if either of (5) and ( 6) hold (with the derivatives acting either on x or on y) with respect to some smooth volume ν, then it holds with respect to any smooth volume (that is, for p ν t where ν is another smooth volume) with possibly different sequences of constants C N and D N .

Proof. Assume ( 5) holds (for all N ), with the partials on either y or x. For convenience, let t (x, y) = log p ν t (x, y). Then we have ,y) where the n i correspond to the indices in α, and N = |α|. Here the partials are understood to act either all on y or all on x. Faà di Bruno's formula gives

∂ α p ν t (x, y) = ∂ n1 ∂ n2 • • • ∂ n N e t(x
∂ α p ν t (x, y) = π∈Π e t(x,y) B∈π ∂ B t (x, y) = p ν t (x, y) π∈Π B∈π ∂ B t (x, y)
where the sum is over all partitions π of {1, . . . , N }, the product is over all blocks in the partition π, and ∂ B is understood to mean ∂ ni The other direction is analogous. Assuming that ( 6) holds (for all N ), and using the same notation as above, we have

∂ α log p ν t (x, y) = π∈Π (-1) |π|-1 (|π| -1)! (p ν t (x, y)) |π| B∈π ∂ B p ν t (x, y), so that |∂ α log p ν t (x, y)| ≤ π∈Π (-1) |π|-1 (|π| -1)! (p ν t (x, y)) |π| B∈π p ν t (x, y)D |B| d(x, y) t + 1 √ t |B| = d(x, y) t + 1 √ t N π∈Π (-1) |π|-1 (|π| -1)! B∈π D |B| .
Thus (5) holds with

C N = π∈Π (-1) |π|-1 (|π| -1)! B∈π D |B| .
Again, the computation is valid with ∂ α acting on either coordinate.

Next, assume that one, and thus both, of ( 5) and ( 6) hold with derivatives on the y-variable. Then if ν is any other smooth volume, the Radon-Nikodym derivative dν dν is smooth. The product rule gives

∂ α y p ν t (x, y) = ∂ n1 y ∂ n2 y • • • ∂ n N y p ν t (x, y) dν dν (y) = A⊂1,...,N ∂ A y p ν t (x, y) ∂ A c y (y) dν dν
where the sum is over all subsets of {1, . . . , N } and for each term, the partial derivatives are divided into A and its complement (with ∂ ∅ y f (y) = f (y)). Since we work on a compact K, each factor ∂ A c y (y) dν dν is bounded on K. Then using (6) on each factor ∂ A y p ν t (x, y), since |A| ≤ |α| = N , we see that ( 6) holds for ∂ α y p ν t (x, y) with possibly different constants. By the above, (5) also holds for ∂ α y p ν t (x, y), with possibly different constants. Finally, if one, and thus both, of ( 5) and ( 6) hold with derivatives on the x-variable, then

∂ α x p ν t (x, y) = (∂ α x p ν t (x, y)) dν dν (y),
and the desired result again follows by smoothness and compactness.

Next, we show that estimates on the y-derivatives of the type in (4) can be localized to a neighborhood of the endpoint y. Lemma 7. For M and p t (x, y) as described above, suppose that, for some η > 0, B, B and B are concentric open balls of radii η/2, (3/2)η, and (7/2)η, respectively, and that B has compact closure. Suppose further that we have an extendable coordinate system on B and that, for any multi-index α, we have [START_REF] Dodziuk | Maximum principle for parabolic inequalities and the heat flow on open manifolds[END_REF] lim sup

t 0 4t log ∂ α y p t (x, y) ≤ -d 2 (x, y),
uniformly for x and y in B (where the partial derivatives are understood with respect to this coordinate system). Then, for any multi-index α,

(8) lim sup t 0 4t log ∂ α y p B t (x, y) ≤ -d 2 (x, y) uniformly over (x, y) ∈ B × B.
Proof. For (x, y) ∈ B × B we let µ x h be the (spacetime) hitting measure of ∂B for the diffusion started from x; that is, µ x h is a (sub-) probability measure on ∂B × (0, ∞) that describes the first place and time the diffusion from x hits ∂B . Then the strong Markov property implies we have the path decomposition [START_REF] Gong | The log-Sobolev inequality on loop space over a compact Riemannian manifold[END_REF] p B t (x, y) = p t (x, y) -

(z,τ )∈∂B ×(0,t) p t-τ (z, y) dµ x h .
Turning our attention to derivatives, since p t is smooth in y for all positive times, we have

∂ i y (z,τ )∈∂B ×(0,t) p t-τ (z, y) dµ x h = (z,τ )∈∂B ×(0,t) ∂ i y p t-τ (z, y) dµ x h
for any i. Iterating this, we have that, for any multi-index α,

(10) ∂ α y (z,τ )∈∂B ×(0,t) p t-τ (z, y) dµ x h = (z,τ )∈∂B ×(0,t) ∂ α y p t-τ (z, y) dµ x h ,
for all y ∈ B and any x. From ( 7) and the construction of B , for any δ > 0, the integrand in the right-hand side of (10) has absolute value bounded by e -(3η) 2 +δ once t is small enough, uniformly in (z, y) ∈ ∂B × B. Using this, and applying ∂ α y to both sides of ( 9), we find that, for any multi-index α,

∂ α y p B t (x, y) -∂ α y p t (x, y) ≤ e -(3η) 2 +δ
for all (x, y) ∈ B × B, if t is small enough. On the other hand, ∂ α y p t (x, y) term satisfies (7) with d 2 (x, y) ≤ (2η) 2 for (x, y) ∈ B × B, so its contribution to the bound on ∂ α y p B t (x, y) is more significant by an exponential factor (which is the underlying idea of localizing the estimate). At any rate, it follows that lim sup

t 0 4t log ∂ α y p B t (x, y) ≤ -d 2 (x, y)
uniformly over (x, y) ∈ B × B, and the result is proven.

Finally, we show how to combine this local result with a localization result for the heat kernel itself to localize derivatives. Lemma 8. In the situation described above, suppose that, K 0 ⊂ U 0 ⊂ K 1 ⊂ U 1 ⊂ M where K 0 and K 1 are compact and U 0 and U 1 are open with compact closure, and suppose that, for some ε > 0, the heat kernel on M satisfies the estimate [START_REF] Güneysu | On the semimartingale property of Brownian bridges on complete manifolds[END_REF] lim sup

t 0 4t log p t (x, U c 1 , y) ≤ -(d(x, y) + ε) 2
uniformly for x and y in K 1 . Suppose further that there exists η > 0 and y 0 ∈ K 0 , such that, if we consider the concentric balls B, B , B centered at y 0 (as in Lemma 7), the closure of B is contained in U 0 , and the conclusion of Lemma 7 holds (including the existence of an appropriate extendable coordinate system). Then for any multi-index α, [START_REF] Hsu | Bismut's gradient formula for vector bundles[END_REF] lim sup

t 0 4t log ∂ α y p t (x, U c 1 , y) ≤ (d(x, y) + ε -3η) 2 uniformly over x ∈ K 0 and y ∈ B.
Proof. Let y 0 ∈ K 0 and the corresponding concentric balls B, B , and B be as in the lemma. Let σ be the first hitting time (of the diffusion X t ) of ∂U 1 , let τ 1 be the first hitting time of ∂B after σ, and let τ 1 be the first hitting time of ∂B after τ 1 . Then, for i > 1, we iteratively define τ i to be the first hitting time of ∂B after τ i-1 and τ i the first hitting time of ∂B after τ i . In the notation of (2), we have y). We understand p B t (x, y) to be zero whenever x ∈ B or t < 0, and we let µ x i be the (spacetime) distribution of (τ i , X τi ) when the diffusion is started from x, so that in particular, µ x i is a sub-probability measure on [0, ∞) × ∂B . Then we have the path decomposition [START_REF] Hsu | On the principle of not feeling the boundary for diffusion processes[END_REF]

∂ α y p t (x, U c 1 , y) = ∂ α y p t (x, y) -∂ α y p U1 t (x,
p t (x, U c 1 , y) = ∞ i=1 (z,τ )∈∂B ×(0,t) p B t-τ (z, y) dµ x i ,
which is valid for any x ∈ K 0 and y ∈ B.

