BMO Martingales and Positive Solutions of Heat Equations

In this paper, we develop a new approach to establish gradient estimates for positive solutions to the heat equation of elliptic or subelliptic operators on Euclidean spaces or on Riemannian manifolds. More precisely, we give some estimates of the gradient of logarithm of a positive solution via the uniform bound of the logarithm of the solution. Moreover, we give a generalized version of Li-Yau's estimate. Our proof is based on the link between PDE and quadratic BSDE. Our method might be useful to study some (nonlinear) PDEs.


Introduction
In this article, we study positive solutions u of a linear parabolic equation where M is either the Euclidean space R n and L is an elliptic or sub-elliptic operator of secondorder L = 1 2 m α=1 A 2 α + A 0 , {A 0 , · · · , A m } is a family of vector fields on R n , or M is a complete manifold of dimension n with Riemannian metric (g ij ), and 2L is the Laplace-Beltrami operator where g denotes the determinate of (g ij ) and (g ij ) is the inverse of the matrix (g ij ).
The well-posedness and the regularity theory for (1.1) are parts of the classical theory in partial differential equations, see [18], [13] and [20] for details. On the other hand, it remains an interesting question to devise precise estimates of a solution u in terms of the (geometric) structures of (1.1). There is already a large number of papers devoted to this question. Among many interesting results, let us cite two of them which are most relevant to the present paper. The first result is a classical result under the name of semigroup domination, first discovered by Donnelly and Li [8], which says that if the Ricci curvature is bounded from below by C, then |∇P t u 0 | ≤ e −Ct P t |∇u 0 | for u 0 ∈ C 1 b (M ) (1.2) where (P t ) t≥0 is the heat semigroup on M , so that the left-hand side is the norm of the gradient of u(t, ·) = P t u 0 a solution to the heat equation with initial data u(0, ·) = u 0 , while the right-hand side P t |∇u 0 | is a solution of (1.3) with initial data |∇u 0 |. The second result is Li-Yau's estimate first established in [19]. If the Ricci curvature is non-negative, and if u is a positive solution of (1.3) then |∇ log u| 2 − 2 ∂ ∂t log u ≤ n t for t > 0 . (1.4) In fact, in the same paper [19], Li and Yau also obtained a gradient estimate for positive solutions in terms of the dimension and a lower bound (which may be negative) of the Ricci curvature, though less precise. Their estimates in negative case have been improved over the years, see for example [25], [26], [2] and [1].
In this paper we prove several new gradient estimates for the positive solutions of (1.1).
Theorem 1.1 Let M be a complete manifold with non-negative Ricci curvature. Suppose u is a positive solution of (1.3) with initial data u 0 > 0, then where || · || ∞ denotes the L ∞ norm on M .
Indeed we will establish a similar estimate for the heat equation with a sub-elliptic operator, under similar curvature conditions, and indeed we will establish a gradient estimate for a complete manifold whose Ricci curvature is bounded from below. We should mention that the kind of estimates such as (1.5) is known in the literature, and it is known that for some constant C > 0 which depends on the manifold M in a complicated way. The interesting feature in our estimate (1.5) is that there is no unknown constant, and (1.5) does not involve the dimension.
Theorem 1.2 Let M be a complete manifold of dimension n with non-negative Ricci curvature. Suppose u is non-negative solution to the heat equation of (1.3) with initial data u 0 > 0. .
By setting C = ∞ in 1) we recover Li-Yau's estimate (1.4). The novelty of the present paper is not so much about the new gradient estimates in Theorem 1.1 and Theorem 1.2, though we believe that they are not known previously even in the case that the underlying manifold is Euclidean space, what is interesting of the present work is the approach we are going to develop in order to discover and prove these gradient estimates. Our approach brings together with the martingale analysis to the study of a class of non-linear PDEs with quadratic growth. Of course the connection between the harmonic analysis, potential theory and martingales is not new, which indeed has a long tradition, standard books may be mentioned in this aspect, such as [9], [10], [11] and etc., what is new in our study is an interesting connection between the BMO martingales and positive solutions of the heat equation (1.3).
To take into account of the positivity, it is better to consider the Hopf transformation of a positive solution u to (1.3), i.e. f = log u, then f itself solves a parabolic equation with quadratic non-linear term, namely The preceding equation (1.6) is an archetypical example of a kind of semi-linear parabolic equations with quadratic growth which has attracted much attention recently associated with backward stochastic differential equations, for example Kobylanski [16], Briand-Hu [6], Delbaen et al. [7] and etc.
The main idea may be described as the following. Suppose f is a smooth solution of the nonlinear equation (1.6), and X t = B t + x where B is a standard Brownian motion on a complete probability space.
, ∇ i is the covariant derivative written in a local orthonormal coordinate system. Then, Itô's lemma applying to f and X may be written as On the other hand, it was a remarkable discovery by Bismut [4] (for a special linear case) and Pardoux-Peng [23] that given the terminal random variable Y T ∈ L 2 (Ω, F T , P), there is actually a unique pair (Y, Z) where Y is a continuous semimartingale and Z is a predictable process which satisfies (1.7). The actual knowledge that Z is the gradient of Y may be restored if Y T = f 0 (X T ). The backward stochastic differential equation (1.7) with a bounded random terminal Y T , which has a non-linear term of quadratic growth and thus is not covered by Pardoux-Peng [23], was resolved by Kobylanski [16]. Observe that the martingale part of Y is the Itô integral of Z against Brownian motion B (which is denoted by Z.B). It can be shown that, if Y is bounded, then Z.B is a BMO martingale up to time T , so that the exponential martingale is a uniformly integrable martingale (up to time T ), as long as h is global Lipschitz continuous. The main technical step in our approach is that, due to the special feature of our non-linear term in (1.6), we can choose h i (z) = z i (one has to go through the detailed computations below to see why this choice of h i is a good one), and making change of probability measure to Q by dQ dP = E(h(Z).B) T , then, under Q, not only Z.B is again a BMO martingale (whereB is the martingale part of B under the new probability Q), but also t → |Z t | 2 is a non-negative submartingale. Next by utilizing the BSDE (1.7), we can see the BMO norm of Z.B under Q is dominated at most 2 ||Y || ∞ , that is Finally the sub-martingale property of |Z t | 2 allows to move |Z s | 2 (for s ∈ (t, T )) out from the time integral on the left-hand side of the previous inequality, which in turn yields the gradient estimate.
Let us now give a heuristic probabilistic proof to Theorem 1.2 to explain from where such estimates come from. Let f = log u, and G = −∆f . Then one can show that and H ≥ 0. We suppose here G > 0. Consider the BSDE: Then then (U, V ) satisfies the following quadratic BSDE: Using BMO martingale techniques, one can prove that there exists a new probability measure Q under whichB t = B t + t 0

