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P R ho d ). , We provide an alternative construction, using the resistance form techniques developed by Kigami in [22] to define a local, regular Dirichlet form on the measure-metric space ( ,d ,µ). Given this Dirichlet form, standard results allow the construction of an asso- T T ciated Markov process, X =(Xt)t 0 with invariant measure µ, which we show is actually ≥ Brownian motion (as defined in [2]) on . The construction used here seems more natu- T ral, allowing us to define the Dirichlet form for Brownian motion directly. Furthermore, the arguments we use to deduce our process satisfies the properties of Brownian motion are more concise, using more recently developed techniques for resistance forms, rather than limiting arguments. Once a Markov process is established on , it is natural to ask whether it has a transition density, and if it does, what form doesT the transition density take? The current literature on measure-metric spaces equipped with a resistance form seems to indicate that an important part of the answer to this question is the volume growth of the space with respect to the resistance metric, see [7] and [25]. However, for certain random subsets it has been shown that they do not satisfy the kind of uniform volume growth that is often assumed. For example, in [19], Hambly and Jones show how for a certain class of random recursive fractals the volume of balls of radius r have fluctuations of order of powers of 1 α ln r− about a leading order polynomial term, r . With the random self-similarity of the CRT ([5]), it is reasonable to expect similar behaviour for the CRT. Indeed, it has already been shown that the CRT and a class of recursive fractals do exhibit the same α 1 θ form of Hausdorff measure function, r (ln ln r− ) , with α = 2, θ = 1 in the case of the CRT (see [13], Corollary 1.2 and [18], Theorem 5.2). Motivated by the random fractal examples, heat kernel bounds for measure-metric spaces with volume fluctuations have been established in [12]; these suggest that to establish global heat kernel estimates for the CRT, it will be useful to determine the volume growth behaviour. Henceforth, define the ball of radius r around the point σ to be ∈ T
B(σ, r) := σ′ : d (σ, σ′) Theorem 1.1 Let ρ be the root of , then T 2r2 E(µ(B(ρ, r))) = 1 e− , r 0. − ∀ ≥ Theorem 1.2 P-a.s., there exist random constants C , C , C , C (0, ) such that 1 2 3 4 ∈ ∞ 2 1 2 1 C1r ln1 r− sup µ(B(σ, r)) C2r ln1 r− , ≤ σ ≤ ∈T 2 and 2 1 1 2 1 C3r ln1 r− − inf µ(B(σ, r)) C4r ln1 ln1 r− , σ ≤ ∈T ≤ for r (0, diam ), where diam is the diameter of ( ,d ) and ln1 x := ln x 1. ∈ T T T T ∨ Locally, we prove the following volume asymptotics, which show that the volume growth of a ball around a particular point has fluctuations about r2 of the order of 1 1 ln ln r− asymptotically. This exactly mirrors the ln ln r− local fluctuations exhibited by the random recursive fractals of [19]. We remark that the lim sup result has also been proved (up to a constant multiple) in the course of deriving the Hausdorff measure function of in [13]. However, we extend the result proved there by deducing the exact T value of the constant. Theorem 1.3 P-a.s., we have µ(B(σ, r)) 8 lim sup 2 1 = 2 , r 0 r ln ln r− π → and also µ(B(σ, r)) lim inf =2, r 0 2 1 1 → r (ln ln r− )− for µ-a.e. σ . ∈ T The global volume bounds of Theorem 1.2 mean that the CRT satisfies the non- uniform volume doubling of [12], P-a.s. Results of that article immediately allow us to deduce the existence of a transition density for the Brownian motion on and the following bounds upon it. T Theorem 1.4 P-a.s., the Brownian motion X =(Xt)t 0 on exists, and furthermore, ≥ T it has a transition density (pt(σ, σ′))σ,σ ,t>0 that satisfies, for some random constants ′∈T C ,C ,C ,C , t > 0 and deterministic θ , θ , θ (0, ), 5 6 7 8 0 1 2 3 ∈ ∞ 3 1/2 θ2 2 1 θ1 d d p (σ, σ′) C t− 3 (ln t− )− exp C ln , (1) t ≥ 5 1 − 6 t 1 t ( ) and 3 1/2 θ3 2 1 1/3 d d − p (σ, σ′) C t− 3 (ln t− ) exp C ln , (2) t ≤ 7 1 − 8 t 1 t ( ) for all σ, σ′ , t (0, t0), where d := d (σ, σ′) and ln1 x := ln x 1. ∈ T ∈ T ∨ This result demonstrates that the heat kernel decays exponentially away from the diagonal and there can be spatial fluctuations of no more than logarithmic order. The following theorem that we prove for the on-diagonal part of the heat kernel shows that global fluctuations of this order do actually occur. Locally, the results we obtain are not precise enough to demonstrate the P-a.s. existence of fluctuations, see Theorem 1.6. However, they do show that there can only be fluctuations of log-logarithmic order, and combined with the annealed result of Proposition 1.7, they prove that log-logarithmic fluctuations occur with positive probability. 3 Theorem 1.5 P-a.s., there exist random constants C9,C10,C11,C12, t1 > 0 and deter- ministic θ (0, ) such that for all t (0, t ), 4 ∈ ∞ ∈ 1 2/3 1 14 2/3 1 1/3 C9t− (ln ln t− )− sup pt(σ, σ) C10t− (ln t− ) , (3) ≤ σ ≤ ∈T 2/3 1 θ4 2/3 1 1/3 C11t− (ln t− )− inf pt(σ, σ) C12t− (ln t− )− . (4) σ ≤ ∈T ≤ Theorem 1.6 P-a.s., for µ-a.e. σ , there exist random constants C13,C14, t2 > 0 such that for all t (0, t ), ∈ T ∈ 2 2/3 1 14 2/3 1 1/3 C t− (ln ln t− )− p (σ, σ) C t− (ln ln t− ) , 13 ≤ t ≤ 14 and also p (σ, σ) lim inf t < . t 0 2/3 1 1/3 → t− (ln ln t− )− ∞ The final estimates we prove are annealed heat kernel bounds at the root of , which 2/3 T show that the expected value of pt(ρ, ρ) is controlled by t− with at most O(1) fluctua- tions as t 0. → Proposition 1.7 Let ρ be the root of , then there exist constants C ,C such that T 15 16 2/3 2/3 C t− E (p (ρ, ρ)) C t− , t (0, 1). 15 ≤ t ≤ 16 ∀ ∈ At this point, a comparison with the results obtained by Barlow and Kumagai for the random walk on the incipient cluster for critical percolation on a regular tree, [8], is pertinent. First, observe that the incipient infinite cluster can be constructed as a particular branching process conditioned to never become extinct and the self-similar CRT (see [2]) can be constructed as the scaling limit of a similar branching process. Note also that the objects studied here and by Barlow and Kumagai are both measure-metric space trees and so similar probabilistic and analytic techniques for estimating the heat kernel may be applied to them. Consequently, it is not surprising that the quenched local heat kernel bounds of [8] exhibit log-logarithmic differences similar to those obtained in this article and furthermore, the annealed heat kernel behaviour at the root is also shown to be the same in both settings. It should be noted though that the volume bounds which are crucial for obtaining these heat kernel bounds are proved in very different ways. Here we use Brownian excursion properties, whereas in [8], branching process arguments are applied. Unlike in [8], we do not prove annealed off-diagonal heat kernel bounds. This is primarily because there is no canonical way of labeling vertices (apart from the root) in the CRT. A more concrete connection between the Brownian motion on the CRT and simple random walks on random graph trees is provided in [11], where it is shown that if n is a family of random graph trees that, when rescaled, converge to the CRT in a suitableT space, then the associated random walks Xn, when rescaled, converge to the Brownian motion on . In turn, this result is closely linked to the convergence of the rescaled T 4 height process (which measures the distance of a simple random walk