We now bound the distributions of the τ i , starting with τ 1 . For any small enough δ, let V be a δ-neighborhood of ∂B . By the conclusion of Lemma 5, which we assume holds, there is a t 0 , such that, for any z ∈ K 1 , the probability that the diffusion, started from z, leaves the ball of radius δ centered at z before t 0 is less than 1/2. It follows that, for any t < t 0 , for any x ∈ K 0 ,

1 2 P x (τ 1 ≤ t) ≤ V p t (x, U c , z) dν(z).
On the other hand, by the triangle inequality and the definitions of U 1 , σ, and τ 1 , we have d(x, U c 1 , z) ≥ d (x, ∂B ) + ε -δ for all z ∈ V and x ∈ K 1 . Note that B is a compact subset of U 0 , and thus for small enough δ, (11) holds for y ∈ V . So for all small enough t,

V p t (x, z) dν(z) ≤ ν(V ) exp - (d(x, ∂B ) + ε -2δ) 2 4t ,
uniformly in x ∈ K 0 . Since ν(V ) is positive for all δ > 0, we conclude that ( 14) lim sup

t 0 4t log P x (τ 1 ≤ t) ≤ -(d(x, ∂B ) + ε) 2 uniformly over x ∈ K 0 .
By the strong Markov property, the distribution of τ i -τ i-1 depends only on X τ i-1 ; in particular, it doesn't depend on i, and similarly for the distribution of τ i -τ i . Moreover, τ i -τ i-1 is conditionally independent of all of the previous increments τ j -τ j-1 and τ j-1 -τ j-1 for j < i given X τ i-1 , and similarly for τ i -τ i . Now, in order for τ i to be less than t, τ i -τ i-1 and τ i-1 -τ i-1 must both be less than t. Further, τ i -τ i-1 can only be less than t if the shifted diffusion {X τ i-1 +t , t ≥ 0} moves a distance 2η from its starting point (at X τ i-1 ∈ ∂B ) in time t or less, and similarly, τ i-1 -τ i-1 can only be less than or equal to t if it moves a distance 2η in time t or less. By independence and the conclusion of Lemma 5, it follows that there exists t 0 such that, for t < t 0 and any i ≥ 2,

P x (τ i -τ 1 < t) < 1 4 i-1 for all x ∈ K 0 .
Combining this with [START_REF]Estimates of derivatives of the heat kernel on a compact Riemannian manifold[END_REF] and again using the independence, we see that, for any δ > 0 there exists t 0 > 0 such that

P x (τ i ≤ t) ≤ 1 4 i-1 exp - (d(x, ∂B ) + ε -δ) 2 4t
whenever 0 < t < t 0 , for all x ∈ K 0 and all i = 1, 2, 3, . . .. Returning to the spacetime hitting measures µ x i in (13), we see that if we consider their restrictions to [0, t] (for t < t 0 ), their masses are absolutely summable, by the above and the summability of the geometric series. More precisely, if we let µ x,t denote their sum, then it is a finite measure on ∂B × [0, t] with total mass bounded from above by ( 15)

F (t) = 1 2 exp - (d(x, ∂B ) + ε -δ) 2 4t
for any x ∈ K 0 . That is, if F x,t0 (t) is the distribution function of the time-marginal of µ x,t0 , viewed as a measure on [0, t 0 ] indexed by x, then F (t) is an upper bound on F x,t0 (t) (for all t ∈ [0, t 0 ]) which holds uniformly in x. Combining this with (13) and the smoothness of p B t (z, y) in y ∈ B for fixed t > 0 (as elaborated on in the proof of Lemma 7), the dominated convergence theorem allows us to show that

∂ α y p t (x, U c 1 , y) = (z,τ )∈∂B ×(0,t) ∂ α y p B t-τ (z, y) dµ x,t .
Using (8), we see that, for any δ, for all small enough t (uniformly in x ∈ K 0 and y ∈ B),

∂ α y p t (x, U c 1 , y) ≤ (z,τ )∈∂B ×(0,t) exp - (d(∂B , y) -δ) 2 4(t -τ ) dµ x,t = τ ∈(0,t) exp - (d(∂B , y) -δ) 2 4(t -τ ) dF x,t0 (τ ),
where we've used the fact that the integrand doesn't depend on z ∈ ∂B to reduce to the time-marginal of µ x,t . Because the integrand is monotone decreasing, we get an upper bound from the upper bound (15) on the distribution function, so that, for some C > 0

∂ α y p t (x, U c 1 , y) ≤ τ ∈×(0,t) exp - (d(∂B , y) -δ) 2 4(t -τ ) dF (τ ) = τ ∈(0,t) exp - (d(∂B , y) -δ) 2 4(t -τ ) exp - (d(x, ∂B ) + ε -δ) 2 4t (d(x, ∂B ) + ε -δ) 2 8t 2 dτ ≤ C τ ∈(0,t) exp - (d(∂B , y) -δ) 2 4(t -τ ) exp - (d(x, ∂B ) + ε -2δ) 2 4t dτ,
where, in the final line, we've absorbed the ((d(x, ∂B ) + ε -δ) 2 )/(8t 2 ) into the second exponential at the cost of doubling δ and multiplying by a constant. Note that, for any constants a, b, t > 0 the maximum of

- a 2 4(t -τ ) - b 2 4τ for τ ∈ (0, t) is -(a + b) 2 /(4t) (achieved at τ = t/(1 + (a/b))).
Then, applying this to the integrand above and using that δ > 0 is arbitrary to clean up the constants, we have that, for any δ > 0, there exists t 0 > 0 such that, for all t ∈ (0, t 0 ), all y ∈ B, and all x ∈ K 0 ,

∂ α y p t (x, U c 1 , y) ≤ exp - (d(x, ∂B ) + d(∂B , y) + ε -δ) 2 4t .
Since the diameter of B is at most 3η, we conclude that, for every α,

lim sup t 0 4t log ∂ α y p t (x, U c 1 , y) ≤ (d(x, y) + ε -3η -δ) 2 uniformly over (x, y) ∈ K 0 × B.
Since δ is arbitrary, we can send it to zero, completing the proof.

Proving the Riemannian bounds

We now return to the situation of Theorems 1 and 2; namely, where M is a (possibly incomplete) Riemannian manifold, ∆ = ∆ LB + Z where Z is conservative, d(•, •) is the Riemannian distance, and p t (x, y) is the corresponding heat kernel with respect to the Riemannian volume. This is a special case of the situation considered in the previous section, so those results apply here.

As is clear from the previous two sections, we mostly work with partial derivatives in coordinates. But since we eventually want to give an estimate for covariant derivatives, we now relate the two. Since covariant differentiation is tensorial in the vector fields along which we differentiate, it's essentailly enough to look at the coordinate vector fields.

∇ N ∂ I f (x) = ∇ N ∂ i 1 ,...,∂ i N f (x) = ∂ i1 • • • ∂ i N f (x) + J:J I F I,J (x)∂ J f (x)
where the sum is over subsequences J of I which are neither empty nor all of I, and each F I,J is a smooth function on a neighborhood of B depending on the Christoffel symbols (of the Levi-Civita connection on M ). Further, there exists c > 0 (depending only on N , B, and the choice of coordinates), such that, for any collection V 1 , . . . , V N of (smooth) vector fields of length 1 or less,

∇ N V1,...,V N f (x) = I∈{1,...,n} N a I (x)∇ N ∂ I f (x),
where the a I are smooth functions on a neighborhood of B, indexed by all sequences of indices I and depending on the V i , such that |a I (x)| < c for all I and all x in a neighborhood of B.

Proof. The first result is trivial for N = 1 and well known for N = 2, where the formula for the (0, 2)-version of the Hessian in coordinates is

∇ 2 ∂j ,∂i f = ∂ j ∂ i f - n k=1 Γ k ji ∂ k f.
We now proceed by induction, assuming the result for any ∇ N ∂ I . Since ∇ N ∂ I is a (0, N )-tensor, we compute its covariant derivative as

∇ ∂j ∇ N ∂ I f = ∂ j ∇ N ∂ I f -∇ N ∇ ∂ j ∂ i 1 ,∂ i 2 ...,∂ i N f -• • • -∇ N ∂ i 1 ...,∂ i N -1 ,∇ ∂ j ∂ i N f
Then using the definition of the Christoffel symbols and multilinearity, we have

∇ ∂j ∇ N ∂ I f = ∂ j ∇ N ∂ I f - n k=1 Γ k ji1 ∇ N ∂ k ,∂ i 2 ...,∂ i N f -• • • -Γ k ji N ∇ N ∂ i 1 ...,∂ i N -1 ,∂ k f .
Using the induction assumption on all N th-order covariant derivatives on the right-hand side and basic calculus, the result follows for ∇ ∂j ∇ N ∂ I f . But since all N + 1st-order covariant derivatives under consideration are of this form for some j and I, the first claim follows.

For the second, since the metric in the coordinate system is smooth and uniformly positive definite, any unit vector can be written as a linear combination of coordinate vector fields with uniformly bounded coefficients. Then the claim follows by linear algebra.