Vs
Us ds is a Brownian motion. Hence from which we deduce that U 0 = T n + E Q [U T ], and which yields the estimate in Theorem 1.2. Even though the above heuristic proof is probabilistic (which can be made rigorous), we prefer to give a pure analytic proof in the last section.
The paper is organized as follows. Next section is devoted to some basic facts about quadratic BSDEs including BMO martingales. Section 3 establishes the gradient estimates for some linear parabolic PDEs on Euclidean space, while Section 4 establishes these estimates on complete manifold. Last section is devoted to establish a generalized Li-Yau estimate via analytic tool.

BSDE and BMO martingales
Let us begin with an interesting result about BSDEs with quadratic growth. The kind of BSDEs we will deal with in this paper has the following form is the Brownian filtration associated with B, and F j are continuous function on R × R m with at most linear growth: there is a constant C 1 ≥ 0 such that According to Peng [24] and as we have seen in the Introduction, if u is a bounded smooth solution to the following non-linear parabolic equation The special feature of (2.2) is that the maximum principle applies, which implies that global solutions (here global means for large t) exist for the initial value problem of the system as long as the initial data is bounded (though, this constraint can be relaxed a bit, but for the simplicity we content ourself to the bounded initial data problem). The maximum principle implies that as long as u is a solution to (2.2) then |u(x, t)| ≤ ||u 0 || ∞ . Therefore, if the initial data u 0 is bounded, and F j are global Lipschitz, then, according to Theorem 6.1 on page 592, [18], u exists for all time, and both u and ∇u are bounded on R m × [0, T ]. The maximum principle for (2.1) however remains true even for a bounded random terminal value (so called non Markovian case), which in turn yields that the martingale part of Y is a BMO martingale. This is the context of the following Proposition 2.1 Suppose that ξ ∈ L ∞ (Ω, F T , P). There exists a unique solution (Y, Z) to (2.1) such that Y is bounded and M = Z.B is a square integrable martingale. Moreover M = Z.B is a BMO martingale up to time T , and Proof. The existence and uniqueness is already given in [16]. The fact that M = Z.B is a BMO martingale up to time T is proved in [22]. Then there exists a constant C 2 > 0 such that under the probability Q, whose solution is given by It particularly implies that ||Y t || ∞ ≤ ||ξ|| ∞ .

