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

Authors: Maxim J. Goldberg; Seonja Kim
Journal: Abstract and Applied Analysis
 (2018)

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## Abstract

In this paper, we consider a general symmetric diffusion semigroupTtft≥0 on a topological space X with a positive σ-finite measure, given, for t>0, by an integral kernel operator: Ttf(x)≜∫X‍ρt(x,y)f(y)dy. As one of the contributions of our paper, we define a diffusion distance whose specification follows naturally from imposing a reasonable Lipschitz condition on diffused versions of arbitrary bounded functions. We next show that the mild assumption we make, 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 Ttf to f is equivalent to local equicontinuity (in t) of the family Ttft≥0. As a corollary of our main result, we show that, for t0>0, Tt+t0f converges locally to Tt0f, as t converges to 0+. In the Appendix, we show that for very general metrics D on X, not necessarily arising from diffusion, ∫X‍ρt(x,y)D(x,y)dy→0a.e., as t→0+. R. Coifman and W. Leeb have assumed a quantitative version of this convergence, uniformly in x, in their recent work introducing a family of multiscale diffusion distances and establishing quantitative results about the equivalence of a bounded function f being Lipschitz, and the rate of convergence of Ttf to f, as t→0+. We do not make such an assumption in the present work.


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## Body

## 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 semigroupTtft≥0 on a topological space X with a positive σ-finite measure (i.e., X is a countable union of measurable sets with finite measure), given, for t>0, by an integral kernel operator: Ttf(x)≜∫X‍ρt(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 Ttf to f, as t→0+ (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 L1 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(1)supx⁡∫Xρtx,ydx,ydy,ast→0+,for their distances d, namely, that(2)supx⁡∫Xρtx,ydx,ydy≤Ctα,for some α>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 factequivalent to their conclusion about the rate of convergence of Ttf to f, as t→0+, 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)α, where D(x,y) is the intrinsic metric of the underlying space and α is a positive number strictly less than 1. The authors emphasize that α cannot be taken to equal 1.In the present paper, we introduce a new family of diffusion distances generated by the diffusion semigroupTtft≥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 Rn, we achieve α=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 Ttf 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 Ttf 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 Ttf to f, as t→0+. 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 Ttf(x)-Ttf(y) for all t small is equivalent to local control of the differences Ttf(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 (Section2), we define our version of a natural diffusion distance Dg in Section 3:(3)Dgx,y≡sup0<t≤1⁡gtρtx,·-ρty,·L1,for g a bounded, nonnegative, increasing function on (0,1], with limt→0+g(t)=0. 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, Tt(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 Rn (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 Section4, we make the assumption that balls of positive radius with respect to the distance Dg 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, Section5, we define our version of local convergence of Tt(f) to f, as well as local equicontinuity of the family Ttft≥0. Both definitions use our distance Dg. 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 t0>0, Tt+t0f converges locally to Tt0f, as t converges to 0+.In the Appendix, we show that, for very general metricsD on X, not necessarily arising from diffusion,(4)∫Xρtx,yDx,ydy→0a.e.,ast→0+.This result is clearly a weaker statement than (2), but has the advantage of holding under virtually no assumptions.


