Stochastic proof of upper bound for the heat kernel coupled with geometric flow, and Ricci flow

We give a proof of Gaussian upper bound for the heat kernel coupled with the Ricci ow. Previous proofs by Lei Ni [5] use Harnack inequality and doubling volume property, also the recent proof by Zhang and Cao [6] uses Sobolev type inequality that is conserved along Ricci ow. We will use a horizontal coupling of curve [1] Arnaudon Thalmaier, C., in order to generalize Harnack inequality with power -for inhomogeneous heat equation - introduced by F.Y Wang. In the case of Ricci ow, we will derive on-diagonal bound of the Heat kernel along Ricci ow (and also for the usual Heat kernel on complete Manifold).


Coupling and Harnack inequality with power
In the first part of this section, we will focus on the operator of the type L t := 1 2 ∆ g(t) , where ∆ g(t) is the Laplace operator associated to a time dependent family of metrics g(t). We will suppose that all considered g(t)-Brownian motion is non explosive. For example when the family of metric comes from the backward Ricci flow, this have been proved in [7]. Let γ(t) be a C 1 (M ) geodesic curve such that γ(0) = x and γ(1) = y and X t (u) be the the horizontal L(t)-diffusion C 1 path space in C 1 ([0, T ], M ) over X 0 , started at u → γ( u T ). Let X t (x) be a g(t)-Brownian motion that start at x, // 0,t the g(t) parallel transport, and W t the damped parallel transport that satisfies the following Stratonovich covariant equation: * d((// 0,t ) −1 (W 0,t )) = − 1 2 (// 0,t ) −1 (Ric g(t) −∂ t (g(t))) #g(t) (W 0,t ) dt Proposition 1.1. The process X t (γ( t T )) satisfies the following stochastic differential equation : Proof. We pass to the Stratonovich differential and obtain the following chain rule formula: * d(X . (γ( . T ))) t 0 = * d(X . (γ( t 0 T ))) t 0 + dX t 0 ( t T ) dt t=t 0 dt 0 .
We use Theorem 3.1 in [1] to identify the last term of the right hand side: Now we come back to the Itô differential equation using the following relation: // −1 s * dY s ), and we obtain We use again Theorem 3.1 in [1] to identify In many situations the Novikov's criterion will be satisfied, so we could expect R t to be a martingale. Define a new probability measure as : Q := R T P.
Proposition 1.2. Suppose that Novikov's criterion is satisfied for N t . Then under Q, the process X t (γ( t T ) is a L t -diffusion that starts at x, that finishes at X T (y).
Proof. One could directly apply Girsanov's theorem. We prefer here to give a direct proof. Let f ∈ C 2 (M, R). Recall that R t is a P-martingale, and P t,Xt(.) 0, t T is an isometry for the metric g(t). Use Itô formula to compute : where M P t is a martingale for P. Also ) is a L t diffusion under the probability Q. It is clear that it finishes at X T (y), so X t (γ( t T ))) can be seen as a coupling between two L t diffusions that started at different point up to one change of probability.
Let α i,j (t) be a family of symmetric 2-tensors on M . We will consider the following heat equation coupled with a geometric flow. • The most famous case is when α i,j (t) := 0, this is the case of constant metric and equation 1.1 is the usual heat equation in M .
is the second fundamental form relatively to the metric g(t), and h is the mean curvature, when the family of metric comes from the mean curvature flow. In all these cases a notion of g(t)-Brownian motion, i.e. a ∆ t diffusion, parallel transport, and damped parallel transport has been given in [3,2].
Let T c be the maximal life time of geometric flow g(t) t∈[0,Tc[ . For all T < T c , let X T t be a g(T − t)-Brownian motion and // T 0,t the associated parallel transport. In this case, for a solution f (t, .) of (1.1), f (T − t, X T t (x)) is a local martingale for any x ∈ M . So the following representation holds for the solution : We introduced a further subscript T referring to the fact that a time reversal step is involved.
Let W T 0,t be the damped parallel transport along the g(T − t)-Brownian motion.Let us recall the covariant differential equation satisfied by this damped parallel transport [3] : All the over subscript T we will mean that the family of metrics is g(T −t). Proposition 1.4. Suppose that there exist α, α ≥ 0 and K, K ≥ 0 such that : −Kg(t) ≤ Ric(t) ≤ Kg(t), then the g(t)-Brownian motion, and the g(T − t)-Brownian motion does not explode before the time T c .
Proof. This is a sufficient condition but it is far from being a necessary one, for the process to don't explode. Use the Itô formula for d t (x 0 , X t ), the comparison theorem of the laplacian of the distance function, and the comparison theorem of stochastic differential equation.
