On some strong ratio limit theorems for heat kernels

We study strong ratio limit properties of the quotients of the heat kernels of subcritical and critical operators which are defined on a noncompact Riemannian manifold.


Introduction
Let M be a connected noncompact Riemannian manifold, and let k M P (x, y, t) be the positive minimal (Dirichlet) heat kernel associated with the parabolic equation where P is a second-order elliptic differential operator on M . The coefficients of P are assumed to be real but P is not necessarily symmetric. By definition, (x, t) → k M P (x, y, t) is the minimal positive solution of (1), subject to the initial data δ y , the Dirac distribution at y ∈ M . We say that the operator P is subcritical (respectively, critical ) in M if for some x = y In this paper we are concerned with the large time behavior of the heat kernel k M P with regards to the criticality versus subcriticality property of the operator P . Since for any fixed x, y ∈ M , x = y, we have that k M P (x, y, ·) ∈ L 1 (R + ) if and only if P is subcritical, it is natural to conjecture that under some assumptions the heat kernel of a subcritical operator P + in M decays (in time) faster than the heat kernel of a critical operator P 0 in M . More precisely, we are interested to study the following conjecture.  The relevance of this conjecture becomes clearer if we recall the relationship of (2) to properties of positive solutions of the elliptic equation (4) P u = 0 on M.
Throughout this paper we always assume that λ 0 ≥ 0 (actually, as it will become clear below, it is enough to assume that λ 0 > −∞).
It is well known that if P is subcritical in M , then P admits a positive minimal Green function G M P (x, y) which is given by On the other hand, if P is critical in M , then P does not admit a positive minimal Green function, but admits a distinguished unique positive solution ϕ ∈ C P (M ) satisfying ϕ(x 0 ) = 1, where x 0 ∈ M is a reference point. Such a solution is called a ground state of the operator P in M [1,16,21]. A ground state is characterized by being a global positive solution of the equation P u = 0 on M of minimal growth in a neighborhood of infinity in M [1]. On the other hand, if P is subcritical in M , then for any fixed y ∈ M , the positive minimal Green function G M P (·, y) is a positive solution of the equation P u = 0 on M \ {y} of minimal growth in a neighborhood of infinity in M . We also note that P is critical in M if and only the equation P u = 0 on M admits (up to a multiplicative constant) a unique positive supersolution. Furthermore, P is critical (respectively, subcritical) in M , if and only if P * (the formal adjoint of P ) is critical (respectively, subcritical) in M . The ground state of P * is denoted by ϕ * .
A critical operator P is said to be positive-critical in M if ϕ * ϕ ∈ L 1 (M ), and null-critical in M if ϕ * ϕ ∈ L 1 (M ). The large time behavior of the heat kernel of a general elliptic operator P (with λ 0 ≥ 0) is governed by the following theorem. Furthermore, As a consequence of this theorem, we see that lim t→∞ e λ0t k M P (x, y, t) always exists. On the other hand, heat kernels might have slow decay (see for example [4] and the references therein). Therefore, it is natural to ask how fast versus slow this limit is approached, and in particular, to examine the validity of Conjecture 1. We note that Theorem 1.1 implies that Conjecture 1 obviously holds true if P 0 is positive-critical.
In [12,Theorems 4.2 and 4.4] M. Murata obtained the exact asymptotic for the heat kernels of nonnegative Schrödinger operators with short-range (real) potentials defined on R d , d ≥ 1. These results imply that Conjecture 1 holds true for such operators.
The aim of the present paper is to discuss Conjecture 1 and closely related problems in the general case, and to obtain some results under minimal assumptions.
Our study is motivated by a recent paper [9] by D. Krejčiřík and E. Zuazua, where it is conjectured that for selfadjoint subcritical and critical operators P + and P 0 , respectively, defined on L 2 (M, dx) one has (8) lim for some positive weight function W . In fact, the above conjecture is proved in [9] for the Dirichlet Laplacian defined on a special class of quasi-cylindrical domains. It turns out that Conjecture 1 is related to the following conjecture raised by E. B. Davies [6] in the self-adjoint case.
