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A Natural Diffusion Distance and Equivalence of Local Convergence and Local Equicontinuity for a General Symmetric Diffusion Semigroup.

1. Introduction

Diffusion semigroups play an important role in analysis, both theoretical and applied. Diffusion semigroups include the heat semigroup and, more generally, as discussed in, e.g., [1], arise from considering large classes of elliptic second-order (partial) differential operators on domains in Euclidean space or on manifolds. For examples of theoretical results involving diffusion semigroups, the interested reader may refer to Sturm [2] and Wu [3]. Some recent applications of diffusion semigroups to dimensionality reduction, data representation, multiscale analysis of complex structures, and the definition and efficient computation of natural diffusion distances can be found in, e.g., [4-11].

A particular important issue in harmonic analysis is to connect the smoothness of a function with the speed of convergence of its diffused version to itself, in the limit as time goes to zero. For the Euclidean setting, see, for example, [12, 13]. In order to consider the smoothness of diffusing functions in more general settings, a distance defined in terms of the diffusion itself seems particularly appropriate.

Defining diffusion distances is of interest in applications as well. As discussed in [5], dimensionality reduction of data and the concomitant issue of finding structures in data are highly important objectives in the fields of information theory, statistics, machine learning, sampling theory, etc. It is often useful to organize the given data as nodes in a weighted graph, where the weights reflect local interaction between data points. Random walks, or diffusion, on graphs may then help understand the interactions among the data points at increasing distance scales. To even consider different distance scales, it is necessary to define an appropriate diffusion distance on the constructed data graph.

In this paper, we consider a general symmetric diffusion semigroup [{[T.sub.t]f}.sub.t[greater than or equal to]0] on a topological space X with a positive [sigma]-finite measure (i.e., X is a countable union of measurable sets with finite measure), given, for t > 0, by an integral kernel operator: [T.sub.t]f(x) [??] [[integral].sub.X] [[rho].sub.t](x, y)f(y)dy. As part of their work in [7,11], Coifman and Leeb introduce a family of multiscale diffusion distances and establish quantitative results about the equivalence of a bounded function f being Lipschitz, and the rate of convergence of [T.sub.t]f to f, as t [right arrow] [0.sup.+] (we are discussing some of their results using a continuous time t for convenience; most of Coifman's and Leeb's derivations are done for dyadically discretized times. Moreover, most of the authors' results are in fact established without the assumption of symmetry and under the weaker condition than positivity of the kernel, namely, an appropriate [L.sub.1] integrability statement (see [11])). To prove the implication that Lipschitz implies an appropriate estimate on the rate of convergence, Coifman and Leeb make a quantitative assumption about the decay of

[mathematical expression not reproducible], (1)

for their distances d, namely, that

[mathematical expression not reproducible], (2)

for some [alpha] > 0. The authors show that their decay assumption holds for semigroups arising in many different settings (for which suitable decay and continuity assumptions are made on diffusion kernels relative to an intrinsic metric D of the underlying space), and even for some examples of nonsymmetric diffusion kernels. Coifman and Leeb also establish that (2) above, in the case of positive diffusion kernels, is in fact equivalent to their conclusion about the rate of convergence of [T.sub.t]f to f, as t [right arrow] [0.sup.+], for a Lipschitz function f. Additionally, Coifman and Leeb show that, in some of the settings they consider (with decay and continuity assumptions on the diffusion kernels relative to an intrinsic metric), their multiscale diffusion distance is equivalent to (localized) D[(x, y).sup.[alpha]], where D(x, y) is the intrinsic metric of the underlying space and a is a positive number strictly less than 1. The authors emphasize that a cannot be taken to equal 1.

In the present paper, we introduce a new family of diffusion distances generated by the diffusion semigroup [{[T.sub.t]f}.sub.t[greater than or equal to]0]. We provide several reasons as to why we think our definition is natural; in particular, we show that, for a convolution diffusion kernel on [R.sup.n], we achieve [alpha] = 1 in the discussion just above; i.e., we can recover (local) Euclidean distance to the "full" power 1.

The implication established in [7, 11] that smoothness of f implies control of the speed of convergence of [T.sub.t]f to f seems to us to be a more notable result than the converse (which the authors establish without assuming the decay of (1)). However, if f is Lipschitz for the multiscale diffusion distance introduced in [7,11], as the authors themselves point out their assumed estimate (2) almost tautologically leads to the desired estimate for the speed of convergence of [T.sub.t]f to f.