Stochastic flows and gradient estimates
Let A 0 , A 1 , · · · , A m be m+1 smooth vector fields on Euclidean space R n , where n is a non-negative integer. Then, we may form a sub-elliptic differential operator of second order in R n : here we add a factor 1 2 in order to save the constant √ 2 in front of Brownian motion which will appear frequently in computations in the remaining of the paper. Our goal is to devise an explicit gradient estimate for a (smooth) positive solution u of the heat equation by utilizing the BSDE associated with the Hopf transformation f = log u, which satisfies the semi-linear parabolic equation

Stochastic flow
The first ingredient in our approach is the theory of stochastic flows defined by the following stochastic differential equation where •d denotes the Stratonovich differential, developed by Baxendale [3], Bismut [5], Eells and Elworthy [12], Malliavin [21], Kunita [17] and etc. The reader may refer to Ikeda and Watanabe [14] for a definite account. To ensure the global existence of a stochastic flow, we require the following condition to be satisfied.
By writing (3.4) in terms of Itô's stochastic integrals, namely where (and thereafter) Einstein's summation convention has been used: repeated indices such as l is summed up from 1 up to n. The existence and uniqueness of a strong solution follow directly from the standard result in Itô's theory, which in turn determines a diffusion process in R n with the infinitesimal generator L.
In fact, more can be said about the unique strong solution, and important consequences are collected here which will be used later on. Suppose w = (w t ) is a standard Brownian motion (started at 0) with its Brownian filtration (F t ) t≥0 on the classical Wiener space (Ω, F, P) of dimension m, so that w = (w t ) t≥0 is the coordinate process on the space Ω of continuous paths in R m with initial zero. Then, there is a measurable mapping ϕ : R + × Ω × R n −→ R n and a probability null set N , which possess the following properties.
1. w → ϕ(t, w, x) is F t -measurable for t ≥ 0 and x ∈ R n , and ϕ(0, w, x) = x for every w ∈ Ω \ N and x ∈ R n .
is smooth and its inverse exists, and the inverse is also smooth.
4. The family {ϕ(t, ·, x) : t ≥ 0, x ∈ R n } is a stochastic flow: for all t, s ≥ 0, x ∈ R n and w ∈ Ω \ N , where θ s : Ω → Ω is the shift operator sending a path w to a path θ s w(t) = w(t + s) for t ≥ 0.

Let
∂x j for i, j ≤ n. Then J i j (0, w, x) = δ i j and J solves the following SDE and its inverse matrix In our computations below, we have to use Itô's integrals rather than Stratonovich's ones. Therefore we would like to rewrite (3.6, 3.7) in terms of Itô's differential, so that and

Structure assumptions
We introduce some technical assumptions on the structure of the Lie algebra generated by the family of vector fields {A 0 , A 1 , · · · , A m }, in addition to Condition 3.1. Recall that A α = A j α ∂ ∂x j , and A j α,β , A j α,β,γ etc. are the corresponding coefficients in Lie brackets

11)
Condition 3.2 There is a constant C 1 ≥ 0 such that for any ξ = (ξ i ) i≤n , θ β = (θ i,β ) i≤n ∈ R n (β = 1, · · · , m), it holds that m α,β=1 The above conditions are satisfied if A is elliptic. Let us now suppose the following Frobenius integrability condition: there exist some bounded smooth coefficients c l β,α (x), such that

The density processes Z α
Let us consider a smooth solution f to the following non-linear parabolic equation where h is a C 1 -function on R × R m , though our archetypical example is f = log u and u is a positive solution to equation (3.3). By Itô's formula for α = 1, · · · , m, and Z = (Z α ). The arguments w and / or x will be suppressed if no confusion may arise. Equivalently (3.14) Our aim in this part is to show that Z is an Itô process, and derives stochastic differential equations for Z (which in turn gives its Doob-Meyer's decomposition).

It follows that
Z α (t, ·, x) = A k α (ϕ(t, ·, x)) ∂f ∂ϕ k (ϕ(t, ·, x)) = A k α (ϕ(t, ·, x))K l k (t, ·, x)Y l (t, ·, x) (3. 16) which implies that Z is a continuous semimartingale. The equation (3.16) is not new, and has been used by many authors in different contexts. We next would like to write down the stochastic differential equations that Z α must satisfy by using the relation (3.16), which is however an easy exercise on integration by parts. Indeed, we have Using the SDEs (3.4, 3.9) and the BSDE (3.15), through a length but completely elementary computation, we establish the following Doob-Meyer's decomposition for Z where repeated indices are added up from 1 to n, We are now in a position to work out the Doob-Meyer's decomposition for which simply follows from Itô's formula and (3.18). In order to simplify our displayed formula, we introduce the following notations: for i, k = 1, · · · , n and β = 1, · · · , m, so that (3.20) Let which is a Brownian motion under probability Q with the Cameron-Martin density Then, by an elementary computation, Proof. In this case