Conjecture 2 (Davies' Conjecture). Let Lu = u t + P (x, ∂ x )u be a parabolic operator which is defined on a noncompact Riemannian manifold M . Fix reference points x 0 , y 0 ∈ M . Then exists and is positive for all x, y ∈ M , Moreover, for any fixed y ∈ M we have a(·, y) ∈ C P −λ0 (M ). Similarly, for a fixed x ∈ M we have a(x, ·) ∈ C P * −λ0 (M ) (see also [19] and the references therein).
Remark 1. Obviously, Davies' Conjecture holds if P is positive-critical. Moreover, it holds true in the symmetric case (for a precise definition of P being symmetric see Section 2) if dim C P (M ) = 1 [3, Corollary 2.7]. In particular, it holds true for a critical symmetric operator. For a probabilistic interpretation of Conjecture 2, see [3].
On the other hand, G. Kozma announced [8] that he constructed a graph G such that for some two vertices x, y ∈ G the sequence {k(x, x, n)/k(y, y, n)} ∞ n=1 of the ratio of the corresponding heat kernel does not converge as n → ∞.
The organization of this paper is as follows. In the following section, we give a precise definition of the operator P in M and introduce the necessary background to study Conjecture 1. In Section 3, we prove (under some additional assumptions) Conjecture 1 in the symmetric case (Theorem 3.1). In particular, Theorem 3.1 provides an affirmative answer to the conjecture in the case of positive perturbations (Corollary 1). The relationship between Davies' conjecture and Conjecture 1 is examined for nonsymmetric operators in Section 4. Two regimes are considered: positive perturbations (Theorem 4.1) and semismall perturbations (Theorem 4.2). We conclude the paper in Section 5, where we ask a general question concerning the equivalence of heat kernels on Riemannian manifolds and provide sufficient conditions for the validity of a principal hypothesis of theorems 2.3, 3.1, and 4.2.

Preliminaries
Let M be a smooth connected noncompact Riemannian manifold of dimension d. We recall the definition of a weighted manifold associated with M . Denote by dx the Riemannian density on M . The divergence and gradient with respect to the Riemannian metric on M are denoted by div and ∇, respectively. Let m be a positive measurable function on M such that m and m −1 are bounded on any compact subset of M . Set dµ := mdx. The couple (M, dµ) is called a weighted manifold over which we consider the Lebesgue spaces L p (M, dµ).
We associate to M an exhaustion, i.e. a sequence of smooth, relatively compact For every j ≥ 1, we denote M * j := Ω \ M j . We consider a second-order elliptic differential operator P which is defined on (M, dµ) by The inner product and the induced norm on T M is denoted by ·, · and | · |, respectively. We assume that D, |B| 2 , |C| 2 ∈ L p loc (M, dµ) for some p > max{n/2, 1}. We say that P is symmetric if B = C = 0 on M . So, in the symmetric case P has the form The reason for this terminology is that the minimal operator constructed from the formal differential operator (11), i.e. the restriction of P to C ∞ 0 (M ), is symmetric in L 2 (M, dµ). The Friedrichs extension of the minimal operator defines a self-adjoint operator in L 2 (M, dµ); it acts weakly as (11) and satisfies Dirichlet boundary conditions on ∂M in a generalized sense. By definition, it is the operator associated with the closure of the quadratic form Q in L 2 (M, dµ) defined by It is well known that for such operators we have where λ 0 is the generalized principal eigenvalue of P introduced in (5). In other words, λ 0 equals to the bottom of the spectrum of the Friedrichs extension if P is symmetric.
Remark 2. Let t n → ∞. By a standard parabolic argument, we may extract a subsequence {t n k } such that for every x, y ∈ M and s < 0 ) denotes the cone of all nonnegative solutions of the equation (1) in M × (a, b). Note that in the selfadjoint case, the above is valid for all s ∈ R [19]. Now we recall some auxiliary results which we will need in the sequel. First, we mention convexity properties of heat kernels.

Lemma 2.1. Consider the following one-parameter family of elliptic operators
, and the corresponding heat kernels satisfy the inequality For a proof of the lemma see [15]. In particular, (14) is proved by applying Hölder's inequality to the Feynmann-Kac formula (see e.g., [22,Lemma B.7.7]).
We also need the following key lemma Lemma 2.2. Assume that λ 0 (P, M ) ≥ 0, and that either P is symmetric or that Davies' conjecture holds for P in M . Then for any fixed x, y ∈ M we have Proof. If P is symmetric, then the function t → k M P (x, x, t) is log-convex, and therefore the lemma follows by a polarization argument (see for example [5,6]).
Suppose now that Davies' conjecture holds for P in M . Then as in the proof of [19, Theorem 3.1], fix y ∈ M and let {t n } be a sequence such that t n → ∞. Consider Remark 2). By our assumption, for any τ we have where b ∈ C P −λ0 (M ), and b does not depend on the sequence {t n }.
On the other hand, , it follows that f solves the initial value problem (backwards in time) In particular, f does not depend on the sequence {t n }. Thus, Finally, It turns out that Lemma 2.2 implies that the case λ 0 (P + , M ) > 0 is easier than the case λ 0 (P + , M ) = 0. Moreover, if λ 0 (P + , M ) > 0, then the assumptions that we need for the validity of Conjecture 1 are weaker. We have Theorem 2.3. Let P 0 be critical operator in M , and let P + be a subcritical operator in M satisfying λ + := λ 0 (P + , M ) > 0. Suppose that either P 0 and P + are symmetric operators, or that Davies' conjecture (Conjecture 2) holds true for both k M P0 and k M P+ . Assume further that for some fixed y 1 ∈ M there exists a positive constant C satisfying the following condition: for each x ∈ M there exists T (x) > 0 such that Then Proof. Fix x ∈ M , and s ∈ R − , and let y 1 ∈ M be the point satisfying (16). We have .
Recall that λ 0 (P 0 , M ) = 0, and by our assumption λ + > 0. By Lemma 2.2 we have Therefore, using (19) and our assumption (16), it follows from (18) that for t sufficiently large we have Since s is an arbitrary negative number, (20) implies that The parabolic Harnack inequality and a standard parabolic regularity argument imply now that By the generalized maximum principle, assumption (16) is satisfied with C = 1 if P + = P 0 + V and V is any nonnegative potential. In Section 5, we discuss some other conditions under which assumption (16) is satisfied.
We shall need also the following Liouville comparison theorem (see [20]).
Theorem 2.4. Let P 0 and P 1 be two symmetric operators defined on M of the form (11). Assume that the following assumptions hold true.
where C > 0 is a positive constant.
Then the operator P 1 is critical in M , and ψ is its ground state. In particular, dim C P1 (M ) = 1 and λ 0 (P 1 , M ) = 0.
Let f, g ∈ C(Ω) be nonnegative functions, we use the notation f ≍ g on Ω if there exists a positive constant C such that for all x ∈ Ω.
In the sequel we shall need also to use results concerning small and semismall perturbations. These notions were introduced in [14] and [13] respectively, and are closely related to the stability of C P (Ω) under perturbation by a potential V .
Moreover, let ϕ be the ground state of P + α 0 V and let y 0 be a fixed reference point in M 1 . Then for any 0 ≤ α < α 0

The symmetric case
In this section we prove the following theorem.
Theorem 3.1. Let the subcritical operator P + and the critical operator P 0 be symmetric in M . Assume that A + and A 0 , the sections on M of End(T M ), and the weights m + and m 0 , corresponding to P + and P 0 , respectively, satisfy the following matrix inequality where C is a positive constant. Assume further that condition (16) holds true. Then Proof. By Theorem 2.3, we may assume that λ 0 (P + , M ) = 0. Assume to the contrary that for some x 0 , y 0 ∈ M there exists a sequence {t n } such that t n → ∞ and Consider the sequence of functions {u n } ∞ n=1 defined by x ∈ M, s ∈ R.