The main reason for our current work is that we wish to avoid making any assumptions about the decay of (1) and still establish a correspondence between some version of smoothness of a function f and convergence of [T.sub.t]f to f, as t [right arrow] [0.sup.+]. Our main contribution is to establish, under almost no assumptions, that local equicontinuity (in t) is equivalent to local convergence; i.e., local control of the differences [T.sub.t]f(x) - [T.sub.t]f(y) for all t small is equivalent to local control of the differences [T.sub.t]f(x) - f(x) for all small t. Here "local" is defined relative to a representative of our family of proposed diffusion distances.

Our paper is organized as follows. Following a notation and assumptions section (Section 2), we define our version of a natural diffusion distance [D.sub.g] in Section 3:

[mathematical expression not reproducible], (3)

for g a bounded, nonnegative, increasing function on (0,1], with [mathematical expression not reproducible]. We are led to our definition by requiring that a diffusion distance has the property that, for all functions f bounded in magnitude by 1, [T.sub.t](f) be Lipschitz with respect to the distance, independent of the particular f (of course, we expect the Lipschitz constant to grow as t goes to 0). This requirement arises from the intuitively reasonable demand that diffusion be smoothing in some sense. We then discuss some other reasons why our resulting distance is natural. In particular, for diffusion semigroups with convolution kernels on [R.sup.n] (this class includes the Poisson and heat kernels), our distance is equivalent to (local) Euclidean or sub-Euclidean distances for certain choices of the function g.

In Section 4, wemaketheassumption that ballsof positive radius with respect to the distance [D.sub.g] have positive measure. We show there is an equivalent topology, which does not depend on the function g, for which a corresponding statement about positive measure is equivalent to our assumption. The latter requirement, in turn, seems to be a mild and reasonable one.

In the main section, Section 5, we define our version of local convergence of [T.sub.t](f) to f, aswell aslocal equicontinuity of the family [{[T.sub.t]f}.sub.t[greater than or equal to]0]. Both definitions use our distance [D.sub.g]. We then establish that local convergence is equivalent to local equicontinuity. We next prove a corollary which extends an a.e. convergence result of Stein in [1]: for [mathematical expression not reproducible] converges locally to [mathematical expression not reproducible], as t converges to [0.sup.+].

In the Appendix, we show that, for very general metrics D on X, not necessarily arising from diffusion,

[[integral].sub.X] [[rho].sub.t] (x,y) D (x,y)dy [right arrow] 0 a.e., as t [right arrow] [0.sup.+]. (4)

This result is clearly a weaker statement than (2), but has the advantage of holding under virtually no assumptions.

2. Notation and Assumptions

Let X be a topological space equipped with a positive [sigma]-finite measure. For t > 0, [[rho].sub.t](x, y) will denote a symmetric kernel on X x X, with [[rho].sub.t] (x, y) [greater than or equal to] 0 for all x, y. We assume that p satisfies the semigroup property:

[[integral].sub.X] [[rho].sub.t] (x,u) [[rho].sub.s] (u, y) du = [[rho].sub.t+s] (x, y), (5)

for all x, y [member of] X, and s,t > 0. In addition, we assume

[[integral].sub.X] [[rho].sub.t] (x, y) dy = 1, (6)

for all x [member of] X and all t > 0. We will refer to a kernel [[rho].sub.t] satisfying the conditions above as a symmetric diffusion kernel (at time t). A typical example for [[rho].sub.t] is the heat kernel on a Riemannian manifold (see [14], for example).

For a function f, say in [L.sub.2](X) (or more generally, for any f where the following definition makes sense), we define the symmetric diffusion operator [T.sub.t], for t > 0, by

[T.sub.t]f(x) [??] [[rho].sub.t] (x,y) f (y) dy. (7)

We define [T.sub.0] to be the identity map. Note that, for all t, [mathematical expression not reproducible], by interpolation.

To avoid degeneracy, e.g., each [T.sub.t] being the averaging operator on a space of finite mass, we make an additional assumption: [T.sub.t](f) [right arrow] f in [L.sub.2], as t [right arrow] [0.sup.+].

