TRANSITION DENSITY ASYMPTOTICS FOR SOME DIFFUSION PROCESSES WITH MULTI-FRACTAL STRUCTURES

We study the asymptotics as t ! 0 of the transition density of a class of (cid:22) -symmetric di(cid:11)usions in the case when the measure (cid:22) has a multi-fractal structure. These di(cid:11)usions include singular time changes of Brownian motion on the unit cube. Research partially supported by a NSERC (Canada) grant and Grant-in-Aid for Scienti(cid:12)c Research (B)(2) 10440029 of Japan.

The transition density p t (x, y) of X satisfies the heat equation ∂ p t /∂t = L p t , and for x ∈ [0, 1] the short time asymptotics of p t (x, x) are given by Now let µ be a measure on K, with closed support K, and consider the Dirichlet form E(f, f ) on L 2 (K, µ). In probabilistic terms the associated process X can be obtained by a time change of X. Set A t = K L a t µ(da), where (L a t ) are the (jointly continuous) local times of X, and let τ t = inf{s : A s > t} be the right-continuous inverse of A. Then (see [9], Theorem 6.2.1), X t = X τt . If dµ/dx = a(x), where a is strictly positive and continuous, then X has a generator Lf (x) = 1 2 a(x) −1 ∆f (x), and the transition density p t (x, y) of X satisfies p t (x, x) ∼ (2a(x)πt) −1/2 , t → 0.
In this paper we wish to study the short time asymptotics of p t (x, x) in the case when µ is singular with respect to Lebesgue measure, but still has closed support equal to K. For the moment we will just discuss the case K = [0, 1], but our results do hold for more general selfsimilar sets. We will assume that the measure µ is "multi-fractal" or self-similar. For [0, 1] examples of measures of this kind are the de Rham p-measures µ = µ (p) , where 0 < p < 1. µ (p) is characterized by the property that, for any n ≥ 1 and 0 ≤ k ≤ 2 n − 1, µ (p) ([k2 −n , k2 −n + 2 −(n+1) ]) = pµ (p) ([k2 −n , (k + 1)2 −n ]).
(This is the measure under which the coefficients x i in the dyadic expansion of x are independent identically distributed random variables with mean 1 − p.) for those x ∈ [0, 1] for which this limit exists. 1) If x is a dyadic rational then d s (x)/2 = a(1)/(log 2 + a(1)) = log(1/p)/ log(2/p).

3) There exist points x at which
In fact our methods handle more general compact self-similar sets K, and include the following: 1) P.c.f. fractals with a 'regular harmonic structure' -see [16].
The unit interval is a special case of 1), and we can treat 2) as a special case of 4). In cases 1) and 3) the underlying diffusion is that given by the harmonic structure, while for 2) it is standard Brownian motion on the unit cube, with normal reflection on the boundary. For 4) it is the diffusion constructed in [2]. We restrict ourselves to self-similar (Bernoulli) measures µ for which the topological support is the whole of K. In case 1) this is the only condition on µ, but in the other cases a further condition (see (2.2)) is needed to ensure that µ does not charge sets of capacity zero.
The main results of this paper are Theorem 3.5 and Corollary 3.6, which give upper and lower bounds on the transition density p t (x, x). Specializing to the case K = [0, 1] we obtain Theorem 1.1.
The essential idea of this paper is to decompose K into regions D (n) i such that the process X takes a time O(e −n ) to cross each of these sets. The self-similarity of K means that these sets are all the same 'shape', but in general different 'sizes'. We therefore expect that, for most This estimate turns out to be correct whenever, on one hand, t is small enough so that P x (X t ∈ D (n) i ) > c > 0, and on the other hand t is large enough so that p t (x, ·) has diffused over a significant proportion of D (n) i . We will see that when x is suitably far from the boundary of D (n) i then (with a few added constants) (1.1) holds.
We can, however, have adjacent regions D , where n i n/(log(2/p i )) and p 1 = 1 − p, p 2 = p. Since p t (x, x) is continuous, (1.1) clearly cannot hold close to 1 2 . In this paper we do not tackle the problem, which seems in general quite hard, of identifying how p t (x, x) behaves in these boundary zones. We are, however, able to show that the sets of bad points (where our upper and lower bounds differ significantly) is small, and this enables us to make the kind of estimates given in Theorem 1.1.
If we set J γ = {x ∈ K : d s (x) exists and equals to γ} then we have a multi-fractal decomposition of K into {J γ } γ and K \∪ γ J γ . A forthcoming paper, [12], studies the Hausdorff dimensions of these sets.