We note that . Therefore, by assumption (27) and Remark 2 it follows that we may subtract a subsequence which we rename by {u n } such that lim n→∞ u n (x, s) = u + (x, s), where u + ∈ H P+ (M × R) and u + 0.
On the other hand, .
By our assumption, λ 0 (P + , M ) = 0, therefore Lemma 2.2 implies that and u + does not depend on s, and hence u + is a positive solution of the elliptic equation P + u = 0 in M and we have On the other hand, by Remark 1 we have where ϕ is the ground state of P 0 . Combining (28) and (29), we obtain On the other hand, by assumption (16) and the parabolic Harnack inequality there exists a positive constant C 1 which depends on P + , P 0 , y 0 , y 1 such that Therefore, (31) and (32) imply that there exists C 0 > 0 such that Consequently, (30) and (33) imply that Therefore, using (25) we obtain where C 2 > 0 is a positive constant. Thus, Theorem 2.4 implies that P + is critical in M which is a contradiction. The last statement of the theorem follows from the parabolic Harnack inequality and parabolic regularity.
By the generalized maximum principle, assumption (16) in Theorem 3.1 is satisfied with C = 1 if P + = P 0 + V , where P 0 is a critical operator on M and V is any nonnegative potential. Note that if the potential is in addition nontrivial, then P + is indeed subcritical in M . Therefore, we have Corollary 1. Let P 0 be a symmetric operator which is critical in M , and let P + := P 0 + V , where V is a nonzero nonnegative potential. Then Consequently, for any compact set K ⋐ M , we arrive at

Davies' conjecture and Conjecture 1
In the present section we discuss the nonsymmetric case. We study two cases where Davies' conjecture imply Conjecture 1. First, we show that in the nonsymmetric case, the result of Corollary 1 for positive perturbations of a critical operator P 0 still holds provided that the validity of Davies' conjecture (Conjecture 2) is assumed instead of the symmetry hypothesis. More precisely, we have x ∈ M, t > 0.
On the other hand, by the generalized maximum principle Therefore, Letting n → ∞ we obtain  Proof. The first part of the proof is similar to the corresponding part in the proof of Theorem 4.1. For completeness we repeat it.
By Theorem 2.3, we may assume that λ 0 (P + , M ) = 0. Assume to the contrary that for some x 0 , y 0 ∈ M there exists a sequence {t n } such that t n → ∞ and Consider the functions v + and v 0 defined by x ∈ M, t > 0.
On the other hand, by assumption (16) we have for t > T (x) .
By our assumption on Davies' conjecture, we have for a fixed x , where u * + and u * 0 are positive solutions of the equation P * + u = 0 and P * 0 u = 0 in M , respectively. By the elliptic Harnack inequality there exists a positive constant C 1 (depending on P * + , P * 0 , y 0 , y 1 but not on x) such that Therefore, (44) and (46) imply that for n sufficiently large (which might depend on x). Therefore, Letting n → ∞ and using (42) and (43), we obtain On the other hand, since V is a semismall perturbation of P * + in M , Theorem 2.6 implies that u 0 (x) ≍ G M P+ (x, y 0 ) in M \B(y 0 , δ), with some positive δ. Consequently, for some C 2 > 0. In other words, u + is a global positive solution which has minimal growth in a neighborhood of infinity in M . Therefore u + is a ground state of the equation P + u = 0 in M , but this contradicts the subcriticality of P + in M .

On the equivalence of heat kernels
In this section we study a general question concerning the equivalence of heat kernels which in turn will give sufficient conditions for the validity of the boundedness assumption (16) which is assumed in theorems 2.3, 3.1 and 4.2.