The symmetric diffusion operator [T.sub.t] has the following properties of a symmetric diffusion semigroup:

(i) [T.sub.0] is the identity

(ii) [T.sub.t+s] = [T.sub.t] [omicron] [T.sub.s], for all s,t [greater than or equal to] 0

(iii) [mathematical expression not reproducible]

(iv) [T.sub.t] is a self-adjoint operator on [L.sub.2](X)

(v) [T.sub.t](f) [right arrow] f in [L.sub.2], as t[right arrow][0.sup.+]

(vi) [T.sub.t](f) [greater than or equal to] 0 if f [greater than or equal to] 0

(vii) [T.sub.t](l) = 1

See Stein's book [1], in which the author derives various harmonic analysis results for symmetric diffusion semigroups without explicitly using kernels.

3. A Natural Diffusion Distance

We now define our diffusion distance.

Definition 1. For a bounded, nonnegative, increasing function g on (0,1], with [mathematical expression not reproducible], and g strictly positive on the interval (0,1], define the distance [D.sub.g] by

[mathematical expression not reproducible]. (8)

It is clear that the distance [D.sub.g] satisfies the triangle inequality. Note that the restriction that g is bounded in the above supremum has the effect of making all "large" distances comparable to a constant, but this is not a drawback for smoothness considerations.

We would now like to discuss why we are using this particular diffusion distance and why we think it is a natural choice. Our starting point is the desire that, for a reasonable diffusion distance d(x,x), [T.sub.t](f) should be "smooth" for t > 0, even for "rough" functions f. This intuitive requirement is suggested by the idea that a diffusion semigroup be smoothing, in some sense. It would further be natural that the smoothness decays, for a general f, as t [right arrow] [0.sup.+]. We are thus led to impose a Lipschitz-like requirement, namely, that, for a diffusion distance d(x, x), and for t > 0,

[mathematical expression not reproducible]. (9)

It is easy to see that

[mathematical expression not reproducible]. (10)

Note that, for any [mathematical expression not reproducible] is decreasing in t, since, for h > 0,

[mathematical expression not reproducible], (11)

using (5) and (6). Letting g(t) = 1/c(t) we thus see that g is increasing, and from (10) we conclude that

[mathematical expression not reproducible]. (12)

This last inequality motivates our Definition 1 of [D.sub.g]. The restriction to t [less than or equal to] 1 is to ensure that [D.sub.g](x, y) is finite for all x and y and is not stringent, due to the fact that [mathematical expression not reproducible] is decreasing in t and that for smoothness purposes we need to only concentrate on points x and y which are near each other.

A further indication of the naturality of our proposed diffusion distance [D.sub.g] is that the [L.sub.1] norm of the difference of two probability densities, [mathematical expression not reproducible], occurring in the definition of [D.sub.g], is the (scaled) total variation distance between the probability distributions [[rho].sub.t](x, x) and [[rho].sub.t](y, x), i.e.,

[mathematical expression not reproducible]. (13)

Here, [[mu].sub.t,x] is the measure given by [[mu].sub.t,x] (A) = [[integral].sub.A] [[rho].sub.t](x,u)du, and [[mu].sub.t,y] is the measure given by [[mu].sub.t,y] (A) = [[integral].sub.A] [[rho].sub.t](y,u)du for measurable A [subset or equal to] X; the supremum is taken over all measurable A [subset or equal to] X (see Chapter 4 of [15]).

As a final argument for the naturality of our proposed diffusion distance, we calculate [D.sub.g] for a special case considered by the authors of [7] (for their own version of diffusion distances). We take X = [R.sup.n], g(t) = [t.sup.[alpha]], and assume that the diffusion kernel has the form [[rho].sub.t](x, y) = [t.sup.-n[beta]] [phi]([t.sup.-[beta]](x - y)). Here, [alpha], [beta] > 0 and [phi] is a nonnegative radial [L.sub.1] function whose gradient is also in [L.sub.1]. The case [beta] = 1/2 is for the heat kernel (with the appropriate [phi]), and the case [beta] = 1 is for the Poisson kernel (with the appropriate [phi]). Now,

[mathematical expression not reproducible], (14)

where we made the change of variables [mathematical expression not reproducible]. Then it is easy to see that h is radial and, for a "generic" [phi], we have the estimates: [mathematical expression not reproducible]. Here, [absolute value of x] is the usual Euclidean norm. Using this observation, and (14), we obtain the following (for this special case).