Dirichlet forms on some self-similar sets with multi-fractal measure 2.1 Self-similar sets
In this section we describe the spaces we consider, and give the properties of the Dirichlet forms on them that we will need. We begin with the definition of a self-similar space: see [1], [18] for more details and examples.
2) For w ∈ Σ, we denote the i-th element in the sequence by w i and write w = w 1 w 2 w 3 · · ·.
We remark that the unit interval is a simple example of a self-similar structure: take N = 2 so that Σ = {1, 2} N and let Then the critical set of L is defined by and the post critical set of L is defined by See [1], Section 5, for the computation of C(L) and P for some simple examples. In the case of the unit interval, with the self-similar structure given above, we have P (L) = {0, 1}.
For m ≥ 0, let We call V 0 the boundary of K. A Bernoulli (probability) measure on K is a measure µ on K such that µ(F i (K)) = µ i > 0, where N i=1 µ i = 1. For u ∈ L 1 (K, µ) we writeū = K udµ. In this paper, we will consider connected self-similar sets (K, S, {F s } s∈S ), with a local regular Dirichlet forms (E, F) on L 2 (K, µ) which satisfy the following assumption.

Assumption 2.3 (a) (E, F ) is a closed local regular Dirichlet form on
Further, (E, F ) satisfies, for some ρ i > 0, i ∈ S, the following self-similarity property: The semigroup (P t ) t≥0 associated with E on L 2 (K, µ) has a jointly continuous density p t (x, y), t > 0, x, y ∈ K. (This is the transition density of the associated diffusion process X with respect to µ.) In the remainder of this section we will discuss the existence of Dirichlet forms satisfying this assumption for the two classes of spaces treated in this paper: p.c.f self-similar sets, and Sierpinski carpets.
First we give some more notation. Set t i = ρ i /µ i , for 1 ≤ i ≤ N . We remark that ρ i can be interpreted as the conductance associated with F i (K) and that t −1 i is the time scaling factor for the diffusion process on F i (K). Let Λ n be defined by Throughout the paper, we denote We call a set of the form . For x ∈ K − V * let Λ r (x) be the length of the word of the Λ r -complex to which x belongs: note that D Λr(x) (x) = D Λr (x).