from the root) on the incipient cluster for critical percolation on a regular tree, see [21], providing further evidence in support of the fact that the observations of the previous paragraph are not merely coincidental. The article is organised as follows. In Section 2, we provide definitions of and give a brief introduction to the main objects of the discussion, namely the normalised Brownian excursion and the CRT. In Section 3 we prove several results about the sample paths of the normalised Brownian excursion, which are then used to obtain quenched and annealed volume bounds in Section 4. In Section 6, we explain how already established results about dendrites and measure-metric spaces may be applied to the CRT to construct a process on and derive the quenched transition density estimates of Theorems 1.4, 1.5 T and 1.6. The annealed heat kernel bounds are proved in Section 7. The proofs of both the quenched and annealed heat kernel bounds use the results established in Section 5 about the upcrossings of the normalised Brownian excursion. Finally, we show that the process we have constructed is actually Brownian motion on in Section 8. T 2 Preliminaries 2.1 Normalised Brownian excursion An important part of the definition of the CRT is the Brownian excursion, normalised to have length 1, a process which we will denote W . In this section, we provide two characterisations of the law of the Brownian excursion and describe the appropriate nor- malisation. We begin by defining the space of excursions, U. First, let U ′ be the space of functions f : R R for which there exists a τ(f) (0, ) such that + → + ∈ ∞ f(t) > 0 t (0, τ(f)). ⇔ ∈ We shall take U := U ′ C(R+, R+), the restriction to the continuous functions contained ∩ (s) in U ′. The space of excursions of length s is then defined to be U := f U : τ(f)= s . Our first description of the law of W involves conditioning the{ Itˆoexcursion∈ law,} which arises from the Poisson process of excursions of a standard Brownian motion. We will denote by B =(Bt)t 0 a standard, 1-dimensional Brownian motion starting from 0, ≥ built on a probability space with probability measure P. Since the Itˆoexcursion law has been widely studied, we shall omit most of the technicalities here. For more details of excursion laws for Markov processes, the reader is referred to [27], Chapter XII, and [28], Chapter VI. 1 Let Lt be the local time of B at 0, and Lt− := inf s > 0 : Ls > t be its right 1 1 { } continuous inverse. Wherever L− = L− , we define et U to be the (positive) excursion t− t at local time t. In particular, 6 ∈ 1 1 BL 1+s , 0 s Lt− L− , e (s) := t− t− t − ≤ ≤1 −1 ( 0, s>Lt− L− . − t− 1 1 The set of excursions of B is denoted by Π := (t, et): L− = Lt− . The key idea is that { t− 6 } Π is a Poisson process on (0, ) U. More specifically, there exists a σ-finite measure, ∞ × 5 n, on U such that, under P, # (Π ) =d N( ), ∩· · where N is a Poisson random measure on (0, ) U with intensity dt n(df). Bearing this result in mind, even though it has infinite∞ mass,× the measure n can be considered to be the “law” of the (unconditional) Brownian excursion. Our second description of the law of the Brownian excursion is the well-known Bismut decomposition. In short, this characterisation of n describes how, if (t, f) is chosen according to the measure 1[0,τ(f)](t)dt n(df), then the law of f(t) is Lebesgue measure on R+ and, conditionally on f(t)= a, the processes (f(s t))s 0 and (f((τ(f) s) t))s 0 are ∧ ≥ − ∨ ≥ independent 3-dimensional Bessel processes run until they last hit a. More precisely, let m be the law of the 3-dimensional Bessel process, and note that, for every a 0 and m-a.e. f C(R , R ), the last hitting time of a by f, defined by T (f) := sup t ≥ 0 : f(t)= a ∈ + + a { ≥ } is finite. The result of interest may now be written as follows: for every non-negative 2 measurable function F on C(R+, R+) , τ(f) n(df) dtF ((f(s t))s 0, (f((τ(f) s) t))s 0) ∧ ≥ − ∨ ≥ Z Z0 ∞ = 2 da m(df1) m(df2)F ((f1(s Ta(f1)))s 0, (f2(s Ta(f2)))s 0). ∧ ≥ ∧ ≥ Z0 Z Z See [27], Theorem XII.4.7 for a proof. We now describe how the excursion measure n can be decomposed into a collection of related measures on the spaces of excursions of a fixed length. For c > 0, the re- normalisation operator Λ : U U is defined by c → 1 Λ (f)(t)= f(ct), t 0, f U. c √c ∀ ≥ ∈ Clearly, if f U, then Λ (f) U (1). Now, according to Itˆo’s description of n, (see ∈ τ(f) ∈ [27], Theorem XII.4.2), we can write, for any measurable A U, ⊆ ∞ ds n(A)= n(s)(A U (s)) , (5) 3 0 ∩ √2πs Z where, for each s, n(s) is a probability measure on U (s) that satisfies n(s) = n(1) Λ , ◦ s for some unique probability measure n(1) on U (1). In particular, we have that n(1) = n( τ(f) = 1). A U (1)-valued process which has law n(1) is said to be a normalised Brownian·| excursion. Henceforth, we assume that W is a normalised Brownian excursion built on the probability space with probability measure P. 2.2 Continuum random tree The connection between trees and excursions is an area that has been of much recent interest. In this section, we look to provide a brief introduction to this link and also a definition of the CRT, which is the object of interest of this article. Given a function f U, we define a distance on [0, τ(f)] by setting ∈ d (s, t) := f(s)+ f(t) 2m (s, t), (6) f − f 6 where m (s, t) := inf f(r): r [s t, s t] . Then, we use the equivalence f { ∈ ∧ ∨ } s t d (s, t)=0, ∼ ⇔ f to define := [0, τ(f)]/ . We can write this as = σ : s [0, τ(f)] , where Tf ∼ Tf { s ∈ } σs := [s] is the equivalence class containing s. It is then elementary (see [15], Section 2) to check that d (σs, σt) := df (s, t), Tf defines a metric on f , and also that ( f ,d ) is a compact real tree in the sense of Tf [15], Definition 2.1 (thereT the term usedT is R-tree). In particular, is a dendrite: an Tf arc-wise connected topological space, containing no subset homeomorphic to the circle. Furthermore, the metric d is a shortest path metric on f , which means that it is Tf additive along (injective) paths of . The root of the treeT is defined to be the Tf Tf equivalence class σ0 and is denoted by ρf . A natural volume measure to put on is the projection of Lebesgue measure on Tf [0, τ(f)]. For open A , let ⊆ Tf µ (A) := ℓ ( t [0, τ(f)] : σ A ) , f { ∈ t ∈ } where, throughout this article, ℓ is the usual 1-dimensional Lebesgue measure. This defines a Borel measure on ( f ,d ), with total mass equal to τ(f). Tf The CRT is then simplyT the random dendrite that we get when the function f is chosen according to the law of the normalised Brownian excursion. This differs from the Aldous CRT, which is based on the random function 2W . Since this extra factor only has the effect of increasing distances by a factor of 2, our results are easily adapted to apply to Aldous’ tree. In keeping with the notation used so far in this section, the measure- metric space should be written ( W ,d W ,µW ), the distance on [0, τ(W )], defined at (6), T T dW , and the root, ρW . However, we shall omit the subscripts W with the understanding that we are discussing the CRT in this case. We note that τ(W ) = 1, P-a.s., and so [0, τ(W )] = [0, 1] and µ is a probability measure on , P-a.s. Finally, it is clear from the T compactness of that the diameter of , diam , is finite P-a.s. T T T 2.3 Other notation The δ-level oscillations of a function y on the interval [s, t] will be written osc(y, [s, t], δ) := sup y(r) y(r′) , r,r′ [s,t]: | − | r ∈r δ | ′− |≤ and we will denote by c constants taking a value in (0, ). We shall also continue to use . ∞ the notation introduced in Theorems 1.2 and 1.4, ln x := ln x 1. 