On the length of chains in a metric space
Introduction
The fundamental solution of the heat equation (or heat kernel) on is given by the Gauss Weierstrass kernel From a probabilistic viewpoint, the heat kernel can be interpreted as the transition probability density of the diffusion generated by the Laplacian Δ. More generally, for any uniformly elliptic, divergence form operator on , Aronson [1] proved that the heat kernel of the corresponding heat equation satisfies for all , where . Here denotes the Lebesgue measure of the Euclidean ball centered at x with radius r.
To prove the above lower bound on , the first step is to obtain the lower bound under the additional restriction that . Such an estimate is called a near diagonal lower bound. In order to obtain the full lower bound from a near diagonal lower bound, one chooses a sequence of points (called a chain) such that and such that for all . Then we use Chapman-Kolmogorov equation to obtain the estimate where . By optimizing over n and the sequence and using the near diagonal lower bound, we obtain the full lower bound on the heat kernel . This method of obtaining full heat kernel lower bound is called the chaining argument.
The use of chaining argument to obtain heat kernel estimates is classical [2], [1]. Such chaining arguments are also used to obtain heat kernel estimates on fractals; see [3] for a general introduction to diffusions on fractals. In this work, we address the natural converse question: Do heat kernel estimates imply the existence of short chains? Our main result provides an upper bound on the length of chains, which in some sense is the best possible. The goal of this work is to obtain sharp quantitative bounds on the connectivity of a metric space. This will be expressed as bounds on the length of chains.
We recall the definition of a chain in a metric space . Definition 1.1 We say that a sequence of points in X is an ε-chain between points if For any and , define where the infimum is taken over all ε-chains between with arbitrary N. Note that if is a geodesic space, then for all .
Throughout this paper, we consider a complete, locally compact separable metric space , equipped with a Radon measure m with full support, i.e., a Borel measure m on X which is finite on any compact set and strictly positive on any non-empty open set. Such a triple is referred to as a metric measure space. Then we set and for and .
Let be a symmetric Dirichlet form on . In other words, the domain is a dense linear subspace of , such that is a non-negative definite symmetric bilinear form which is closed ( is a Hilbert space under the inner product ) and Markovian (the unit contraction operates on ; and for any ). Recall that is called regular if is dense both in and in . Here is the space of -valued continuous functions on X with compact support.
For a function , let denote the support of the measure , i.e., the smallest closed subset F of X with ; note that coincides with the closure of in X if u is continuous. Recall that is called strongly local if for any with , compact and v is constant m-almost everywhere in a neighborhood of . The pair of a metric measure space and a strongly local, regular symmetric Dirichlet form on is termed a metric measure Dirichlet space, or an MMD space. We refer to [8], [7] for a comprehensive account of the theory of symmetric Dirichlet forms.
We recall the definition of energy measures associated to an MMD space. Note that for any by [8, Theorem 1.4.2-(ii)] and that and in norm in by [8, Theorem 1.4.2-(iii)].
Definition 1.2 cf. [8, (3.2.14) and (3.2.15)] Let be an MMD space. The energy measure of is defined, first for as the unique (-valued) Borel measure on X with the property that and then by for each Borel subset A of K for general . The notion of energy measure can be extended to the local Dirichlet space , which is defined as For any and for any relatively compact open set , we define where is as in the definition of . Since is strongly local, the value of does not depend on the choice of , and is therefore well defined. Since X is locally compact, this defines a Radon measure on X.
Definition 1.3 Capacity between sets For subsets , we define and the capacity as
The main result of this work provides upper bounds on based on analytic conditions on an MMD space that we introduce now. Henceforth, we fix a continuous increasing bijection such that for all , for some constants and . Throughout this work, the function Ψ is meant to denote the space time scaling of the process. Definition 1.4 We recall the following properties that an MMD space may satisfy: We say that satisfies the volume doubling Property (VD) if there exists such that We say that satisfies the Poincaré inequality , if there exist constants such that for all , and where . If all balls are relatively compact, then the above inequality can be extended to all . We say that satisfies the capacity estimate if there exist such that for all , , Definition 1.5 Let be an MMD space, and let denote its associated Markov semigroup. A family of non-negative Borel measurable functions on is called the heat kernel of , if is the integral kernel of the operator for any , that is, for any and for any , We remark that not every MMD space has a heat kernel. The existence of heat kernel is an issue in general.
Our main result is the following upper bound on . Theorem 1.6 Let be an MMD space that satisfies (VD), , and , where Ψ satisfies (1.3). Then there exists such that for all and for all that satisfy , we have In particular, for all , we have Remark 1.7 If , (1.4) implies the chain condition for all . If , then (1.4) and the triangle inequality imply that for all with . By letting , we give a new proof of the known fact that must necessarily hold. Let with . Consider the Dirichlet form corresponding to the Brownian motion on with Lebesgue measure as the symmetric measure, and the snowflake metric , where denotes the Euclidean distance (cf. [18, Definition 1.2.8] for the terminology ‘snowflake metric’). In this case, it is easy to obtain for all with . Hence, the bound (1.4) is sharp for all . Theorem 1.6 provides a new proof to an estimate due to Hambly and Kumagai [11, Lemma 3.3]. Based on the results in [11], the estimate (1.5) was introduced by Grigor'yan and Telcs in [10, (1.8)] to obtain sharp estimates of the heat kernel (cf. Corollary 2.9).
Corollary 1.8
Let be an MMD space that satisfies the following sub-Gaussian estimate on its heat kernel : there exists such that for all , where . Then the metric d satisfies the chain condition: there exists such that
Remark 1.9
- (a)
The chain condition (1.7) admits the following characterization. Let be metric space such that the open balls are relatively compact for all . Then satisfies the chain condition if and only if there exists a geodesic metric ρ such that d is bi-Lipschitz equivalent to ρ [15, Proposition A.1]. Recall that is geodesic, if for any two points , there exists a function such that for all .
- (b)
Corollary 1.8 was previously known only for the case . By a version of Varadhan's asymptotic formula in [14, Theorem 1.1], we obtain that d is bi-Lipschitz equivalent to the intrinsic metric. Hence by the remark above, d satisfies the chain condition. However, the intrinsic metric vanishes identically for the case . This suggests the need for a completely different approach when .
- (c)
The chain condition plays an essential role in the proof of singularity of energy measures in [15] for spaces satisfying the sub-Gaussian heat kernel estimate.
Recall that the parameter β in Corollary 2.9 is called the walk dimension. The following result can be viewed as a generalization of the result that the walk dimension is at least two. Corollary 1.10 Let be an MMD space that satisfies (VD), , and , where Ψ satisfies (1.3). Then there exists such that
Section snippets
Connectedness
We recall the notion of a net in a metric space. Definition 2.1 Let be a metric space and let . A maximal ε-separated subset is called an ε-net; in other words, V satisfies the following properties: V is ε-separated; that is, whenever and . (maximality) If and W is ε-separated, then .
Lemma 2.2
Let be an MMD space that satisfies (VD), , where Ψ satisfies (1.3). Then
Acknowledgements
I thank Takashi Kumagai for inspiring this work by asking if Corollary 1.8 could be true. It is a pleasure to acknowledge Naotaka Kajino and Takashi Kumagai for helpful references and discussions. This work was carried out at the Research Institute for Mathematical Sciences, Kyoto University and at Kobe University. I am grateful to Takashi Kumagai, Naotaka Kajino, Jun Kigami, and Ryoki Fukushima for support and hospitality during the visit. The author thanks the anonymous referee for a careful
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Research partially supported by NSERC (Canada).