Proof of Theorem 1. Note that the condition on K, namely that {z : d(x, z) + d(z, y) < d(x, y) + ε} has compact closure for all (x, y) ∈ K, is symmetric in x and y, that is, it is preserved under reflection across the diagonal in M × M . Therefore, we can enlarge K by taking the union with its reflection to get a symmetric set which satisfies the same condition, and obviously it's enough to prove the theorem for this (potentially) larger set. Thus, without loss of generality, we assume that K is symmetric for the rest of the proof.

Because Z is conservative, there exists a smooth measure ν such that p ν t (x, y) is symmetric. To elaborate, recall that ∆ LB = div µ • grad, where div µ is the divergence with respect to the Riemannian volume µ and grad is the Riemannian gradient. Any other smooth volume ν can be written as e f µ for some smooth f , and then we can write our operator in divergence form with respect to ν as

∆ = div ν • grad -grad f + Z,
where grad f is a vector field acting on smooth functions. Since div ν • grad is symmetric with respect to ν and Z -grad f is only when it is identically 0, we see that ∆ admits a symmetrizing volume if and only if Z = grad f for some smooth f , in which case the symmetrizing measure is ν = e f µ (unique up to constant rescaling). Thus, we now let ν be this measure for the given Z, and we note that p ν t satisfies the assumptions of the previous section and is also symmetric.

Let y 0 be any point on M . Then there exists some η such that the concentric balls B, B , and B around y 0 are such that B has compact closure and is contained in a single extendable coordinate system. Using a bump function, B can be isometrically included in a Riemannian manifold M diffeomorphic to R n with a Riemannian metric and smooth volume that satisfy the assumptions under which Léandre proved (4). In particular, ( 7) is satisfied for p ν,M t (x, y) uniformly for (x, y) in any compact subset of M × M , and thus for x and y in B . It follows, by Lemma 7, that, for any multi-index α, [START_REF] Hsu | Heat kernel on noncomplete manifolds[END_REF] lim sup

t 0 4t log ∂ α y p ν,B t (x, y) ≤ -d 2 (x, y)
uniformly over (x, y) ∈ B × B. But since p ν,B t (x, y) only depends on (the Riemannian structure and the volume on) B , it this holds for p ν,B t (x, y) with B viewed as an open ball in the original M as well. Also note that, for fixed y 0 , if [START_REF] Hsu | Heat kernel on noncomplete manifolds[END_REF] holds for some η, then it also holds for any smaller η. Now take any (x 0 , y 0 ) ∈ K. By the triangle inequality and continuity of the distance, there is some δ > 0 such that, if N x0 (δ) is an open δ-ball around x 0 and N y0 (δ) is an open δ-ball around y 0 (here we use N instead of B for these balls to avoid notational confusion with B, B , and B ), then

U 1 = x∈Nx 0 (δ),y∈Ny 0 (δ) z : d(x, z) + d(z, y) < d(x, y) + ε 2
is open with compact closure (because it is a subset of {z : d(x 0 , z) + d(z, y 0 ) < d(x 0 , y 0 ) + ε} for small enough δ). Now let

K 0 = N x0 (δ/2) ∪ N y0 (δ/2), U 0 = N x0 (δ) ∪ N y0 (δ),
and

K 1 = U 0 . Then K 0 ⊂ U 0 ⊂ K 1 ⊂ U 1 ⊂ M
where K 0 and K 1 are compact and U 0 and U 1 are open with compact closure. Further, again by the triangle inequality, we have that there exists ε > 0 such that

+ ε for all x and y in K 1 . Because ∆ is symmetric with respect to ν and K 1 is a compact subset of U 1 , we can apply the result of Bailleul-Norris, so that (2) yields

lim sup t 0 4t log p ν t (x, U c 1 , y) ≤ -(d(x, y) + ε ) 2
uniformly for x and y in K 1 . After possibly shrinking η, we have that B ⊂ U 0 , which means we've shown that all of the assumptions of Lemma 8 are satisfied. Further, after possibly shrinking η again, we have 3η < ε /2. We conclude that, for any multi-index α,

t 0 4t log ∂ α y p ν t (x, U c 1 , y) ≤ -(d(x, y) + ε -3η) 2 < -d(x, y) + ε 2 2 uniformly over x ∈ K 0 and y ∈ B. (18) lim sup
Because U 1 has compact closure, we can isometrically include it in a compact Riemannian manifold M (say, by a smooth doubling construction), equipped with a measure that extends ν and a conservative vector field that extends Z; we just extend the smooth f in ν = e f µ, and the associated heat kernel also remains symmetric. Moreover, we can choose M such that [START_REF] Léandre | Majoration en temps petit de la densité d'une diffusion dégénérée[END_REF] holds with respect to the distance on M as well. (In particular, we can use a bump function to conformally rescale the metric outside of U 1 to make paths from x to y that leave U 1 "too long," and for all (x, y) ∈ K 1 by smoothness and compactness.) Then the argument leading to [START_REF] Li | Estimations optimales du noyau de la chaleur sur les groupes de type Heisenberg[END_REF] applies to p ν,M t (x, U c 1 , y) as well (and perhaps more easily, since M is compact). Because

p ν t (x, y) = p ν,U1 t (x, y) + p ν t (x, U c 1 , y), p ν,M t (x, y) = p ν,U1 t (x, y) + p ν,M t (x, U c
1 , y), and p ν,U1 t (x, y) is the same whether it's viewed as a subset of M or of M , we conclude that, for any δ > 0 and any multi-index α, there exists t 0 > 0, such that

∂ α y p ν t (x, y) -∂ α y p ν,M t (x, y) ≤ exp   - d(x, y) + ε 2 2 -δ 4t   
for all x ∈ K 0 , y ∈ B, and t ∈ (0, t 0 ]. By Lemma 9, for each N , there is some t 0 > 0 such that

(19) ∇ N V1,...,V N p ν t (x, y) -∇ N V1,...,V N p ν,M t (x, y) ≤ exp   - d(x, y) + ε 2 2 -δ 4t   
for any collection of unit vector fields V 1 , . . . , V N (acting on y) on B, and any x ∈ K 0 , y ∈ B, and t ∈ (0, t 0 ] (perhaps after decreasing δ).

Moreover, because M is compact and p ν,M t is symmetric (so that we can exchange x and y derivatives) we know that p ν,M t satisfies (20)

∇ N y p ν,M t (x, y) ≤ D N d(x, y) t + 1 √ t N p ν,M t (x, y)
for t ∈ (0, 1], uniformly over (x, y) ∈ M , for some sequence of constants. Since both p ν,M t and p ν t satisfy ( 2) and ( 3), we see that they also satisfy [START_REF] Li | On hypoelliptic bridge[END_REF] with N = 0 (that is, such a bound holds for the heat kernels themselves, not just their derivatives). It follows that 4t log p ν,M t (x, y) → -d 2 (x, y) and p ν,M t (x, y) p ν t (x, y)

→ 1 as t 0, uniformly for x ∈ K 0 and y ∈ B. In light of this and ( 19) (recall that δ can be chosen as small as needed), we can replace p ν,M t with p ν t on both sides of (20) at the cost of possibly adjusting the D N . We conclude that, for any N , there is a constant D N and a t 0 > 0 such that

(21) ∇ N y p ν t (x, y) ≤ D N d(x, y) t + 1 √ t N p ν t (x, y)
for all t ∈ (0, t 0 ), x ∈ K 0 , and y ∈ B. Because x is in the interior of K 0 and B is open, and (x, y) is an arbitrary point in K, we've proven that any point in K has an open neighborhood on which (21) holds. By compactness of K, this means that we can find D N and t 0 such that [START_REF] Ludewig | Strong short-time asymptotics and convolution approximation of the heat kernel[END_REF] holds on all of K. Further, as already noted, we are free to let t 0 = 1 by smoothness, at the cost of again adjusting the D N . Now by symmetry (of both K and the heat kernel with respect to ν) we can replace the x-derivatives by y-derivatives, at which point we use Lemma 6 to replace p ν t with p t (since we no longer need the symmetry) and then again to deduce (5) from [START_REF] Colin De Verdière | Small-time asymptotics of hypoelliptic heat kernels near the diagonal, nilpotentization and related results[END_REF]. This proves the theorem.

Proof of Theorem 2. As in the proof of Theorem 1, we assume K is symmetric (recall that Cut(M ) is symmetric, so this is compatible with the assumptions), and we construct K 0 ⊂ U 0 ⊂ K 1 ⊂ U 1 ⊂ M in the same way.