## 2. Notation and Assumptions

LetX be a topological space equipped with a positive σ-finite measure. For t>0, ρt(x,y) will denote asymmetric kernel on X×X, with ρt(x,y)≥0 for all x,y. We assume that ρ satisfies the semigroup property:(5)∫Xρtx,uρsu,ydu=ρt+sx,y,for all x,y∈X, and s,t>0. In addition, we assume(6)∫Xρtx,ydy=1,for all x∈X and all t>0. We will refer to a kernel ρt satisfying the conditions above as a symmetric diffusion kernel (at time t). A typical example for ρt is the heat kernel on a Riemannian manifold (see [14], for example).For a functionf, say in L2(X) (or more generally, for any f where the following definition makes sense), we define the symmetric diffusion operator Tt, for t>0, by(7)Ttfx≜∫Xρtx,yfydy.We defineT0 to be the identity map. Note that, for all t, TtfL1≤fL1, by Fubini’s theorem, that clearly TtfL∞≤fL∞, and hence TtfLp≤fLp, for 1≤p≤∞, by interpolation.To avoid degeneracy, e.g., eachTt being the averaging operator on a space of finite mass, we make an additional assumption: Tt(f)→f in L2, as t→0+.The symmetric diffusion operatorTt has the following properties of asymmetric diffusion semigroup:(i)
T0 is the identity(ii)
Tt+s=Tt∘Ts, for all s,t≥0(iii)
Tt(f)Lp≤fLp, for 1≤p≤∞(iv)
Tt is a self-adjoint operator on L2(X)(v)
Tt(f)→f in L2, as t→0+(vi)
Tt(f)≥0 if f≥0(vii)
Tt(1)=1See 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 functiong on (0,1], with limt→0+g(t)=0, and g strictly positive on the interval (0,1], define the distance Dg by(8)Dgx,y≡sup0<t≤1⁡gtρtx,·-ρty,·L1.It is clear that the distanceDg 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 distanced(·,·), Tt(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→0+. We are thus led to impose a Lipschitz-like requirement, namely, that, for a diffusion distance d(·,·), and for t>0,(9)ct≡supx≠y,fL∞=1⁡Ttfx-Ttfydx,y<∞.It is easy to see that(10)ct=supx≠y⁡ρtx,·-ρty,·L1dx,y.Note that, for any x and y, ρt(x,·)-ρt(y,·)L1 is decreasing in t, since, for h>0,(11)ρt+hx,·-ρt+hy,·L1=∫Xρt+hx,u-ρt+hy,udu=∫X∫Xρtx,vρhv,u-ρty,vρhv,udvdu≤∫Xρtx,v-ρty,v∫Xρhv,ududv=∫Xρtx,v-ρty,vdv=ρtx,·-ρty,·L1,using (5) and (6). Letting g(t)=1/c(t) we thus see that g is increasing, and from (10) we conclude that(12)dx,y≥gtρtx,·-ρty,·L1,t>0.This last inequality motivates our Definition 1 of Dg. The restriction to t≤1 is to ensure that Dg(x,y) is finite for all x and y and is not stringent, due to the fact that ρt(x,·)-ρt(y,·)L1 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 distanceDg is that the L1 norm of the difference of two probability densities, ρt(x,·)-ρt(y,·)L1, occurring in the definition of Dg, is the (scaled) total variation distance between the probability distributions ρt(x,·) and ρt(y,·), i.e.,(13)12ρtx,·-ρty,·L1=supA⁡μt,xA-μt,yA.Here, μt,x is the measure given by μt,x(A)=∫A‍ρt(x,u)du, and μt,y is the measure given by μt,y(A)=∫A‍ρt(y,u)du for measurable A⊆X; the supremum is taken over all measurable A⊆X (see Chapter 4 of [15]).As a final argument for the naturality of our proposed diffusion distance, we calculateDg for a special case considered by the authors of [7] (for their own version of diffusion distances). We take X=Rn, g(t)=tα, and assume that the diffusion kernel has the form ρt(x,y)=t-nβϕt-β(x-y). Here, α,β>0 and ϕ is a nonnegative radial L1 function whose gradient is also in L1. The case β=1/2 is for the heat kernel (with the appropriate ϕ), and the case β=1 is for the Poisson kernel (with the appropriate ϕ). Now,(14)ρtx,·-ρty,·L1=∫Rnt-nβϕt-βx-u-ϕt-βy-udu=∫Rnϕw+t-βy-x-ϕwdw,where we made the change of variables w=t-β(x-u). Let h(z)=∫Rn‍ϕw+z-ϕ(w)dw. Then it is easy to see that h is radial and, for a “generic” ϕ, we have the estimates: h(z)~z if z≤1, and h(z)~1 if z>1. Here, · is the usual Euclidean norm. Using this observation, and (14), we obtain the following (for this special case).Proposition 2.
Forx-y≤1, Dg(x,y)~x-yα/β if α≤β, and Dg(x,y)~x-y if α≥β. For x-y>1, Dg(x,y)~1.Proof.
Using the notation for the special case above, we need to estimatesup0<t≤1tαht-β(y-x).
Let us first consider the situation whenx-y>1. Then, for 0<t≤1, t-βy-x≥1, so sup0<t≤1tαht-β(y-x)~1 using the estimate for h mentioned before the proposition.
Next, consider the situation whenx-y≤1. Let t0=x-y1/β. Note that 0<t0≤1.
Whent0≤t≤1, we have that t-βx-y≤1, so(15)tαht-βy-x~tα-βx-y. If α≤β, the maximum of the right hand side occurs at t=t0 and equals(16)t0α=x-yα/β. If α≥β, the maximum of the right hand side occurs at t=1 and equals x-y.
When0<t≤t0, we have that t-βx-y≥1, so(17)tαht-βy-x~tα, and the maximum of the right hand side occurs at t=t0 and equals(18)t0α=x-yα/β. Note that if α≥β, since x-y≤1, x-yα/β≤x-y.
Combining the above discussions for the two ranges of values oft, the result follows.Thus, for this special case ofX=Rn, g(t)=tα, and ρt(x,y)=t-nβϕt-β(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 α and β). This result seems appropriate.