1 ∨ 3 Brownian excursion properties In this section, we use sample path properties of standard 1-dimensional Brownian motion and the 3-dimensional Bessel process to deduce various sample path properties for the normalised Brownian excursion. The definitions of the random variables B and W and measures n and n(1) should be recalled from Section 2.1. 7 Lemma 3.1 P-a.s., osc(W, [0, 1], δ) lim sup = √2. δ 0 √δ ln δ 1 → − Proof: The proof of this lemma from L´evy’s 1937 result on the modulus of continuity of a standard Brownian motion in R (see [27], Theorem I.2.7, for example) and the Poisson process description of the excursion law n is standard, and so we omit it. Lemma 3.2 P-a.s., inft [0,1 δ] osc(W, [t, t + δ], δ) lim sup ∈ − < . (7) δ 0 δ(ln δ 1) 1 ∞ → − − Proof: We start by proving the correspondingp result for a standard Brownian motion. Fix a constant c and then, for n 0, 1 ≥ n n n n 1 P inf osc(B, [j2− , (j + 1)2− ], 2− ) c12− 2 n− 2 j=0,...,2n 1 ≥ − n n n n 1 n = P osc(B, [j2− , (j + 1)2− ], 2− ) c 2− 2 n− 2 , j =0,..., 2 1 ≥ 1 − 2n n n n 1 = P osc(B, [0, 2− ], 2− ) c 2− 2 n− 2 , ≥ 1 using the independent increments of a Brownian motion for the second equality. The probability in this expression is bounded above by c3n c1 4n c2 P sup B = P T 1 c e− 1 , t 1+ n 1 BE (0,1) 2 2 t [0,2 n] | | ≥ 2 2 n 2 ≤ c1 ≤ − ∈ − ! for some constants c2,c3, where TBE (0,1) represents the exit time of a standard Brownian motion from a Euclidean ball of radius 1 about the origin. The distribution of this random variable is known explicitly, see [10], and the above tail estimate is readily deduced from x the expression that is given there. Using the fact that 1 x e− for x 0 and summing over n, we have − ≤ ≥ c3n ∞ n 1 ∞ n − c2 n n n 2 c2e 1 P inf osc(B, [j2− , (j + 1)2− ], 2− ) c12− 2 n− 2 e− , j=0,...,2n 1 ≥ ≤ n=0 − n=0 X X which is finite for c1 chosen suitably large. Hence Borel-Cantelli implies that, P-a.s., there exists a constant c4 such that n n n n 1 inf osc(B, [j2− , (j + 1)2− ], 2− ) c42− 2 n− 2 , n 0. j=0,...,2n 1 − ≤ ∀ ≥ (n+1) n Let δ (0, 1], then δ [2− , 2− ] for some n 0. Hence ∈ ∈ ≥ n n inf osc(B, [t, t + δ], δ) inf osc(B, [t, t +2− ], 2− ) n t [0,1 δ] ≤ t [0,1 2− ] ∈ − ∈ − n n n inf osc(B, [j2− , (j + 1)2− ], 2− ) j=0,...,2n 1 ≤ − n 1 c 2− 2 n− 2 ≤ 4 c δ(ln δ 1) 1, ≤ 5 − − p 8 which proves (7) holds when W is replaced by B. By rescaling, an analogous result holds for any interval with rational endpoints. By countability and a monotonicity argument, this is easily extended to P-a.s., inft [r,s δ] osc(B, [t, t + δ], δ) lim sup ∈ − < , 0 r Lemma 3.3 P-a.s., 1 1 Wt<δ dt 8 0 { } lim sup 2 1 = 2 , δ 0 δ ln ln δ− π → R and 1 1 Wt<δ dt lim inf 0 { } =2. δ 0 δ2 ln ln δ 1 → R − Proof: By applying the Bismut decomposition of n, see Section 2.1, to complete the 1 proof, it will be sufficient to check that the lemma holds when 1 Wt<δ dt is replaced 0 { } by ∞ ∞ R 1 R1<δ dt + 1 R2<δ dt, (8) { t } { t } Z0 Z0 where, under the probability measure P, the processes R1 and R2 are independent 3- dimensional Bessel processes. In the remainder of the proof we will write the first and second summands of (8) as T 1(δ) and T 2(δ) respectively, and also T (δ) := T 1(δ)+ T 2(δ). The lim sup asymptotic behaviour of the occupation time of 3-dimensional Brownian motion in a ball was considered in [10], Theorem 3, where it was shown that, P-a.s., T 1(δ) 8 lim sup 2 1 = 2 . δ 0 δ ln ln δ− π → 2 1 2 From this result we clearly have lim supδ 0 T (δ)/δ ln ln δ− 8/π . The identical upper → ≥ bound will follow from a simple Borel-Cantelli argument, in a similar manner to the proof of [10], Theorem 3, if we can