Remark 1.5. For the backward Ricci flow, the g(t)-Brownian motion does not explode [7], but the condition of the sufficient existence of the Ricci flow in complete Riemannian manifolds as given by Shi [8] theorem 1.1, that is the boundedness of the initial Riemannian tensor (for the metric g(0)) also gives a bound of the Ric tensor along the flow (for bounded time). So the conditions for non explosion of the g(t)-Brownian motion given in the above proposition will be satisfied if the initial metric satisfy Shi condition for the complete manifolds. Proposition 1.6. Suppose that there exist C ∈ R such that in a matrix sens: Then R t is a martingale, and for β ≥ 1 Moreover suppose that there existC ∈ R such that in a matrix sens: then R T t is a martingale and for β ≥ 1 Proof. Let v ∈ T x M . Then we use the isometry property of the parallel transport i.e. // s : By Gronwall's lemma we get So by Nokinov's criterion, R t is a martingale. Let β ≥ 1, By the same computation we have Thus R T t is a martingale. Given β ≥ 1 we have similarly, does not explode (we do not need proposition 1.4, but [7]) and the condition of the above proposition is satisfied withC = 0 and We are now ready to give the Harnack inequality with power. Let f be a solution of (1.1) and let P 0,T be the inhomogeneous heat kernel associated to (1.1), i.e.
. Theorem 1.8. Suppose that the g(T − t)-Brownian motion X T t does not explode, and that the process R T t is a martingale. Then : for all α > 1 and f 0 ∈ C b (M ) we have : . Moreover if there existsC ∈ R such that in a matrix sens: then we have: Proof. We writeX T t := X T t (γ( t T )), and use 1.2, and Holder inequality.
. The last part in the theorem is an application of proposition 1.6.
We will denote by µ t the volume measure associated to the metric g(t), and for A a Borelian, V t (A) := A 1 dµ t , and B t (x, r) the ball for the metric g(t) of center x and radius r. . Corollary 1.9. Suppose that the hypothesis of theorem 1.8 is satisfied, and that there existC ∈ R such that Ric g(t) +α(t) ≥Cg(t). Let f 0 ∈ L α (µ 0 ). Moreover suppose that there exists a function τ : [0, T ] → R such that : Proof.
we apply theorem 1.8 and get : We integrate both sides along the ball B T x, , with respect to the measure µ t , in y and obtain : We have that d dt µ t (y) = 1 2 trace g(t) (α(t))(y)dµ t (y), and by Stokes theorem we have : We deduce that : .
and we conclude by a classical density argument that the same inequality is true forf 0 ∈ L α (µ 0 ) .
After a time reversal, P * (y, t − s, x, t) satisfies the following heat equation : Suppose that there exist functions τ (t) and τ (t)such that : By Feynman Kac formula, we conclude that : s 0 τ (t−s+u)du P (y, s, x, t).
We now fix t. Let P (y, s, x, t) be the fundamental solution, defined by : We have in particular: Theorem 2.1. Suppose that the g( t 2 + s)-Brownian motion does not explode before the time t 2 and that there exists C ∈ R such that ∀s ∈ [0, t 2 ]: Ric g(s) −α(s) ≥ Cg(s).
Assume that g(t − s)-Brownian motion does not explode before the time t 2 and that there existC ∈ R such that ∀s ∈ [0, t 2 ]: Then the fundamental solution of (1.1) that we note P (x, t, y, 0) satisfies : )) 1 2 Proof. By the Chapman Kolmogorov formula we have: Recall that P (x, t 2 + s, z, t 2 ) is the fundamental solution, that starts at δ x at time s = 0, of : Then we have : According to the proof of corollary 1.9, we get that for f 0 ∈ C b (M ) ∩ L 2 (µ t 2 ) : . Given x 0 ∈ M and n ∈ N, we apply the above inequality to f 0 (y) := P (x, t, y, t 2 ) ∧ (nII B(x 0 ,n) (y)) to obtain : Letting n goes to infinity, we obtain that the heat kernel is twice integrable, and that: Recall that: where P (y, t 2 , x, t 2 ) is the heat kernel at time t 2 , that start at time 0 at δ y , of the following equation: We also have: Here the family of metric is s → g( t 2 − s), so many changes of sign are involved. However, the proof of the following is the same as the proof of corollary 1.9. We get for Similarly we can show the square integrability of the kernel and the following inequality: We obtain : Remark 2.4. For the Ricci flow, in dimension 3, there is a result of Hamilton that says : if the Ricci curvature is non negative at time 0, it is still non negative for all time before the critical time. In arbitrary dimension n, if we suppose that the Ricci curvature is positive at all times, we have the following one diagonal estimate (use the above theorem 2.1 with C = 0 and C = 0) : Recall that in the case of Ricci flow : τ (s) = − 1 2 inf M R(s, .) and τ (s) = − 1 2 sup M R(s, .) so we have :

Grigor'yan tricks, one diagonal estimate to Gaussian estimate, the Ricci flow case
In this section we use the one diagonal estimate of the previous section to derive a Gaussian type estimate of the heat kernel coupled with Ricci flow (for complete manifold with non negative Ricci curvature) . The proof involves in several steps. In particular, we use a modification of Grigor'yan tricks to control exponential integrability of the square of the heat kernel, combined to an adapted version of Hamilton entropy estimate to control the difference of the heat kernel at two points. This type of strategy, is a modification of different arguments which appears in the literature on the Ricci flow ( Hamilton, Lei Ni , Cao-Zhang). Unfortunately, we have not been able to cover the case of general manifold (without assumption of non negativity of the Ricci curvature).