## 4. A Geometric Assumption about the Measure onX

We make the following reasonable assumption about our distanceDg: for any x0∈X and any ϵ>0, (19)Bx0,ϵ≡x:Dgx0,x<ϵ,the ball of radius ϵ and center x0, has positive measure.To justify the statement that this assumption is indeed reasonable, we first define another family of subsets ofX. For any x0∈X, t>0, and ϵ>0, let(20)Nx0,t,ϵ≡x:ρtx,·-ρtx0,·L1<ϵ.We then have the following equivalence of topologies induced by the setsBx0,ϵ and Nx0,t,ϵ:Proposition 3.
For anyx0∈X and any ϵ>0, there exist t>0 and δ>0 such that Nx0,t,δ⊆Bx0,ϵ. Conversely, for any x0∈X, t>0, and ϵ>0, there exists a δ>0 such that Bx0,δ⊆Nx0,t,ϵ.Proof.
Fix anx0∈X and an ϵ>0. We first show that there exist t>0 and δ>0 such that Nx0,t,δ⊆Bx0,ϵ.
Since we made the assumption thatlimt→0+g(t)=0 for the function g used in defining the distance Dg, there exists a 0<t<1 with g(t)<ϵ/4. Let δ=ϵ/(2M), where M=sup0<s≤1g(s)=g(1). Now, pick an arbitrary x∈Nx0,t,δ.
For0<s≤t, since g in increasing, we see that(21)gsρsx,·-ρsx0,·L1≤2gt<ϵ2, Using the fact that the L1 norm of ρs(x,·) is 1 for any s and x.
Now consider the case whent≤s≤1. Note that, by definition of Nx0,t,δ, we have that ρt(x,·)-ρt(x0,·)L1<δ=ϵ/(2M). Then, for this range of s, we observe that(22)gsρsx,·-ρsx0,·L1≤Mρtx,·-ρtx0,·L1<Mϵ2M=ϵ2, where we have used that ρs(x,·)-ρs(x0,·)L1 is decreasing in s; see (11).
We conclude (see (8)) that(23)Dgx,x0=sup0<s≤1⁡gsρsx,·-ρsy,·L1<ϵ, and hence x∈Bx0,ϵ.
For the converse, fixx0∈X, t>0 and ϵ>0. We will show that there exists a δ>0 such that Bx0,δ⊆Nx0,t,ϵ.
Since, for anyx, ρs(x,·)-ρs(x0,·)L1 is decreasing in s (see (11)), we clearly have that Nx0,s1,ϵ⊆Nx0,s2,ϵ for any 0<s1<s2. Thus, we may assume 0<t<1. Let δ=ϵg(t). Then, for any x∈Bx0,δ, we have that Dgx,x0<ϵg(t). Hence, using Definition 1 of the distance Dg, we obtain(24)gtρtx,·-ρtx0,·L1≤Dgx,x0<ϵgt. Thus, ρt(x,·)-ρt(x0,·)L1<ϵ, and we have that x∈Nx0,t,ϵ.Returning to our assumption that, for anyx0∈X and any ϵ>0, Bx0,ϵ has positive measure, Proposition 3 shows that it is equivalent to require the following: for any x0∈X, t>0, and ϵ>0, the set Nx0,t,ϵ has positive measure. Note that the definition of the sets Nx0,t,ϵ is more “universal” than that of the balls Bx0,ϵ, since the former do not involve the function g.The assumption that, for anyx0∈X, t>0, and ϵ>0, the set Nx0,t,ϵ 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 ϵ>0, the set of points in our space X which have not diffused more than ϵ away (in the L1 sense) from the diffused point x0, 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=x0 is “enough”) and the continuous case (every point x0 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,Tt is a symmetric diffusion operator as defined in Section 2.Definition 4.