P.c.f. self-similar sets and their Dirichlet forms
We call the self-similar set (K, S, {F s } s∈S ) a p.c.f fractal set if the post critical set P (L) is a finite set -p.c.f. here stands for 'post critically finite'. This condition implies that K is finitely ramified.
These sets were introduced by Kigami ( [16]). In [16], [18], [20] it is shown that, provided a 'non-degenerate harmonic structure' exists, then a closed regular local Dirichlet form satisfying (2.1) exists, with the property that E(f, f ) = 0, f ∈ F, implies that f is constant. (For work on the existence of non-degenerate harmonic structures see [25], [23].) In [16] the additional hypothesis of 'regularity' of the harmonic structure was imposed: in our context this means that the conductivities ρ i satisfy We now summarise how the remainder of Assumption 2.3 is proved in this case. Because the resolvent operator is compact (see [18], [20]) and P t f = f if and only if f is constant, there is a spectral gap so that (2.3) holds.
Let L µ be the self-adjoint operator on L 2 (K, µ) associated with the Dirichlet form (E, F ), and let {λ n } n be the eigenvalues of −L µ and {ϕ n } n be the normalized eigenfunctions. In [18], it is proved that ϕ n is continuous and where κ depends only on the Dirichlet form and K. Thus, by Mercer's theorem, and the right hand side converges uniformly. This proves joint continuity of the transition density, and completes the verification of Assumption 2.3.
Let n µ (x) = #{λ : λ is an eigenvalue of −L µ ≤ x.}. In [19], [18] it is proved that, if d e s (µ) > 0 is the unique positive number satisfying In the case when (2.4) and (2.5) holds, let ν be the Bernoulli measure satisfying Then max µ d e s (µ)/2 (where µ is taken to be a Bernoulli measure on K) is attained only at ν, and the maximum value is σ/(σ+1). For this special case, (i.e. µ = ν) detailed estimates on p t (x, y) are obtained in [13]. We remark that if (2.4) holds then (2.2) is satisfied for any Bernoulli measure µ (with µ i > 0), and that d e s (µ) < 2. In general, however, it is possible to have d e s (µ) > 2.

Sierpinski carpets and their Dirichlet forms
Here (see [5]) (SC1) and (SC2) are essential, while (SC3) and (SC4) are included for technical convenience. The main difference from p.c.f. self-similar sets is that Sierpinski carpets are infinitely ramified: the critical set C(L) in Definition 2.2 is infinite, and K cannot be disconnected by removing a finite number of points. In fact, for the classical Sierpinski carpet can be included as an example of a Sierpinski carpet by taking N = l d .
We write ν for the Bernoulli measure with weights ν i = 1/N : ν is a multiple of the Hausdorff measure on K. In [2], [21], [5], [14] a non-degenerate Dirichlet form E on L 2 (K, ν) is constructed on these spaces, with the property that E is invariant under local isometries of K -and in particular E is the same on each k-complex. The uniqueness of E is an open problem -see [5]. If E were unique then (2.1) would follow immediately. However, without requiring uniqueness, in [21] (see also Remark 5.11 of [5] and [17]) a compactness argument is used to construct a Dirichlet form E with the same invariances as E and in addition satisfying (2.1) in the case when, for a constant ρ K depending on K, Let t K = t i = Nρ K , and let X = ( X t , t ≥ 0) be the diffusion associated with E and L 2 (K, ν). We define d w = log t K / log l, the walk dimension of K, and d s = 2 log N/ log t K , the spectral As X satisfies the same local isotropy condition as the processes studied in [2], [5], the techniques of those papers apply to X and lead to the same estimates for the Green's function and transition density of the process.
Let µ be a Bernoulli measure satisfying (2.2). We now verify Assumption 2.3. For functions for all x. We have the following estimate of the 1-order Green's kernel for the process X. The proof follows from the estimates and methods of [1], [4], [5].

Proposition 2.4
There exists a Green's kernel g 1 (x, y) which is continuous on K × K \{x = y} (and on K × K when d s < 2), and satisfies the following: We now wish to consider E on the space L 2 (K, µ), and to do this we use an argument due to Osada [24].
For an open set B ⊂ K, define the capacity of B by . The capacity of any set F ⊂ K is defined as the infimum of the capacity of open sets which contain F . We say that µ charges no set of zero capacity if the following holds: Proof. If d s < 2 then points have strictly positive capacity, and the result is immediate. We prove the result for d s > 2: the proof for d s = 2 is similar. It is well-known that for each compact set M ⊂ K, (2.7) Using Proposition 2.4, Cap(M ) ≥ µ(M )/c 3 for each compact set M , thus for each Borel set, which completes the proof.
As µ is a Radon measure which charges no set of zero capacity, it is a smooth measure in the sense of [9] (p. 80). Thus, it is a Revuz measure for some positive continuous additive functional A (see section 5.1 of [9]) and we can time change by the inverse of A -see section 6.2 of [9]. Let S denote the quasi support of µ (see p. 168 of [9]) for the Dirichlet form on L 2 (K, ν). Note that S ⊂ K and µ( and F e is the extended Dirichlet space associated with (E, F) (p. 35 of [9]). By Theorem 5.1.5 and Theorem 6.2.1 of [9], ( E, F ) is a closed regular Dirichlet form on L 2 (K, µ). The next proposition shows that E = E, so that the Dirichet form is not affected by the time change.