The difference with the proof of Theorem 1 is that now we use bump function to include U 1 in a manifold M for which the heat kernel satisfies the following expansion: there exists some t 0 > 0 and a function c 0 (x, y), smooth on a neighborhood of K such that c 0 > 0 on K and the heat kernel satisfies the expansion

(22) p ν,M t (x, y) = 1 t n/2 e -d 2 (x,y) 4t (c 0 (x, y) + r(t, x, y))
where the remainder r(t, x, y) satisfies the bound sup (t,x,y)∈(0,t0]×K

∂ α x ∂ β y ∂ k t r(t, x, y) < ∞
for all multi-indices α and β and all k = 0, 1, 2, . . .. That this is possible follows from the well-known paper of Ben Arous [START_REF] Arous | Développement asymptotique du noyau de la chaleur hypoelliptique hors du cut-locus[END_REF], although one should account for any topological difficulties in the inclusion, as done in [START_REF] Neel | Uniform, localized asymptotics for sub-riemannian heat kernels and diffusions[END_REF]. Alternatively, Ludewig [START_REF] Ludewig | Strong short-time asymptotics and convolution approximation of the heat kernel[END_REF] proves that [START_REF] Malliavin | Short time behavior of the heat kernel and its logarithmic derivatives[END_REF] holds for compact M (or even a general complete M ), when one considers the specialization from his vector bundle context to the scalar Laplacian. (Actually, both authors give an expansion to any number of terms, but we only need the leading coefficient c 0 at present.)

Either way, we have such an M so that [START_REF] Li | On hypoelliptic bridge[END_REF] holds. Since the right-hand side of ( 19) is exponentially small compared to the remainder in [START_REF] Malliavin | Short time behavior of the heat kernel and its logarithmic derivatives[END_REF], we have that [START_REF] Malliavin | Short time behavior of the heat kernel and its logarithmic derivatives[END_REF] holds for p ν at the cost of altering the remainder but still having it satisfy the the same bound. Thus

log p ν t (x, y) = - n 2 log t - d 2 (x, y) 4t
+ log (c 0 (x, y) + r(t, x, y)) .

Taking derivatives yields -4t∇ N y log p ν t (x, y) -∇ N y d 2 (x, y) = -4t∇ N y log (c 0 (x, y) + r(t, x, y)) . Now since c 0 > 0 and r has uniformly bounded derivatives, ∇ N y log (c 0 (x, y) + r(t, x, y)) is uniformly bounded for (t, x, y) ∈ (0, t 0 ] × K and thus the right-hand side of the above is uniformly O(t). By symmetry, we have that Since dν dµ (y) is smooth and positive for y ∈ π 2 (K), the right-hand side is O(t) uniformly, just as we saw for ∇ N y log (c 0 (x, y) + r(t, x, y)) above. This shows that Theorem 2 holds for either x or y derivatives, as desired.

-4t∇ N x log p ν t (x, y) -∇ N x d 2 (x, y) = O(t)
Remark 10. As mentioned, weaker versions of these theorems can be proven using results on Riemannian Brownian motion from the 90s. In particular, assume that, in Theorems 1 and 2, Z ≡ 0 and K satisfies the stronger assumption that there exists ε > 0 such that, for all (x, y) ∈ K, In this case, we can use that (1) itself holds on compact manifolds, along with Lemmas 6 and 9 to move between partial derivatives and covariant derivatives as well as between logarithmic derivatives and "regular" derivatives of p t , in place of Léandre's result [START_REF] Chavel | Riemannian geometry[END_REF]. That gives the necessary input for Lemma 7. Moreover, a pair of papers of Hsu [START_REF] Hsu | On the principle of not feeling the boundary for diffusion processes[END_REF][START_REF] Hsu | Heat kernel on noncomplete manifolds[END_REF] can be used in place of the results of Bailleul-Norris (2) and (3), giving the necessary input for Lemma 8 with a larger U 1 than before, reflecting the weaker localization condition for K.

Given this, the proofs of Theorems 1 and 2 proceed in essentially the same way. In Theorem 1 we localize (1) as proven in Hsu [START_REF]Estimates of derivatives of the heat kernel on a compact Riemannian manifold[END_REF] and Stroock-Turetsky [START_REF] Stroock | Upper bounds on derivatives of the logarithm of the heat kernel[END_REF] , with no need to refer to [START_REF] Neel | Uniform, localized asymptotics for sub-riemannian heat kernels and diffusions[END_REF] for the case of non-zero Z, and similarly in Theorem 2, we can localize the results of [START_REF] Malliavin | Short time behavior of the heat kernel and its logarithmic derivatives[END_REF] without using the expansion [START_REF] Malliavin | Short time behavior of the heat kernel and its logarithmic derivatives[END_REF] to handle the case of non-zero Z, although at the cost of weakening the O(t) rate of convergence.

Sharpness of the upper bound and explicit examples

In [START_REF] Neel | Uniform, localized asymptotics for sub-riemannian heat kernels and diffusions[END_REF], an expression for the leading (that is, the 1/t N order) term in the asymptotics of an N thorder derivative of log p t (x, y) is given. Briefly, if Γ is the set of midpoints of minimal geodesics from x to y, there is a probability measure m 0 on Γ such that, for any smooth vector fields Z 1 , . . . , Z N in a neighborhood of y, [START_REF] Mckean | Stochastic integrals[END_REF] lim

t 0 t N Z N y • • • Z 1 y log p t (x, y) = - d(x, y) 2 N κ m0 Z 1 y d(•, y), . . . , Z N y d(•, y) .
where κ m0 is the joint cumulant with respect to m 0 and the Z i y d(•, y) are viewed as random variables on Γ. Note that if (x, y) ∈ Cut(M ), then Γ is a single point, m 0 is a point mass, and any cumulant of order 2 or more is 0, which is consistent with Theorem 2. However, since this cumulant will generally be non-zero for (x, y) ∈ Cut(M ), it's clear that the upper bound in (1) is sharp (that is, the constant is not explicitly given, but 1/t N is the correct order of blow up) even on a compact Riemannian manifold (so this is not a matter of completeness or incompleteness). To underscore this, and also to illustrate the pointwise result just mentioned, we treat the simple example of the circle explicitly.

If we consider S 1 ∼ = R/Z with the standard heat kernel (that is, with Z ≡ 0), then for any x and y (understood mod 1), we have

p S 1 t (x, y) = 1 √ 4πt ∞ n=-∞ e -(y-x+n) 2 4t
.

Here the symmetry between x and y is explicit, and one can take spatial derivatives on either variable. By translation invariance, without loss of generality we can let x = 0, in the sense that it is the image of 0 under the quotient map, in which case the cut locus of x is the point y = 1 2 , again in the sense that it is the image of 1 2 under the quotient map. More geometrically, y is the antipodal point of x and there are two minimizing geodesics from x to y, corresponding to the two ways to go halfway around the circle.

To compute the log-Hessian of p t (0, 1/2), let y = 1 2 + α, for small α (which is trivially normal coordinates centered at 1 2 ). Then explicit computation gives that,

∂ α p S 1 t 0, 1 2 + α = 1 √ 4πt ∞ n=-∞ - 1 2 + α + n 2t e -( 1 2 +α+n ) 2 4t
and

∂ 2 α p S 1 t 0, 1 2 + α = 1 √ 4πt ∞ n=-∞ - 1 2t + 1 2 + α + n 2 4t 2 e -( 1 2 +α+n ) 2 4t
.

Letting α = 0, we see that only two terms from the sum contribute (up to an exponentially small error), corresponding to the two minimizing geodesics. In particular, for any ε > 0, we have

p S 1 t 0, 1 2 = 1 √ 4πt 2e -1 16t + O e -9-ε 16t , ∂ α p S 1 t 0, 1 2 = 0 and ∂ 2 α p S 1 t 0, 1 2 = 1 √ 4πt - 1 2t + 1 16t 2 2e -1 16t + O e -9-ε 16t . Then since ∂ 2 y log p t (x, y) = ∂ 2 y pt(x,y) pt(x,y) - ∂ypt(x,y) pt(x,y) 2
, we have

∂ 2 x log p S 1 t 0, 1 2 = ∂ 2 y log p S 1 t 0, 1 2 = - 1 2t + 1 16t 2 + O e -8-ε 16t
for any ε > 0.

We note that here we have computed the logarithmic Hessian of the heat kernel directly (that is, by elementary methods, independent of any of the preceding results of this paper). In reference to [START_REF] Mckean | Stochastic integrals[END_REF], note that, by symmetry, m 0 assigns probability 1/2 each to the 2 midpoints of the 2 minimizing geodesics from 0 to 1/2, and thus the random variable ∂ α d •, 1 2 under m 0 takes values ±1 each with probability 1/2. So the variance (note that the joint cumulant of 2 random variables is their covariance) in ( 23) is 1, and we have

lim t 0 t 2 ∇ 2
x log p S 1 t 0, 1 2 = d 2 0, 1 2 4 = 1 16 using ∂ α ⊗ ∂ α as a basis for bilinear forms on T x M , in agreement with the above. Also, note that this is not special to this simple example. Any time (on any Riemannian manifold and for any x and y points such that the conditions of Theorem 1 hold) there are exactly 2 nonconjugate minimizing geodesics between x and y, one will have similar behavior for the logarithmic Hessian.