Proposition 2. For [mathematical expression not reproducible].

Proof. Using the notation for the special case above, we need to estimate [sup.sub.0<t[less than or equal to]1] [t.sup.[alpha]]h([t.sup.-[beta]](y - x)).

Let us first consider the situation when [absolute value of x- y] > 1. Then, for [mathematical expression not reproducible] using the estimate for h mentioned before the proposition.

Next, consider the situation when [absolute value of x- y] [less than or equal to] 1. Let [t.sub.0] = [[absolute value of x - y].sup.1/[beta]]. Note that 0 < [t.sub.0] [less than or equal to] 1.

When [mathematical expression not reproducible], so

[t.sup.[alpha]] h ([t.sup.-[beta]] (y - x)) ~ [t.sup.[alpha]-[beta]] [absolute value of x - y]. (15)

If [alpha] [less than or equal to] [beta], the maximum of the right hand side occurs at t = [t.sub.0] and equals

[t.sup.[alpha].sub.0] = [[absolute value of x - y].sup.[alpha]/[beta]]. (16)

If [alpha] [greater than or equal to] [beta], the maximum of the right hand side occurs at t = 1 and equals [absolute value of x- y].

When 0 < t [less than or equal to] [t.sub.0], we have that [t.sup.-[beta]] [absolute value of x- y] [greater than or equal to] 1, so

[t.sup.[alpha]] h([t.sup.-[beta]] (y - x)) ~ [t.sup.[alpha]], (17)

and the maximum of the right hand side occurs at t = [t.sub.0] and equals

[t.sup.[alpha].sub.0] = [[absolute value (of x - y)].sup.[alpha]/[beta]]. (18)

Note that if [mathematical expression not reproducible].

Combining the above discussions for the two ranges of values of t, the result follows.

Thus, for this special case of X = [R.sup.n], g(t) = [t.sup.[alpha]], and [[rho].sub.t](x,y) = [t.sup.-n[beta]] [phi]([t.sup.-[beta]](x - y)), which includes both the heat kernel and the Poisson kernel, our definition of diffusion distance gives (local) Euclidean or sub-Euclidean distance (depending on the relative sizes of [alpha] and [beta]). This result seems appropriate.

4. A Geometric Assumption about the Measure on X

We make the following reasonable assumption about our distance [D.sub.g]: for any [x.sub.0] [member of] X and any [epsilon] > 0,

B([x.sub.0], [epsilon]) [equivalent to] [x : [D.sub.g] ([x.sub.0], x) < [epsilon]}, (19)

the ball of radius [epsilon] and center [x.sub.0], has positive measure.

To justify the statement that this assumption is indeed reasonable, we first define another family of subsets of X. For any [x.sub.0] [member of] X, t > 0, and [epsilon] > 0, let

[mathematical expression not reproducible]. (20)

We then have the following equivalence of topologies induced by the sets B([x.sub.0], e) and N([x.sub.0], t, [epsilon]):

Proposition 3. For any [x.sub.0] [member of] X and any [epsilon] > 0, there exist t > 0 and [delta] > 0 such that N([x.sub.0], t, [delta]) [subset or equal to] B([x.sub.0], [epsilon]). Conversely, for any [x.sub.0] [member of] X, t >0, and [epsilon] > 0, there exists a [delta] > 0 such that B([x.sub.0], [delta]) [subset or equal to] N([x.sub.0], t, [epsilon]).

Proof. Fix an [x.sub.0] [member of] X and an [epsilon] > 0. We first show that there exist t > 0 and [delta] > 0 such that N([x.sub.0], t, [delta]) [subset or equal to] B([x.sub.0], [epsilon]).

Since we made the assumption that [mathematical expression not reproducible] for the function g used in defining the distance [D.sub.g], there exists [mathematical expression not reproducible].

For 0 < s [less than or equal to] t, since g in increasing, we see that

[mathematical expression not reproducible], (21)

Using the fact that the [L.sub.1] norm of [[rho].sub.s](x, x) is 1 for any s and x.

Now consider the case when t [less than or equal to] s [less than or equal to] 1. Note that, by definition of N([x.sub.0], t, [delta]), we have that [mathematical expression not reproducible]. Then, for this range of s, we observe that

[mathematical expression not reproducible], (22)

where we have used that [mathematical expression not reproducible] is decreasing in s; see (11).