Gradient estimates
Recall that f is a smooth solution to the non-linear parabolic equation (3.12), where the nonlinear term h(Y, Z) has at most quadratic growth. In order to devise explicit estimate for A α f (α = 1, · · · , m), we assume the following condition to be satisfied.
Condition 3.5 h(y, z) depends only on (y, |z| 2 ), i.e. there is a continuously differentiable function denoted again by h so that h(y, z) = h(y, |z| 2 ), and we assume that Then, under Conditions 1) -4), according to (3.23), we have We are now in a position to prove the following gradient estimate.
for any positive solution u of (3.2).
Proof. Apply the computations in the preceding sub-section to f = log u, and h(y, z) = − 1 2 |z| 2 . Then, under the probability Q (defined by (3.21)) and therefore On the other hand M t = e Kt |Z t | 2 is a submartingale, thus one has Putting (3.27, 3.28) together to obtain (3.26).
We can proceed as above. Under the probability Q It is important to note that if then h y (y, z) ≥ 0, so that from Lemma 2.2 in [7] On the other hand M t = e Kt |Z t | 2 is a submartingale, thus one has and therefore Putting (3.29, 3.30) together to obtain which yields the following estimate.

Heat equation on complete manifold
In this section, we study positive solutions of the heat equation where M is a complete manifold of dimension n, ∆ is the Beltrami-Laplace operator. In a local coordinate system so that the Riemann metric ds 2 = g ij dx i dx j and where g = det(g ij ) and (g ij ) is the inverse matrix of (g ij ). We prove the following Theorem 4.1 Suppose the Ricci curvature Ric ≥ −K for some K ≥ 0, and suppose u is a positive solution of (4.1) with initial data u 0 > 0. Then The preceding theorem is proved by using similar computations as in the proof of Theorem 3.7 but working on the orthonormal frame bundle O(M ) over M .
Recall that a point γ = (x, e) ∈ O(M ), where (e 1 , · · · , e n ) is an orthonormal basis of the tangent space T x M at x ∈ M . Let π : γ = (x, e) → x be the natural projection from O(M ) to M . O(M ) is a principal fibre bundle with its structure group O(n). For the general facts on differential geometry, we refer to Kobayashi and Nomizu [15]. Suppose x = (x 1 , · · · , x n ) is a local coordinate system on M , then it induces a local coordinate system γ = (x k , e i j ) on O(M ) so that e j = e i j ∂ ∂x i . If L is a vector field, thenL denotes the horizontal in a local coordinate system, where Γ k ij are the Christoffel symbols associated with the Levi-Civita connection, and L = L i ∂ ∂x i . For α = 1, · · · , n and γ = (x, e), thenL α denotes the horizontal lifting of e α , that isL The system {L 1 , · · · ,L n } is called the system of canonical horizontal vector fields. The mapping L : R n → Γ(T O(M )) where L ξ = ξ αL α , is defined globally, and is independent of the choice of a local coordinate system. Therefore The following relations will be used in what follows.
We need the following geometric facts, whose proofs are elementary.

Gradient estimate
Suppose now f satisfies the non-linear heat equation (4.14) Letf be the horizontal lifting of f i.e.f = f • π, so thatf satisfies the parabolic equation on O(M ): ∂f ∂x k (t, X t ) = (L αf )(t, (X t , E(t))) (4.16) for α = 1, · · · , n. Then so that, according to (4.12), We will consider (4.18) as a backward stochastic differential equation. Applying Itô's formula toL αf and the stochastic flow ϕ(t, ·, γ) to obtain so that Define a probability Q by dQ But on the other hand e Kt 2 |Z t | 2 is submartingale under Q, so that which yields that ||f 0 || ∞ and hence (4.2).

Li-Yau's estimate
In this section we prove Theorem 1.2. Thus, u is a positive solution to the heat equation where M is a complete Riemannian manifold of dimension n, with non-negative Ricci curvature. Then f = log u is a solution to the semi-linear heat equation Taking derivative in the parabolic equation ( Since ∆f is the trace of the Hessian ∇∇f so that |∇∇f | 2 ≥ 1 n (∆f ) 2 , thus, since the Ricci curvature is non-negative, then By combining (5.3) and (5.5) together, we obtain Next we apply the maximum principle to and therefore By applying the maximum principle, we obtain the estimates in Theorem 1.2. This follows by integrating the gradient estimates in Theorem 1.2 along geodesics, see [2] for details.