6. The non-symmetric and sub-Riemannian cases Now suppose Z is such that ∆ no longer admits a symmetrizing measure. Then y-derivatives of the heat kernel can still be localized, and thus are seen to satisfy the same estimates as above, under either the condition on the distance to infinity in Theorem 1 (if the asymmetry is "not too bad" in the sense of the sector condition of [START_REF] Bailleul | Diffusion in small time in incomplete sub-Riemannian manifolds[END_REF]) or the condition described in Remark 10 (in general). However, without the symmetry, y-derivatives cannot be transferred to x-derivatives. Indeed, when exchanging the role of the variables in general, one needs to take the adjoint of the operator (this is the relationship between the forward and backward Kolmogorov equations). The point is that the adjoint may have a potential term (that is, 0th-order term), which is not covered by the preceding arguments and is less easy to localize.

To quickly illustrate the point, we let (u, v) be Cartesian coordinates on R 2 . If we consider ∆ ∂ 2 u + ∂ 2 v + a(u, v)∂ u + b(u, v)∂ v then the (formal) adjoint, with respect to the standard Euclidean area, is

∆ * = ∂ 2 u + ∂ 2 v -(u, v)∂ u -b(x, y)∂ v -(∂ u a) (u, v) -(∂ v b) (u, v).
Thus we now have a potential, and to find a stochastic representation for the heat kernel, corresponding to ∆ * , we should look at the Feynman-Kac formula [START_REF] Neel | Uniform, localized asymptotics for sub-riemannian heat kernels and diffusions[END_REF] p t (x, y) = E x exp t 0

(∂ u a + ∂ v b) (X s ) ds
where X t is the diffusion with generator ∂ 2 u +∂ 2 v -(u, v)∂ u -b(x, y)∂ v started from x. On a non-compact manifold, the integrand on the right-hand side need not be bounded, so even if the probability of a path leaving some open U and eventually arriving at y can be estimated to be small, as above, the contribution of such a path to p t is weighted by the Feynman-Kac factor, which depends on the entire trajectory and which we have no a priori control over. Similarly, trying to treat the potential via an h-transform runs into the problem that it's not clear in general why the heat equation for the adjoint should admit a positive solution. Indeed, it is not clear that we should expect (1) to hold on a noncompact Riemannian manifold without some additional assumptions to control the potential. In that spirit, one can see [START_REF]Hessian formulas and estimates for parabolic Schrödinger operators[END_REF] for recent results in this direction on complete (but not necessarily compact) Riemannian manifolds under additional geometric assumptions (for example, curvature bounds, the existence of a pole, etc.). (Note also that the case where the path integral in [START_REF] Neel | Uniform, localized asymptotics for sub-riemannian heat kernels and diffusions[END_REF] only depends on the endpoints and thus can be localized is exactly the case when Z admits a potential, and this is directly related to the density of the symmetrizing measure.)

In a different direction, suppose p t (x, y) is the heat kernel associated to a sub-Laplacian on a sub-Riemannian manifold M . Then the localization for p t and its y-derivatives still works (and thus under a symmetry assumption, it also works for x-derivatives), but we don't have a global estimate to localize. That is, no estimates comparable to (1) are known to hold for compact sub-Riemannian manifolds (or for structures on R n given by bounded vector fields with bounded derivatives of all orders, as in the work of Léandre and Ben Arous used above). More precisely, such bounds are known to hold away from abnormal minimizers (see [START_REF] Neel | Uniform, localized asymptotics for sub-riemannian heat kernels and diffusions[END_REF]), but on a properly sub-Riemannian manifold (that is, not one that happens to be Riemannian), the diagonal is always abnormal, and thus one does not have appropriate estimates on any K that intersects the diagonal. (Note that more than the asymptotic results of [START_REF] Colin De Verdière | Small-time asymptotics of hypoelliptic heat kernels near the diagonal, nilpotentization and related results[END_REF] are needed in the present context.) Derivative bounds for the special case of H-type groups were established in [START_REF] Eldredge | Precise estimates for the subelliptic heat kernel on H-type groups[END_REF] and [START_REF] Li | Estimations optimales du noyau de la chaleur sur les groupes de type Heisenberg[END_REF], including bounds for the log-gradient of the heat kernel in [START_REF] Li | Estimations optimales du noyau de la chaleur sur les groupes de type Heisenberg[END_REF], but this comes nowhere near the complete generality we have in the Riemannian situation.

That said, note that the semi-martingale property for the bridge process on compact, step-2 sub-Riemannian manifolds was established in [START_REF] Li | On hypoelliptic bridge[END_REF], so there is hope for future advances.

Lemma 9 .

 9 Let B ⊂ M be an open ball in M with compact closure and extendable coordinates ∂ 1 , . . . , ∂ n , and suppose that f is a smooth function on a neighborhood of B. Then if I = (i 1 , . . . , i N ) is a sequence of indices from 1, . . . , n, we have

  d(x, ∞) + d(y, ∞) > d(x, y) + ε, where d(x, A) = inf {d(x, z) : z ∈ A} and d(x, ∞) = sup {d(x, A) : A closed and M \ A relatively compact} .

  • • • ∂ ni |B| where i 1 , . . . , i |B| are the elements of B and |B| is the cardinality of the block B. Applying (5) and using that the sum of the |B| over all B ∈ π is necessarily

	π∈Π B∈π	C |B|	d(x, y) t	+	1 √ t	|B|	= p ν t (x, y)	d(x, y) t	+	1 √ t	N	π∈Π B∈π

N , we have |∂ α p ν t (x, y)| ≤ p ν t (x, y) C |B| , and thus (6) holds with D N = π∈Π B∈π C |B| .

Acknowledgements

This work was partially supported by a grant from the Simons Foundation (#524713 to Robert Neel).


GAS TURBINES AND BIODIESEL :

A CLARIFICATION OF THE RELATIVE NO X INDICES OF FAME, GASOIL AND NATURAL GAS Pierre A. Glaude, René Fournet, Roda Bounaceur DCPR, Nancy-Université, CNRS 1, rue Grandville -BP 20451 -54001 NANCY Cedex -France Michel Moliere GE Energy Product-Europe, 20 avenue de Maréchal Juin, BP 379, 90007 Belfort, France ABSTRACT There is currently a sustained interest in biofuels as they represent a potential alternative to petroleum derived fuels. Biofuels are likely to help decrease greenhouse gases emissions and the dependence on oil resources. Biodiesels are Fatty Acid Methyl Esters (FAMEs) that are mainly derived from vegetable oils; their compositions depend from the parent vegetables: rapeseed ("RME"), soybean ("SME"), sunflower, palm etc. A fraction of biodiesel has also an animal origin ("tallow").

A key factor for the use of biofuels in gas turbines is their Emissions Indices (NOx, CO, VOC, PM) in comparison with those of conventional "petroleum gasoils". While biodiesels reduce carbon-containing pollutants, experimental data from diesel engines show a slight increase in NOx. The literature relating to gas turbines is very scarce. Two recent, independent field tests carried out in Europe (RME) and in the USA (SME) showed slightly lower NOx while a lab test on a microturbine showed the opposite effect. To clarify the NOx index of biodiesels in gas turbines, a study has been undertaken, taking gasoil and natural gas (NG) as reference fuels. In this study, a calculation of the flame temperature developed by the 3 classes of fuels has been performed and the effect of their respective compositions has been investigated. The five FAMEs studied were RME, SME and methyl esters of sunflower, palm and tallow; these are representative of most widespread vegetable and animal oil bases worldwide. The software THERGAS has been used to calculate the enthalpy and free energy properties of the fuels and GASEQ for the flame temperature (T f ), acknowledging the fact that "thermal NOx" represents the predominant form of NOx in gas turbines. To complete the approach to structural effects, we have modeled two NG compositions (rich and weak gas) and three types of gasoil using variable blends of eleven linear/branched/cyclic molecules.

The results are consistent with the two recent field tests and show that the FAMEs lie close to petroleum gasoils and higher than NG in terms of NOx emission. The composition of the biodiesel and regular diesel fuel influences their combustion heat: methyl esters with double bonds see a slight increase of their T f and their NOx index while that of gasoil is sensitive to the aromatic content.

INTRODUCTION

Biomass-derived fuels can be used in many combustion equipments and may represent a valuable alternative to fossil fuels for reducing greenhouse gases (GHG). A key consideration for the success of any alternative liquid fuel is its emission signature in comparison with diesel oil which is the paradigmatic liquid fuel. In this respect, NOx emission has a critical importance as far as gas turbines (GT) are concerned. While a relatively high number of studies have been devoted to the combustion of biofuels in diesel engines, few work and scarce information are currently available for biofuels in gas turbine applications.