We conclude (see (8)) that

[mathematical expression not reproducible] (23)

and hence x [member of] B([x.sub.0], [epsilon]).

For the converse, fix [x.sub.0] [member of] X, t > 0 and [epsilon] > 0. We will show that there exists a [delta] > 0 such that B([x.sub.0], [delta]) [subset or equal to] N([x.sub.0], t, [epsilon]).

Since, for any [mathematical expression not reproducible] is decreasing in s (see (11)), we clearly have that [mathematical expression not reproducible]. Thus, we may assume 0 < t < 1. Let [delta] = [epsilon]g(t). Then, for any x [member of] B([x.sub.0], [delta]), we have that [D.sub.g](x,[x.sub.0]) < [epsilon]g(t). Hence, using Definition 1 of the distance [D.sub.g], we obtain

[mathematical expression not reproducible]. (24)

Thus, [mathematical expression not reproducible], and we have that x [member of] N([x.sub.0], t, [epsilon]).

Returning to our assumption that, for any [x.sub.0] [member of] X and any [epsilon] > 0, B([x.sub.0], [epsilon]) has positive measure, Proposition 3 shows that it is equivalent to require the following: for any [x.sub.0] [member of] X, t >0, and [epsilon] > 0, the set N([x.sub.0], t, [epsilon]) has positive measure. Note that the definition of the sets N([x.sub.0], t, [epsilon]) is more "universal" than that of the balls B([x.sub.0], [epsilon]), since the former do not involve the function g.

The assumption that, for any [x.sub.0] [member of] X, t > 0, and [epsilon] > 0, the set N([x.sub.0], t, [epsilon]) has positive measure appears to us to be a very natural, and mild, one. In words, this requirement is saying that, for any time t > 0 and any [epsilon] > 0, the set of points in our space X which have not diffused more than e away (in the [L.sub.1] sense) from the diffused point [x.sub.0], at time t, is not "thin" with respect to the underlying measure on X. This assumption seems reasonable in both the discrete case (each point has positive mass, and x = [x.sub.0] is "enough") and the continuous case (every point [x.sub.0] has "many" arbitrarily close points in the sense of diffusion).

5. Local Convergence Is Equivalent to Local Equicontinuity

In this section, we define local convergence and local equicontinuity for our situation and show that the two concepts are equivalent under our assumptions.

In what follows, [T.sub.t] is a symmetric diffusion operator as defined in Section 2.

Definition 4. Let f [member of] [L.sub.p], 1 [less than or equal to] p [less than or equal to] [infinity]. Note that f is actually an equivalence class of functions on the space X. Suppose there exists a particular representative of this equivalence class, which we will also call f, such that this representative f is defined at every point of X, and for every [epsilon] > 0, there exist [t.sub.0] > 0 and [delta] > 0 so that [absolute value of [T.sub.t]f(x) - f(x)] < [epsilon], for all t with 0 < t [less than or equal to] [t.sub.0] and all x [member of] B([x.sub.0], [delta]). We then say [T.sub.t]f converges to f locally at [x.sub.0].

We also make the following.

Definition 5. Let f [member of] [L.sub.p], 1 [less than or equal to] p [less than or equal to] [infinity]. Suppose there exists a particular representative of the equivalence class specified by f and which we will also call f, such that this representative f is defined at every point of X, and for every [epsilon] > 0, there exist [t.sub.0] > 0 and [delta] > 0 with the property that, for all x [member of] B([x.sub.0], [delta]), we have [mathematical expression not reproducible]. We then say the family [{[T.sub.t]f}.sub.t[greater than or equal to]0] is locally equicontinuous (in t) at [x.sub.0].

Our main result is the following.

Proposition 6. For f [member of] [L.sub.2] [intersection] [L.sub.[infinity]] and any [x.sub.0] [member of] X, the following are equivalent:

(i) [T.sub.t]f converges to (the representative) f locally at [x.sub.0]

(ii) The family [{[T.sub.t]f}.sub.t[greater than or equal to]0] is locally equicontinuous at [x.sub.0]

Moreover, if a representative f satisfies one of these statements, the same representative satisfies the other statement.

Proof. We first show that local convergence at [x.sub.0] implies local equicontinuity at [x.sub.0]. We thus begin by assuming that [T.sub.t]f converges to a representative f locally at [x.sub.0].