Letf∈Lp, 1≤p≤∞. 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 ϵ>0, there exist t0>0 and δ>0 so that Ttf(x)-f(x)<ϵ, for all t with 0<t≤t0 and all x∈Bx0,δ. We then say Ttf converges to f locally at x0.We also make the following.Definition 5.
Letf∈Lp, 1≤p≤∞. 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 ϵ>0, there exist t0>0 and δ>0 with the property that, for all x∈Bx0,δ, we have f(x)-f(x0)<ϵ and for all t with 0<t≤t0, Ttf(x)-Ttf(x0)<ϵ. We then say the family Ttft≥0 is locally equicontinuous (in t) at x0.Our main result is the following.Proposition 6.
Forf∈L2∩L∞ and any x0∈X, the following are equivalent: (i)
Ttf converges to (the representative) f locally at x0(ii)
The familyTtft≥0 is locally equicontinuous at x0
Moreover, if a representativef satisfies one of these statements, the same representative satisfies the other statement.Proof.
We first show that local convergence atx0 implies local equicontinuity at x0. We thus begin by assuming that Ttf converges to a representative f locally at x0.
First, we establish continuity of this representativef at x0. Fix ϵ>0. By the assumption, there exist 1≥t0>0 and δ>0 such that Ttf(x)-f(x)<ϵ/3, for all t with 0<t≤t0 and all x∈Bx0,δ. Then, for any x∈Bx0,δ, using the definition of the distance Dg, we see that(25)fx-fx0≤fx-Tt0fx+Tt0fx-Tt0fx0+Tt0fx0-fx0≤2ϵ3+ρt0x,·-ρt0x0,·L1fL∞=2ϵ3+gt0ρt0x,·-ρt0x0,·L1fL∞gt0≤2ϵ3+Dgx,x0gt0fL∞. Since we assumed that g(t)>0 if t>0, we have that δ′=minδ,ϵ/3g(t0)/fL∞>0. Thus, if x∈Bx0,δ′,(26)fx-fx0<2ϵ3+ϵ3gt0fL∞fL∞gt0=ϵ, and continuity of f at x0 is shown.
Next, note that(27)Ttfx-Ttfx0≤Ttfx-fx+fx-fx0+fx0-Ttfx0. Let t0 and δ>0 be as above, i.e., Ttf(x)-f(x)<ϵ/3, for all t with 0<t≤t0 and all x∈Bx0,δ. Since we have already shown that f is continuous at x0, there exists a δ′′>0 such that f(x)-f(x0)<ϵ/3 for x∈Bx0,δ′′. Let δ′′′=min(δ,δ′′). Then, for x∈Bx0,δ′′′ and 0<t≤t0, we see that Ttf(x)-Ttf(x0)<ϵ. Hence, the local equicontinuity of the family Ttft≥0 at x0 follows.
Conversely, we now show that local equicontinuity atx0 implies local convergence at x0. We thus begin by assuming that the family Ttft≥0 is equicontinuous at x0.
Fixϵ>0. By the assumption, there exist 1≥t0>0 and δ>0 such that, for the representative f, f(x)-f(x0)<ϵ/5 and Ttf(x)-Ttf(x0)<ϵ/5, for all x∈Bx0,δ and all t with 0<t≤t0. In Section 4, we made the assumption that all balls of positive radius have positive measure. Using Stein’s Maximal Theorem (see Chapter III, §3 in [1]), limt→0+Ttf=f a.e. So there is a y0∈Bx0,δ such that limt→0+Ttf(y0)=f(y0). Now, for x∈Bx0,δ,(28)Ttfx-fx≤Ttfx-Ttfy0+Ttfy0-fy0+fy0-fx.