Diesel Engines: As far as diesel engines are concerned, two interesting reviews of the effect of FAME on emissions have been published recently [START_REF] Lapuerta | Effect of biodiesel fuels on diesel engine emissions[END_REF]- [START_REF] Szibist | Biodiesel combustion, emissions and emission control[END_REF]. Most works show a slight increase in NOx emissions when using biofuels instead of conventional diesel fuel while some of them found that NOx increases only in certain operating conditions. Some authors did not find any effect with respect to diesel fuel or even found a decrease in NOx emissions when using biofuels. The increase in NOx is mainly explained by an advanced injection time that is tied with the physical properties of FAMEs. The higher compressibility bulk modulus of biofuels leads to a quicker pressure rise produced by the fuel pump which propagates towards the injectors [START_REF] Lapuerta | Effect of biodiesel fuels on diesel engine emissions[END_REF] [START_REF] Szibist | Biodiesel combustion, emissions and emission control[END_REF]. Some authors investigated the influence of the type of biodiesel fuel. Graboski et al. [START_REF] Graboski | The effect of biodiesel composition on engine emission from a DDC series 60 diesel engine[END_REF] found that the NOx emission decreased when the mean carbon chain length increased and that the presence of double bonds impacted NOx emissions: NOx was found to increase linearly with the iodine number, which quantifies the number of multiple bonds in the fuel. The authors suggested that this effect could be due to differences in both physical and chemical properties of the FAMEs. Knothe et al. [START_REF] Knothe | Exhaust emissions of biodiesel, petrodiesel, neat methyl esters, and alkanes in a new technology engine[END_REF] compared pure methyl esters behavior in engines to conventional diesel fuel and found a decrease in NOx with saturated C 12 and C 16 methyl esters while mono-unsaturated C 18 increased slightly the emission. The authors explained their results by a change in the adiabatic flame temperature. Based on numerical simulations of ideal reactors, Ban-Weiss et al. [START_REF] Ban-Weiss | A numerical investigation into the anomalous slight NO x increase when burning biodiesel: a new (old) theory[END_REF] proposed recently that the increase in the NOx emission when using FAMEs would be due to higher flame temperature, especially when the fuel contains double bonds. The NOx formation in such conditions is due to the thermal Zeldovich mechanism, which is predominant at high temperature. Their calculation are however based on C 3 and C 4 hydrocarbons, and methyl butanoate and methyl butenoate that are far to be representative of diesel or biodiesel fuels. Nabi et al. [START_REF] Nabi | Improvement of engine emissions with conventional diesel fuel and diesel-biodiesel blends[END_REF] calculated adiabatic flame temperature for a real diesel fuel and neem oil from the elementary C/H/O analyses of both fuels. They found that the Vegetable Oil had a slightly lower flame temperature than the conventional fuel. Contrary to diesel engine where ignition time during the cycle is an important parameter, flame temperature is the essential driver for the NOx emission in gas turbines.

Gas turbines:

As already stressed, there is a very reduced number of data available for the combustion of biodiesel in gas turbines. In 2006, a field test has been performed on a 40 MW, E class gas turbine (Frame 6B) using a rapeseed methyl ester (RME), which led to the following emissions [START_REF] Molière | Gas turbines in alternative fuel applications: biodiesel field test[END_REF]: -247 mg/Nm 3 on NG -367 mg/Nm 3 on DF -360 mg/Nm 3 on RME In 2007, a similar field test has been performed on a 80 MW, E class gas turbine (Frame 7EA) with gasoil-soy methyl ester blends (SME) [START_REF]GE Energy Successfully conduct first-of-a-kind biodiesel field tests[END_REF]. The results showed again a lower NOx figure with BF3 which was about 83% that of DF. Since numerous papers published by the diesel engine community pointed higher NOx with biodiesel, it was considered rather necessary to clarify these isolated results. On the other hand, a recent comparison of biofuel and diesel distillate #2 in a microturbine generator (30kW Capstone C30) showed an increase in NOx emissions by switching from gasoil to biofuel [START_REF] Bolszo | Emissions optimization of a biodiesel fired gas turbine[END_REF]. The most sensitive parameter seems to be a change in the atomization patterns of the fuel.

NUMERICAL APPROACH

In order to elaborate a rational statement about the NOx indices of biodiesel and diesel fuel, a thermochemical approach has been used to evaluate the adiabatic flame temperatures. NOx production is directly linked to temperature and this parameter allows a simple comparison of the different fuels, whatever the real combustion device is. The following fuels were investigated:

-individual chemical compounds belonging to the chemical families involved in gas turbine fuels in order to assess the influence of each chemical family, -various FAME blends that are representative of the most common biodiesel productions worldwide, -various blend models representative of petroleum gasoil, -various natural gas compositions. This approach was intended to get at the same time (i) a comparison of the emission indices of the different fuel classes (ii) an analysis of the influence of their chemical structures on flame temperature and thus on NOx emission.

Composition of Biodiesel fuels (FAME)

The world production of vegetable oil ("VO") is rapidly increasing and has been estimated to 101 Mt in 2003, when it reached only 86 Mt in 1999. Estimations are around 130 Mt for 2010. Another important production of fat material is 20 to 25 Mt of animal greases per year [START_REF] Ballerini | Les biocarburants, état des lieux, perspectives & enjeux du développement[END_REF]. Soybean oil is the main production (more than 30% of the total amount), with palm oil (ca 28%). Rapeseed and sunflower oils are the next important productions, around 13% and 9%, respectively. Soybean and palm oil have enjoyed the most important production increases during the past years. Regional specificities can be noted: soybean oils are produced in America while sunflower and rapeseed are essentially produced in Europe and palm oils in Asia. VOs essentially consist of triglycerides, i.e. triesters of glycerol and fatty acids. Table 1 displays the proportions of the different fatty acids that are contained in these four major VOs. As far as animal greases are concerned, five categories can be distinguished: lard from pigs, tallow from bovines and sheeps, poultry grease and fish oils. The latter cannot be used in combustion because of its high content of unsaturated molecules that causes the formation of gums and polymers. However, due to their natural origin, vegetable oils also contain cholesterol, fatty alcohols that can be free or esterified by fatty acids, and other minor compounds [START_REF] Karleskind | Raffinage des corps gras[END_REF] that can be classified as follows, by decreasing content:

-triglycerides (or triesters of fatty acids), -free fatty acids, -minor components (sterols, phospholipids…) -traces of metals (calcium, potassium…) Some of these components can create operational issues in thermal equipments; in addition, vegetable oils are generally very viscous fluids. Therefore, while they can be used as raw products in boilers, VO are often processed to produce biodiesel that suit better the operational constraints of diesel or gas turbine engines. For that purpose, a transesterification reaction transforms the heavy triglycerides into monoesters of methanol: the Fatty Acid Methyl Ester. Interestingly FAMEs have physical and thermal properties (viscosity, LHV) close enough to that of petroleum diesel for a use in combustion without significant hardware modifications.

Based on Table 1, four VO-derived biodiesels will be considered in this study; they are representative of the most important biodiesel classes worldwide. Table 2 displays the composition of these mixtures. Tallow and lard consist mainly of C 14 to C 20 fatty acids but contain higher proportions of saturated acids than VOs. Typically, they involve more than 50% of saturates and less than 5% of polyunsaturated acids, which explains that they have a higher viscosity than VO-derived FAMEs. Based on a typical composition of tallow [START_REF]Institut Technique d'Etude et de Recherche sur les corps Gras[END_REF], the fifth case (BF5) can be taken as representative of a biodiesel yielded by the transesterification of an animal grease.