First, we establish continuity of this representative f at [x.sub.0]. Fix [epsilon] > 0. By the assumption, there exist [mathematical expression not reproducible]. Then, for any x [member of] B([x.sub.0], [delta]), using the definition of the distance [D.sub.g], we see that

[mathematical expression not reproducible] . (25)

Since we assumed that [mathematical expression not reproducible],

[mathematical expression not reproducible], (26)

and continuity of f at [x.sub.0] is shown.

Next, note that

[mathematical expression not reproducible]. (27)

Let [t.sub.0] and [delta] > 0 be as above, i.e., [absolute value of [T.sub.t]f(x) - f(x)] < [epsilon]/3, for all t with 0 < t [less than or equal to] [t.sub.0] and all x [member of] B([x.sub.0], [delta]). Since we have already shown that f is continuous at [mathematical expression not reproducible]. Hence, the local equicontinuity of the family [{[T.sub.t]f}.sub.t[greater than or equal to]0] at [x.sub.0] follows.

Conversely, we now show that local equicontinuity at [x.sub.0] implies local convergence at [x.sub.0]. We thus begin by assuming that the family [{[T.sub.t]f}.sub.t[greater than or equal to]0] is equicontinuous at [x.sub.0].

Fix [epsilon] > 0. By the assumption, there exist 1 [greater than or equal to] [t.sub.0] > 0 and [delta] > 0 such that, for the representative f, [absolute value of f(x) - f([x.sub.0])] < [epsilon]/5 and [absolute value of [T.sub.t]f(x) - [t.sub.t]f([x.sub.0])] < [epsilon]/5, for all x [member of] B([x.sub.0], [delta]) and all t with 0 < t [less than or equal to] [t.sub.0]. In Section 4, we made the assumption that all balls of positive radius have positive measure. Using Stein's Maximal Theorem (see Chapter III, [section]3 in [1]), [mathematical expression not reproducible]. So there is a [mathematical expression not reproducible],

[mathematical expression not reproducible]. (28)

We estimate the first term on the right hand side of the above inequality as follows:

[mathematical expression not reproducible], (29)

for all 0 < t [less than or equal to] [t.sub.0], since x, [y.sub.0] [member of] B([x.sub.0], [delta]). For the second term, we use that [mathematical expression not reproducible]. Finally, for the third term, we see that

[mathematical expression not reproducible], (30)

since x, [y.sub.0] [member of] B([x.sub.0], [delta]).

Thus, for all t with 0 < t [less than or equal to] min([t.sub.0],[t.sub.1]), and for any x [member of] B([x.sub.0], [delta]), we obtain that [absolute value of [T.sub.t]f(x) - f(x)] < [epsilon], which concludes the proof of the converse.

In the proof above, we used Stein's Maximal Theorem (see Chapter III, [section]3 in [1]) to state that [mathematical expression not reproducible]. Stein's a.e. convergence result, for f [member of] [L.sub.2] say, is the main place in our paper where the symmetry of the operators [T.sub.t] is needed: Stein requires symmetry to prove his Maximal Theorem.

We immediately have the following.

Corollary 7. Let [mathematical expression not reproducible].

Proof. By Proposition 6, it suffices to show that [mathematical expression not reproducible] is locally equicontinuous at [mathematical expression not reproducible]. For any t [greater than or equal to] 0, we have that

[mathematical expression not reproducible], (31)

using the definition of the distance [D.sub.g] and the function G, that G is increasing, and inequality (11). Then, for [mathematical expression not reproducible] , and we have shown local equicontinuity at [x.sub.0].

Using our notation, Stein in [1] mentions that [mathematical expression not reproducible], since he proves that [T.sub.t]f is a real-analytic function of t > 0 for almost all x. Corollary 7 extends Stein's result (under our assumption discussed in Section 4) to show local convergence with respect to the distance [D.sub.g].

6. Conclusions and Future Work

In this paper, we have defined a diffusion distance which is natural if one imposes a reasonable Lipschitz condition on diffused versions of arbitrary bounded functions. We have next shown that the mild assumption that balls of positive radius have positive measure is equivalent to a similar, and an even milder looking, geometric demand. In the main part of the paper, we establish that local convergence of [T.sub.t]f to (a representative) f at a point is equivalent to local equicontinuity of the family [{[T.sub.t]f}.sub.t[greater than or equal to]0] at that point.