We start this section by the following entropy estimate.
Proof. By the homogeneity of the desired inequality under multiplication by a constant, and the homogeneity of the heat equation under the same operation, we can suppose that f > 1, in the proof.
Using an orthonormal frame and Weitzenbock formula, we have the following identity: Thus, in the case of Ricci flow we get : By a direct computation we have : Let where X t s (x) is a g(t − s)-Brownian motion started at x. If h(s) := t/2−s 2 then by Itô formula, it is easy to see that N s is a super-martingale. So we have : Lemma 3.2. Suppose that the family of metrics g(t) comes from the Ricci flow, and let B be a Borelian in M . Then: Moreover if Ric g(.) ≥ 0 then for all x, y ∈ M and r > 0 we have : Proof. A simple computation computation shows that: In the case of a Ricci flow this becomes d dt µ t (dx) = − 1 2 R(x, t)µ t (dx). Thus, the first inequality of the lemma follows from a direct integration. For the second point, it's clear that Ric ≥ 0 yields that d t (x, y) is non increasing in time. So B 0 (x, r) ⊂ B t (x, r), which clearly gives The above lemma immediately yields the following remark. Remark 3.3. If Ric g(.) ≥ 0 , and(g)(t) = − Ric g(t) then we have: Proposition 3.4. Let g(t) be a solution of Ricci flow such that Ric g (t) ≥ 0, and r > 0 , t 0 > t ≥ 0, A ≥ 1. Let also : and Λ(t) = t 0 inf x∈M (R(s, x))ds. Then for f (t, x) a solution of (1.1) we have for t 2 < t 1 < t 0 : M f 2 (t 1 , y)e ξ(y,t 1 ) µ t 1 (dy) ≤ e −(Λ(t 1 )−Λ(t 2 )) M f 2 (t 2 , y)e ξ(y,t 2 ) µ t 2 (dy).
Proof. By direct computation and using intensively that Ric g(t) ≥ 0. Let Proposition 3.5. Under the same assumptions as in the above proposition, and if, ρ < r we have: Proof. .
Then remark that the definition of I r (t) is independent of t 0 and of the corresponding ξ, so we can pass to the limit when t 0 ց t 1 in the following : to obtain the desired result.
We apply the above proposition to the heat kernel P (x, t, y, 0) of the equation (2.1) that also satisfy (1.1).
Theorem 3.6. Ifġ(t) = −Ric g(t) and Ric g(t) ≥ 0, and the following technical assumption is satisfied for the initial manifold • H1 : if M is not compact, we suppose that there exists a uniform constant c n > 0 such that V ol(B g(0) (x, r)) ≥ c n r n (that is a non collapsing condition ) • H2 : all the curvature tensor is bounded for the metric (M, g(0).
Then for all a > 1 there exist two positive explicit constants q a , m a depending only on a and on the dimension, such that we have the following heat kernel estimate : Proof. Let f (t, x) := P (x, t, y, 0) be the heat kernel of (1.1) that is the solution of equation (2.1). Then we have by the proof of theorem 2.1: .
Let a > 1 be a constant. Following Gregorian we define : r k := ( 1 2 + t k+2 )r and t k := 1 a k . Thus proposition 3.5 can be applied to r k+1 < r k and t k+1 < t k , yielding to the same estimate as before : .
Applying recursively this inequality, we have for all k : (3.1) where in the last inequality we use that Ric g(t) ≥ 0 so R(x, t) ≥ 0.
We have lim k−→∞ I r k (t k ) = 0, by Levis asymptotic of heat kernel. This can be seen, using a probabilistic argument: use that for small t, P x (τ r < t) ≤ Ce − r 2t , where τ r := inf t > 0, d t (X t (x), x) = r, and (M \Bt(x,r)) ≤ cst e − r 2t t n 2 and the right hand side goes to 0 when t goes to 0. (we used H1 to get a global bound of the heat kernel i.e. remark 3.3). So we can pass to the limit when k goes to infinity in equation (3.1) to get : . Also By Bishop-Gromov theorem in the case Ric ≥ 0 we have Iterating the above inequality we get : So we have : There exists a constant m (a,A) such that a i+1 A(a−1)(i+3) 4 ≥ m (a,A) (i + 2), and thus we get :  So there exists z 0 ∈ B t (y 0 , t 4 ) such that: We denote q a a constant that depends only of the parameter a and the dimensions, that possibly changes from line by line. By the above lemma (comparison of volume) we have : P (z 0 , t, x 0 , 0) ≤ q a ψ(t) e