show that for every ε > 0, there exists a constant c6 such that 2 2 ( π ε)t P(T (δ) > δ t) c e− 8 − , (9) ≤ 6 for every strictly positive t and δ. To prove this first note that from [10], Theorem 1, 1 2 π2t/8 we have P(T (δ) > δ t) e− for every t,δ > 0. Consequently, using an argument analogous to the proof of≤ [29], Lemma 6.5, we are able to obtain, for k N, ∈ δ2t P(T (δ) > δ2t) = P(T 1(δ) > δ2t s)dP(T 1(δ) s)+ P(T 1(δ) > δ2t) − ≤ Z0 k iδ2t/k = P(T 1(δ) > δ2t s)dP(T 1(δ) s)+ P(T 1(δ) > δ2t) 2 − ≤ i=1 (i 1)δ t/k X Z − 9 k P(T 1(δ) > δ2t(1 i/k))P(T 1(δ) > δ2t(i 1)/k) ≤ − − i=1 X + P(T 1(δ) > δ2t) π2t(1 1/k)/8 (k + 1)e− − . ≤ By choosing k suitably large, we have the upper bound at (9), and the lim sup result follows. For the lim inf result, we can refer directly to [29], Theorem 6.8, where it was 2 1 1 shown that lim infδ 0 T (δ)/δ (ln ln δ− )− = 2, P-a.s. → 4 Volume results In this section, we apply properties of the normalised Brownian excursion, including those introduced in Section 3, to deduce the results about the volume growth on the CRT that were stated in the introduction. We start by proving the annealed volume result of Theorem 1.1; this follows easily from the expected occupation time of [0,r) for a normalised Brownian excursion, for which an explicit expression is known. Proof of Theorem 1.1: By definition, we have that 1 µ(B(ρ, r)) = 1 Ws We now proceed to deduce the global upper volume bound for . The three main ingredients in the proof are the modulus of continuity result provedT in Lemma 3.1, a bound on the tail of the distribution of the volume of a ball about the root, and the invariance under re-rooting of the CRT. This final property is also important in proving the local volume growth results. Before continuing, we define precisely what we mean by re-rooting and state the invariance result that we will use. (s) (s) Given W and s [0, 1], we define the shifted process W =(Wt )0 t 1 by ∈ ≤ ≤ (s) Ws + Ws+t 2m(s,s + t), 0 t 1 s Wt := − ≤ ≤ − Ws + Ws+t 1 2m(s + t 1,s), 1 s t 1, − − − − ≤ ≤ where m = mW is the minimum function defined in Section 2.2. It is known that, for any fixed s, the process W (s) has the same distribution as W , (see [26], Proposition 4.9). From this fact, an elementary application of Fubini’s theorem allows it to be deduced that, for any measurable A U (1), ⊆ E(ℓ s [0, 1] : W (s) A )= P(W A), (11) { ∈ ∈ } ∈ 10 where, as in Section 2.2, ℓ is Lebesgue measure on [0,1]. Written down in terms of excursions, it is not immediately clear what this result is telling us about the CRT. However, if we observe that the real tree associated with W (s) has exactly the same structure as , apart from the fact that it is rooted at σs instead of ρ, and recall that µ is the imageT of ℓ under the map s σ , then we are able to obtain to conclude from 7→ s (11) that, heuristically, any property of the random real tree that holds when the root is assumed to be ρ will also hold when we consider the rootT to be σ, for µ-a.e. σ . More precisely, if we use the notation σ to represent the real tree with root σ ∈ T , T T ∈ T then E(µ σ : σ A )= P( A), (12) { ∈ T T ∈ } T ∈ for any measurable subset, A, of the space of compact rooted real trees. The following result provides an exponential tail bound for the distribution of the volume of a ball of radius r about the root that will be useful in the proof of Proposition 4.2. Lemma 4.1 There exist constants c ,c such that, for all r> 0,λ 1, 7 8 ≥ 2 c8λ P µ(B(ρ, r)) r λ c e− . ≥ ≤ 7 Proof: First, observe that we can consider n(1) to the law of the excursion of Brownian motion straddling a fixed time, conditional on this excursion having length 1, see [27], Proposition XII.3.4. Hence it follows from [9], Theorem 9 that, for r> 0, 1 k 2k E 1 Wt