It may well be useful to have a quantitative estimate on the rate of convergence of [T.sub.t] f to f under the assumption that f is Lipschitz, say, with respect to some distance d (where d may be our [D.sub.g]). As essentially pointed out in the papers [7,11], a key issue is whether, and how rapidly,

[mathematical expression not reproducible]. (32)

In the Appendix below, we show that, for very general metrics D on X, not necessarily arising from diffusion,

[mathematical expression not reproducible]. (33)

This result is certainly far from establishing the convergence in (32), much less a quantitative estimate.

We plan to continue exploring for which (diffusion) distances the convergence in (32) holds and an estimate can be obtained.

https://doi.org/10.1155/2018/6281504

Appendix

Proposition 8. Let D be a metric on X with the following properties:

(1) [sup.sub.x,y[member of]X] D(x,y) < [infinity]

(2) X is separable with respect to the metric D, i.e., it contains a countable dense subset

(3) There exists a [delta] > 0 so that m[B(x, [delta])] < [infinity] for every x [member of] X (the bound need not be uniform in x). Here, m[B(x,[delta])] denotes the measure of the ball B(x,[delta]) = {y : D(x,y) < [delta]}

Then,

[mathematical expression not reproducible]. (A.1)

To prove the proposition, we first establish the following.

Lemma 9. For any [x.sub.0] [member of] X, if r > 0 is such that m[B([x.sub.0], r)] < [infinity], then

[mathematical expression not reproducible], (A.2)

for almost all x [member of] B([x.sub.0], r).

Proof. Let f(y) = D([x.sub.0], y) [chi] (y), where [chi](y) is the characteristic function of the ball B([x.sub.0], r). Since m[B([x.sub.0], r)] < [infinity], we see that f [member of] [L.sub.2](X). Using Stein's Maximal Theorem (see Chapter III, [section]3 in [1]), we conclude that

[mathematical expression not reproducible].

In particular, for some set C [subset or equal to] B([x.sub.0],r), with m[B([x.sub.0],r)\C] = 0,

[mathematical expression not reproducible]. (A.4)

We would be done if the integration were over all of X, not just B([x.sub.0], r).

To this end, we apply Stein's Maximal Theorem to the [L.sub.2] function [chi](y) to see that there is a set D [subset or equal to] B([x.sub.0], r), with m[B([x.sub.0],r) \ D] = 0, so that

[mathematical expression not reproducible]. (A.5)

For x [member of] D, since

[mathematical expression not reproducible], (A.6)

we conclude that

[mathematical expression not reproducible], (A.7)

where [B.sup.c]([x.sub.0], r) is the complement of B([x.sub.0], r).

Since we assumed that [sup.sub.x,y[member of]X] D(x, y) < [infinity], we obtain that, for every x [member of] D,

[mathematical expression not reproducible]. (A.8)

Combining (A.4) and (A.8), we conclude that, for x [member of] C n D [subset or equal to] B([x.sub.0], r),

[mathematical expression not reproducible]. (A.9)

Note that the set C [intersection] D is of full measure in B([x.sub.0], r):

[mathematical expression not reproducible]. (A.10)

The lemma is proved.

We now turn to proving the proposition. Choose a positive integer N so that 1/N < [delta], where [delta] > 0 is such that m[B(x, [delta])] < [infinity] for every x [member of] X (Assumption (3)). Clearly, for every n [greater than or equal to] N and x [member of] X,

m [B (x, 1/n)] [less than or equal to] m[B(x,[delta])] < [infinity]. (A.11)

Let [{[x.sub.k]}.sup..[infinity].sub.k=1] be a countable, dense subset of X relative to the distance D (Assumption (2)). For every k = 1, 2,..., and n = N, N + 1,..., apply Lemma 9 to obtain a set E(k, n) such that E(k,n) [subset or equal to] B([x.sub.k], 1/n), m[B([x.sub.k], 1/n) \ E(k,n)] = 0, and, for x [member of] E(k, n),

[mathematical expression not reproducible]. (A.12)

Let

[mathematical expression not reproducible]. (A.13)

Since [{[x.sub.k]}.sup.[infinity].sub.k=1] is dense in X,

[mathematical expression not reproducible]. (A.14)

Hence,

[mathematical expression not reproducible]. (A.15)

Since

[mathematical expression not reproducible], (A.16)

we see that

m [[E.sup.c.sub.n]] = 0. (A.17)

Now, let

[mathematical expression not reproducible]. (A.18)

Since [E.sup.c] = [[union].sup.[infinity].sub.n=N] [E.sup.c.sub.n] and m[[E.sup.c.sub.n]] = 0, we see that m[[E.sup.c] = 0.