We estimate the first term on the right hand side of the above inequality as follows:(29)Ttfx-Ttfy0≤Ttfx-Ttfx0+Ttfx0-Ttfy0<2ϵ5, for all 0<t≤t0, since x,y0∈Bx0,δ. For the second term, we use that limt→0+Ttf(y0)=f(y0): there exists a t1>0 such that Ttf(y0)-f(y0)<ϵ/5, for all t satisfying 0<t≤t1. Finally, for the third term, we see that(30)fy0-fx≤fy0-fx0+fx0-fx<2ϵ5, since x,y0∈Bx0,δ.
Thus, for allt with 0<t≤min(t0,t1), and for any x∈Bx0,δ, we obtain that Ttf(x)-f(x)<ϵ, which concludes the proof of the converse.In the proof above, we used Stein’s Maximal Theorem (see Chapter III, §3 in [1]) to state that limt→0+Ttf=f a.e. Stein’s a.e. convergence result, for f∈L2 say, is the main place in our paper where the symmetry of the operators Tt is needed: Stein requires symmetry to prove his Maximal Theorem.We immediately have the following.Corollary 7.
Letf∈L2∩L∞. Fix t0>0. Then for any x0∈X, TtTt0f=Tt+t0f converges locally to Tt0f at x0.Proof.
By Proposition6, it suffices to show that Tt+t0ft≥0 is locally equicontinuous at x0. Fix ϵ>0. Let G(t)=g(t) for 0<t≤1 and G(t)=g(1) for t>1. For any t≥0, we have that(31)Tt+t0fx-Tt+t0fx0≤ρt+t0x,·-ρt+t0x0,·L1fL∞=Gt+t0ρt+t0x,·-ρt+t0x0,·L1Gt+t0fL∞≤Dgx,x0Gt+t0fL∞≤Dgx,x0Gt0fL∞, using the definition of the distance Dg and the function G, that G is increasing, and inequality (11). Then, for δ=G(t0)/fL∞ϵ, we see that x∈Bx0,δ implies that Tt+t0f(x)-Tt+t0f(x0)≤ϵ, and we have shown local equicontinuity at x0.Using our notation, Stein in [1] mentions that limt→0+Tt+t0f(x)=Tt0f(x) for almost all x, since he proves that Ttf 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 Dg.


## 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 ofTtf to (a representative) f at a point is equivalent to local equicontinuity of the family Ttft≥0 at that point.It may well be useful to have a quantitative estimate on the rate of convergence ofTtf to f under the assumption that f is Lipschitz, say, with respect to some distance d (where d may be our Dg). As essentially pointed out in the papers [7, 11], a key issue is whether, and how rapidly,(32)supx⁡∫Xρtx,ydx,ydy→0,ast→0+.In the Appendix below, we show that, for very general metricsD on X, not necessarily arising from diffusion,(33)∫Xρtx,yDx,ydy→0a.e.,ast→0+.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.



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