Compositions of Conventional Diesel Fuels ("DF")

Table 3 summarizes the 3 blends (DF1 to 3) selected as models for petroleum diesels. Diesel fuels are actually complex mixtures of hundreds of molecules, most of them containing from 7 to 15 carbon atoms. Since modeling in detail such blends is currently unpractical, synthetic mixtures have been proposed in the literature. Although n-heptane has been widely modeled [START_REF] Colin | Detailed chemistry-based auto-ignition model including low temperature phenomena applied to 3-D engine calculations[END_REF], its chemical and physical properties remain far from a real diesel fuel. A more refined blend model (DF1) has been recently proposed which contains four linear, branched and cyclic molecules [START_REF] Mati | The oxidation of a diesel fuel at 1-10 atm: Experimental study in a JSR and detailed chemical kinetic modelling[END_REF] that account for all the chemical families involved in a real diesel fuel used in engines. Another representative surrogate (DF2) is proposed based on the analysis of a commercial diesel fuel [START_REF] Guibet | Carburants et Moteurs[END_REF]. A single molecule represents each chemical family; C 10 isomers are considered as the most representative molecular weights. One can note that for a conventional diesel fuel, H/C is in the range of 1.9-2.1 [START_REF] Guibet | Carburants et Moteurs[END_REF]. A last blend (DF3) represents a non conventional gasoil called Light Cycle Oil (LCO), that is rich in aromatic and polycyclic molecules [START_REF] Molière | Gas Turbines in alternative fuel applications: The utilization of highly aromatic fuels in power generation[END_REF]. In this case, C 14 isomers are considered since the mean molecular weight is higher for this type of fuel. Compositions of Natural Gas fuels ("NG")

The composition of commercial natural gas depends on the production field and actually from the particular extraction well it stems from. In particular, NGs can contain different amounts of hydrocarbons heavier than methane that impact on their LHV. Two NG compositions have been considered that correspond to pure methane gas and to a "rich gas" [START_REF] Baukal | The John Zink Combustion Handbook[END_REF] respectively. They are displayed in Table 4.

CALCULATION METHOD Thermochemical data

The thermochemical data used in the calculation of the adiabatic flame temperature of the different fuels have been computed by using the software THERGAS [START_REF] Muller | THERGAS: a computer program for the evaluation of thermochemical data of molecules and free radicals in the gas phase[END_REF] and based on the group additivity method proposed by Benson [START_REF] Benson | Thermochemical kinetics[END_REF].

The data used herein for the group contributions are mostly those proposed by Domalski and Hearing [START_REF] Domalski | Estimation of the Thermodynamic Properties of C-H-N-O-S-Halogen Compounds at 298.15 K[END_REF]. Since most of the natural unsaturated fatty acids have a cis conformation of their double bond, all FAMEs have been considered as cis isomers (a cis double-bond induces an enthalpy of 1 kcal mol -1 higher than a trans double bond which is quite insignificant). Table 5 summarizes the standard data in gas phase calculated at 298 K for species involved in the surrogate fuels studied here.

Flame temperature calculation

The adiabatic flame temperature (Tf) at constant pressure is evaluated by calculating the temperature at which the enthalpy of the combusted air/fuel mixture is minimal, air and fuel being considered as perfect gases. The calculation is performed using the software GASEQ developed by C.

Morley [START_REF] Morley | Chemical equilibria in perfect gases[END_REF]. The method is based on the minimization of the Gibbs Free Energy G of the mixture at constant pressure by fixing a set of reaction products. The temperature is adjusted iteratively until the equilibrium composition has an enthalpy which is the same as that of the reactants. -Case 1: a standard case considering a stoichiometric mixture of combustible in air, at 300 K under 1 atm. These conditions are far from gas turbine conditions but comparable with data from other sources.

-Case 2: a case representative of an F-class gas turbine, i.e. a stoichiometric mixture under a temperature of 400°C (673 K) and a pressure of 16 atm.

-Case 3: a standard gas turbine case, representative of an E-class, i.e. a stoichiometric mixture, under a temperature of 350°C (623 K) and a pressure of 11.5 atm. Note that these calculations does not allow any prediction of the NOx emission in a real gas turbine, but only of a trend when substituting a fuel to another.

RESULTS AND DISCUSSION 1. Parametric Study of T f

Figure 1 displays the adiabatic flame temperature calculated in Case 1 for molecules of different chemical families as a function of the H/C ratio. Focusing first on diesel fuels, species containing between 10 and 20 carbon atoms were studied, plus benzene. Aliphatic alkanes, alkenes, dienes, and trienes were investigated, such as alkylcyclohexanes, alkylbenzenes, and alkylnaphthalene as components of conventional gasoils. In the case of FAMEs, temperatures were calculated for linear saturated methyl esters, and for those containing one, two, and three double bonds. The results show that T f increased when unsaturated bonds were present in the molecule, while the H/C ratio decreased. T f rises from alkanes to trienes, and from saturated FAMEs to FAMEs containing three double bonds. A key result of these calculations is that FAMEs have always lower T f data than the corresponding hydrocarbons (carbon chain without an ester function) because the oxygenated function in which the carbon atom is already oxidized decreases the enthalpy of reaction (see arrows in figure 1). In other terms, the presence of the ester function, in which a carbon atom is already bound to two oxygen atoms within a carboxylic group, adds virtually nothing to the heat of the combustion and a methyl ester involving N carbon atoms has about the same thermal effect (and the same stoichiometric air consumption) as a similar hydrocarbon involving (N-1) carbon atoms. This equals to say that a (1/N) fraction of the carbon content of the FAME molecule is already oxidized and is "lost" for the generation of heat. At the same time, the CO 2 molecule generated by this carboxylic group tends to decrease T f as it consumes sensible heat among the reaction products. This is the key explanation of the systematically lower T f data of FAMEs as compared with gasoil of equivalent composition. FAMEs with aromatic chains have not been modeled in this work but would have shown the same effect; such FAMEs are not representative of normal biodiesels, except for those deriving from used frying oil (yellow greases) and having undergone strong pyrolysis. Reversely, an increasing proportion of aromatic species in the diesel fuel will dramatically increase the flame temperature. However, natural gas fuels have always lower T f data even when they contain significant amounts of C 2+ alkanes.

T f data of the fuel selection

Table 6 gives the calculated values of the adiabatic flame temperatures (T f ) for the complex mixtures studied as surrogates of the real fuels: the 5 FAMEs, the 3 DFs and the 2 NGs. a) NG fuels: Natural gas fuels lead to lower T f values than diesel fuels as their higher H/C ratio implies a lower enthalpy of combustion than that of heavier molecules. NG2 contains alkanes heavier than methane and leads then to higher T f values than NG1. b) DF fuels: The 3 diesel fuel compositions lead to close values of Tf. DF2, the second surrogate for commercial diesel fuel, contains a little bit more of diaromatic molecules than DF1 and leads to slightly higher temperatures. DF3 represent a heavier blend such as LCO and is rich in aromatics and polycyclic molecules. It leads to the highest temperature and induces the worse NOx index. c) BF fuels: Biodiesel fuels lead to adiabatic flame temperature data that are higher than those for NG and close to those of DF. This outcome is consistent with calculations performed for single molecules and with the recent results procured by the above-mentioned field tests. As compared with DFs, usual biodiesel fuels lead to equal or slightly lower Tf values when comparing to diesel fuels. The five biofuels have a very similar behavior. In particular, there is a limited difference between FAMEs obtained from rapeseed (BF1) and soybean (BF3). A larger amount of unsaturated bonds increases however slightly the enthalpy of combustion. BF2 and BF3 contain the largest amounts of insaturated bonds and lead to the highest Tf,ad values while BF4 (palm oil) and BF5 (tallow) lead to smaller Tf,ad values due to their higher content of saturates (C 16 and C 18 ). In summary, FAMEs have a level of LHV equal or lower than conventional diesel fuels and, within the FAME family, the LHV of BF4 and BF5 have the lowest ones; T f follows the same trend as the LHV.

The overall results obtained for realistic mixtures contradict those of Ban-Weiss et al. [START_REF] Ban-Weiss | A numerical investigation into the anomalous slight NO x increase when burning biodiesel: a new (old) theory[END_REF], who proposed higher adiabatic flame temperatures for biofuels than for conventional diesel fuel but compared in fact small FAMEs to small alkanes, without considering the large amount of unsaturated species contained in a diesel fuel, especially aromatics and naphthenoaromatics. On the other hand, our results are in accordance with those of Nabi and al. calculated from the elementary analyses of a diesel fuel and a neem oil [START_REF] Nabi | Improvement of engine emissions with conventional diesel fuel and diesel-biodiesel blends[END_REF].

CONCLUSIONS

This study was intended to clarify the question of how biodiesel impacts on the NOx emission of gas. The adiabatic flame temperature T f represents the highest temperature that a combustion can produce for a specific mixture whatever the combustion device. Thus, T f can be considered as the major determinant of NOx emissions in gas turbine and has been taken as criterion.

Calculations of Tf for various types of molecules contained in biodiesels, gasoil and natural gas and for surrogate fuel mixtures allowed to rank these 3 fuel classes and to point out interesting effects each fuel class. The results show that diesel fuels tend to generate the highest temperatures, natural gas the lowest and biodiesel lie inbetween. The variability of the composition of gasoils can substantially change flame temperature, while biofuels are less sensitive to composition variations. This ranking complies with the finding of two recent field tests that show, for gas turbines, a slight decrease of NOx when passing from gasoil to biodiesel. They highlight a main difference between gas turbines and diesel engines for which the trend is reversely an increase of NOx when switching from petroleum diesel to biodiesel. The calculations performed with variable FAME distributions show no significant effect on the exothermicity of the combustion and thus on NOx. This result is interesting as it tends to predict an even behavior of biodiesels worldwide. It must be stressed that the conclusions of this approach are valid for invariant combustion conditions (constant geometry and air distribution in combustion chambers; unchanged fuel nozzle sizes and atomization conditions; prompt and organic NOx neglected). Finally OEM's shall be consulted for an authorized statement on actual emissions.