To finish the proof of the proposition, we will show that, for every x [member of] E, hence a.e.,

[mathematical expression not reproducible]. (A.19)

Fix x [member of] E. Choose any [mathematical expression not reproducible]. But then, by definition of the set E([k.sub.0], K),

[mathematical expression not reproducible]. (A.20)

Now, using the triangle inequality,

[mathematical expression not reproducible]. (A.21)

Hence,

[mathematical expression not reproducible]. (A.22)

Since [member of] > 0 is arbitrary,

[mathematical expression not reproducible], (A.23)

and we are done.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

We are both grateful and indebted to Raphy Coifman for his continued willingness to discuss mathematics with us. The first author was partially supported by Faculty Development Funding from Ramapo College of New Jersey.

References

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[2] K. T. Sturm, "Diffusion processes and heat kernels on metric spaces," Annals of Probability, vol. 26, no. 1, pp. 1-55, 1998.

[3] H.-T. Wu, "Embedding Riemannian manifolds by the heat kernel of the connection Laplacian," Advances in Mathematics, vol. 304, pp. 1055-1079, 2017.

[4] M. Belkin and P. Niyogi, "Laplacian eigenmaps for dimensionality reduction and data representation," Neural Computation, vol. 15, no. 6, pp. 1373-1396, 2003.

[5] R. R. Coifman and S. Lafon, "Diffusion maps," Applied and Computational Harmonic Analysis, vol. 21, no. 1, pp. 5-30, 2006.

[6] R. R. Coifman, S. Lafon, A. B. Lee et al., "Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps," Proceedings of the National Academy of Sciences of the United States of America, vol. 102, no. 21, pp. 7426-7431, 2005.

[7] R. R. Coifman and W. E. Leeb, "Earth Mover's distance and equivalent metrics for spaces with semigroups," Tech. Rep. YALEU/DCS/TR-1481, 2013.

[8] R. R. Coifman and M. Maggioni, "Diffusion wavelets," Applied and Computational Harmonic Analysis, vol. 21, no. 1, pp. 53-94, 2006.

[9] M. J. Goldberg and S. Kim, "Some Remarks on Diffusion Distances," Journal of Applied Mathematics, vol. 2010, Article ID 464815, 17 pages, 2010.

[10] M. J. Goldberg and S. Kim, "An efficient tree-based computation of a metric comparable to a natural diffusion distance," Applied and Computational Harmonic Analysis , vol. 33, no. 2, pp. 261-281, 2012.

[11] W. Leeb and R. Coifman, "Holder-Lipschitz norms and their duals on spaces with semigroups, with applications to earth mover's distance," Journal of Fourier Analysis and Applications, vol. 22, no. 4, pp. 910-953, 2016.

[12] P. L. Butzer and H. Berens, Semi-Groups of Operators and Approximation, Springer, Berlin, Germany, 1967.

[13] H. Triebel, Theory of Function Spaces, vol. 78 of Monographs in Mathematics, Birkhouser, Boston, Mass, USA, 1983.

[14] I. Chavel, "Eigenvalues in Riemannian geometry," in Pure and Applied Mathematics #115, Academic Press, Orlando, Fla, USA, 1984.

[15] D. A. Levin, Y. Peres, and E. L. Wilmer, Markov Chains and Mixing Times, American Mathematical Society, Providence, RI, USA, 2009.

Maxim J. Goldberg (iD) (1) and Seonja Kim (iD) (2)

(1) Theoretical and Applied Sci., Ramapo College of NJ, Mahwah, NJ 07430, USA

(2) Mathematics Dept., Middlesex County College, Edison, NJ 08818, USA

Correspondence should be addressed to Maxim J. Goldberg; mgoldber@ramapo.edu

Received 27 July 2018; Accepted 17 September 2018; Published 2 October 2018

Academic Editor: Khalil Ezzinbi
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Title Annotation:Research Article
Author:Goldberg, Maxim J.; Kim, Seonja
Publication:Abstract and Applied Analysis
Article Type:Report
Date:Jan 1, 2018
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