Definition 5.1. Let P i , i = 1, 2, be two elliptic operators on M satisfying λ 0 (P i , M ) ≥ 0 for i = 1, 2. We say that the heat kernels k M P1 (x, y, t) and k M P2 (x, y, t) are equiv- for some fixed y 0 ∈ M ). There is an intensive literature dealing with (almost optimal) conditions under which two positive (minimal) Green functions are equivalent or semiequivalent (see [2,13,14,17] and the references therein). On the other hand, sufficient conditions for the equivalence of heat kernels are known only in a few cases (see [10,11,23]). In particular, it seems that the answer to the following conjecture is not known. Conjecture 3. Let P 1 and P 2 be two subcritical operators of the form (10) which are defined on a Riemannian manifold M such that P 1 = P 2 outside a compact set in M . Then k M P1 and k M P2 are equivalent. It is well known that certain 3-G inequalities imply the equivalence of Green functions, and the notions of small and semismall perturbations is based on this fact. Moreover, small (respectively, semismall) perturbations are sufficient conditions and in some sense also necessary conditions for the equivalence (respectively, semiequivalence) of the Green functions [13,14,17]. We introduce an analog definition for heat kernels (cf. [23]).
Definition 5.2. Let P be a subcritical operator in M . We say that a potential V is a k-bounded perturbation (respectively, k-semibounded perturbation) with respect to the heat kernel k M P (x, y, t) if there exists a positive constant C such that the following 3-k inequality is satisfied: for all x, y ∈ M (respectively, for a fixed y ∈ M , and all x ∈ M ) and t > 0.
The following result shows that the notion of k-(semi)boundedness is naturally related to the (semi)equivalence of heat kernels. Theorem 5.3. Let P be a subcritical operator in M , and assume that the potential V is k-bounded perturbation (respectively, k-semibounded perturbation) with respect to the heat kernel k M P (x, y, t). Then there exists c > 0 such that for any |ε| < c the heat kernels k M P +εV (x, y, t) and k M P (x, y, t) are equivalent (respectively, semiequivalent).
Proof. Consider the iterated kernels  The lower bound C 1 k M P (x, y, t) ≤ k M P +εV (x, y, t) (for |ε| small enough) follows from the upper bound using (52) and (51).
Corollary 2. Assume that P and V satisfy the conditions of Theorem 5.3, and suppose further that V is nonnegative. Then there exists c > 0 such that for any ε > −c the heat kernels k M P +εV (x, y, t) and k M P (x, y, t) are equivalent (respectively, semiequivalent).
Proof. By Theorem 5.3 the heat kernels k M P +εV (x, y, t) and k M P (x, y, t) are equivalent (respectively, semiequivalent) for any |ε| < c. Recall that by the generalized maximum principle, k M P +εV (x, y, t) ≤ k M P (x, y, t) ∀ε > 0.
On the other hand, using also Lemma 2.1, we obtain the desired equivalence also for ε ≥ c.
Theorem 5.4. Let P 0 be a critical operator in M . Assume that V = V + − V − is a potential such that V ± ≥ 0 and P + := P 0 + V is subcritical in M . Assume further that V − is k-semibounded perturbation with respect to the heat kernel k M P+ (x, y 1 , t). Then condition (16) is satisfied uniformly in x. That is, there exist positive constants C and T such that (53) k M P+ (x, y 1 , t) ≤ Ck M P0 (x, y 1 , t) ∀x ∈ M, t > T.
(grant No. 419/07) founded by the Israeli Academy of Sciences and Humanities. M. F. was also partially supported by a fellowship of the UNESCO fund.
Remark 4 (Added on 17.5.2010). After the paper has been published, we realized that the generalized principal eigenvalue is characterized by the following formula (55) lim t→∞ log k M P (x, y, t) t = −λ 0 (P, M ).
The above characterization of λ 0 is well-known in the symmetric case, see for example [7,Theorem 10.24]. The needed upper bound for the validity of (55) for general elliptic operators of the form (10) follows directly from Theorem 1.1, while the lower bound follows from the large time behavior of the heat kernel in a smooth bounded domain using a standard exhaustion argument (cf. the proof of [7, Theorem 10.24]).
Consequently, if P 0 is a critical operator in M , and P + is a subcritical operator in M satisfying λ + := λ 0 (P + , M ) > 0, then Conjecture 1 holds true without any further assumption (cf. Theorem 2.3).