Figure 1 :

 1 Figure 1: Adiabatic flame temperatures of chemical species contained in fuels as a function of H/C ratio These calculations allowed to compare the range of temperatures that are reached by the various chemical families to quantify the influence of the composition on the flame temperature and then on the NOx index. For each surrogate mixture defined above, three calculations have been performed:

  Alkylbenzenes and alkylnaphthalenes lead to high T f values and low H/C ratio. The T f data of naphthenes (here: alkylcyclohexanes) are not sensitive to the size of the alkyl group and remain close to 2281 K, with a H/C ratio of 2. It is worth noting that all lines are converging toward the point (T f = 2281 K; H/C = 2) when the carbon atom number increases since C n H 2n is the limit formula for all studied chemical families when the alkyl group grows in length i.e. when n becomes infinite, whereby overwhelming the weight of the other chemical functions. This limit case that corresponds to H/C = 2 gives T = 2281 K. Methane and other components of natural gas were not represented in the figure to avoid wider scales. Methane leads to T f = 2226 K and H/C = 4, ethane to 2259 K and 3, respectively.

Figure 2 :

 2 Figure 2: Range of adiabatic flame temperature of fuels Real fuels involve different proportions of each classes, with an average H/C ratio of 1.8.Figure 2 summarizes the temperature range reached by the components of the different fuels of interest in gas turbines, i.e. diesel fuels (DFs), biofuels (FAMEs), and natural gas (NGs). The darker area delimits the T f range of what can be considered the most usual composition range of each fuel class: alkylbenzenes for DFs, unsaturated methyl esters with two double bonds for FAME's and methane for NG. It appears that the flame temperature of DFs and FAMEs are close even though DF tend to have higher values. The composition of fuels is very sensitive: a gasoil rich in alkanes and naphtenes could have a lower flame temperature than a very unsaturated biofuel.Reversely, an increasing proportion of aromatic species in the diesel fuel will dramatically increase the flame temperature. However, natural gas fuels have always lower T f data even when they contain significant amounts of C 2+ alkanes.

Figure 2

 2 summarizes the temperature range reached by the components of the different fuels of interest in gas turbines, i.e. diesel fuels (DFs), biofuels (FAMEs), and natural gas (NGs). The darker area delimits the T f range of what can be considered the most usual composition range of each fuel class: alkylbenzenes for DFs, unsaturated methyl esters with two double bonds for FAME's and methane for NG. It appears that the flame temperature of DFs and FAMEs are close even though DF tend to have higher values. The composition of fuels is very sensitive: a gasoil rich in alkanes and naphtenes could have a lower flame temperature than a very unsaturated biofuel.

Table 1 :

 1 Typical composition of fatty acids in VOs (wt%)[START_REF] Ballerini | Les biocarburants, état des lieux, perspectives & enjeux du développement[END_REF].

		VO1 Rapeseed	VO2 Sunflower	VO3 Soybean	VO4 Palm
	C14:0				1
	C16:0	5	6	10	44
	C18:0	2.5	5	4	6
	C18:1	59	18	23	38
	C18:2	21	69	53	10
	C18:3	9	<0.5	8	<0.5
	C20:0	<0.5	<0.5	<0.5	<0.5
	C20:1	1	<0.5	<0.5
	C22:0	<0.5	1	<0.5
	C22:1	1

Table 2 :

 2 Compositions of the 5 selected biodiesels (% mol) (BF1 to 4 derive from the VO1 to 4 of table 1; BF5 from tallow )

	Name myristic acid palmitic acid stearic acid oleic acid linoleic acid linolenic acid arachidic acid gadoleic acid behenic acid erucic acid	Formula C14:0 C16:0 C18:0 C18:1, ∆9 C18:2, ∆9,12 C18:3, ∆9, 12, 15 C20:0 C20:1, ∆9 C22:0 C22:1, ∆13	BF 1 Rapeseed 0 5 3 59 21 9 0.5 1 0.5 1	BF 2 Sunflower 0 6 5 18 69 1 0.5 0.5 0 0	BF 3 Soybean 0 10 4 23 53 8 1 0.5 0.5 0	BF 4 Palm 1 44 6 38 10 0.5 0.5 0 0 0	BF 5 Tallow 4 27 24 40 3 0.5 0.5 1 0 0

Table 3 :

 3 Composition of the synthetic diesel fuels (mol%)

	Name	Formula DF 1 DF 2 DF 3
	n-decane	n-C10H22	31
	n-tetradecane	n-C14H30	14
	n-hexadecane	n-C16H34 23.5
	n-propyl (PCYH) cyclohexane	C9H18	26.9
	n-butyl cyclohexane (BCYH)	C10H20	39
	n-propyl benzene (PBZ)	C9H12	22.9
	n-butyl benzene (BBZ)	C10H14	16
	n-octyl benzene (OBZ)	C14H22	29
	naphthalene (NP)	C10H8	7.7
	α-methyl naphthalene (MNP)	C11H10	14
	α-butyl naphthalene (BNP)	C14H18	55
	H/C ratio		1.86 1.80 1.51

Table 5 :

 5 Ideal gas phase thermodynamics properties

		∆	f H	o 298	K	is expressed in kcal.mol -1 and o 298 S	K	and	) C p °are given in cal.mol -1 .K -1 . T (
																C p °	(T	)
	Species	∆	f H	o 298	K	S	o 298	K	300 K	400 K		500 K	600 K	800 K	1000 K	1500 K
	N2				0.00	45.77	6.95	7.01		7.08	7.19	7.50	7.83	8.32
	CH4			-17.92	44.52	8.49	9.81		11.15	12.50	15.13	17.43	21.07
	C2H6			-20.29	54.85	12.59	15.76		18.67	21.31	25.80	29.37	36.24
	C3H8			-24.88	64.51	17.64	22.60		26.99	30.85	37.10	41.83	49.95
	C4H10			-30.23	74.12	23.36	29.69		35.31	40.25	48.24	54.24	64.51
	n-C10H22			-59.81 129.96	56.05	71.35		84.71	96.29 114.75 128.16 149.00
	n-C14H20			-79.62 167.35	77.94	99.18 117.69 133.68 159.11 177.51 205.99
	n-C16H34			-89.49 186.10	88.88 113.06 134.14 152.35 181.27 202.13 234.41
	PCYH			-45.40	99.73	43.36	58.44		71.56	82.89 100.78 113.48 131.56
	BCYH			-50.35 109.07	48.86	65.38		79.80	92.25 111.89 125.81 145.82
	PBZ				1.91	95.61	36.28	47.53		57.18	65.38	78.03	86.74	99.77
	BBZ			-3.38 104.76	42.07	54.85		65.81	75.13	89.50	99.43 114.54
	OBZ			-22.83 142.37	63.65	82.30		98.39 112.12 133.48 148.40 170.97
	NP			35.91	82.35	31.99	42.94		52.06	59.58	70.73	78.26	93.19
	MNP			28.24	92.58	37.14	49.34		59.65	68.27	81.32	90.25 105.91
	BNP			13.13 119.41	54.44	70.57		84.59	96.59 114.95 126.67 133.62
	C14:0		-157.81 192.42	87.49 109.90 129.63 146.74 173.56 191.28 201.94
	C16:0		-167.70 211.13	98.44 123.81 146.10 165.43 195.74 215.94 230.42
	C18:0		-177.60 229.85 109.37 137.73 162.59 184.12 217.90 240.60 258.92
	C18:1		-149.48 226.47 104.94 132.69 157.02 178.07 210.95 232.85 249.88
	C18:2		-120.84 224.25 101.97 128.98 152.66 173.14 205.10 226.31 241.97
	C18:3			-92.20 222.02	99.00 125.28 148.30 168.20 199.25 219.78 234.02
	C20:0		-187.49 248.56 120.34 151.59 179.04 202.82 240.10 265.22 287.41
	C20:1		-159.38 245.18 115.87 146.60 173.50 196.75 233.12 257.52 278.38
	C22:0		-197.38 267.27 131.27 165.52 195.53 221.50 262.26 289.90 315.91
	C22:1		-169.27 263.90 126.82 160.51 189.98 215.44 255.29 282.17 306.87

Table 4 :

 4 Composition of the natural gas fuels (mol%)

	Formula CH4 C2H6 C3H8 C4H10 N2	NG 1 Alaska 100	NG 2 Libya 70 15 10 4 1