Proposition 2.6 Assume (2.2).
Then Note that as µ i > 0 for all i ∈ S, µ(S n (x)) = 0 for n ≥ 0. To prove (2.8) it is enough to prove that If K = [0, 1] d this is just the classical Wiener test (see [15] or [22]); the result used here follows, using Proposition 2.4, by exactly the same arguments.
Using Kelvin's principle (see Section 2.2 of [9]), we have for each compact set M ⊂ K, where the infimum is taken over the positive Radon measures m with m(M ) = 1. Now, take an arbitrary compact set So, using Proposition 2.4 as before, we have Here we used the fact that l d f −dw µ * = ρ −1 K µ * < 1 (due to assumption (2.2)) in the last inequality. We thus obtain a closed local regular Dirichlet form (E, F ) on L 2 (K, µ) with the property (2.1): write X for the associated diffusion.
We next show that (2.3) holds. If d s < 2 then this is easy to verify directly. We omit the argument for d s = 2: it follows by similar arguments to those for d s > 2.
For A ⊂ K write T A = inf{t ≥ 0 : X t ∈ A}. Let x ∈ K − V * , and σ n = T Dn(x) c . Let g(x, y) be the (0th order) Green's function for X killed on hitting D n (x) c . From [5] we have that g(x, y) ≤ c|x − y| dw−d f . As the 0th-order Green's function is not affected by the time change, by a calculation similar to that in Lemma 2.5 (2.11) Define for λ > 0. Using (2.11), the argument of [5] Proposition 6.14 goes through (with some modification) to prove that there exists β > 0 such that Finally, we prove the joint continuity of the transition density. As P t is a self-adjoint compact operator on L 2 (K, µ), there exist ϕ i that form a complete orthonormal system in L 2 (K, µ) with . (x, y).
Further, the convergence is absolute and takes place in L ∞ (K × K) (see [5] Proposition 6.15).
for some γ 1 > 0 depending only on the Dirichlet form and µ (the detailed argument will be given for the killed process on D Λm(x) in Proposition 5.1). Note that this upper bound is not sharp, but it is enough to deduce the continuity of p t (x, y). By this estimate, we see that Thus we can take a version of transition density as p t (x, y) = n e −λnt ϕ n (x)ϕ n (y), and the convergence is uniform. This proves joint continuity of the density.
We thus obtain a Dirichlet form on L 2 (K, µ) which satisfies Assumption 2.3.

Main Theorems
In the following, we identify S n and the set of all n-complexes. For x, y ∈ K, we say Π = {x, x 1 , · · · , x l , y} is an m-walk of length |Π| = l + 1 if x 1 , · · · , x l ∈ S m , x ∈ F x 1 (K), y ∈ F x l (K) and F x i , F x i+1 are adjacent m-complexes for 1 ≤ i ≤ l − 1. For simplicity, we will assume the following in the p.c.f. self-similar set case. We remark that if the self-similar structure L is changed to L l = (K, S l , {F w } w∈S l ), then for sufficiently large l, L l satisfies Assumption 3.1. For x ∈ K \V * , let n r,j (x) be the shortest number of steps by a (Λ r (x) + j)-walk from x to ∂D Λr (x). Further, for x ∈ K \ V * , define p r (x) = min{k : x / ∈ D Λr(x)+k (∂D Λr (x))}.
Note that p n−1 (x) ≤ p n (x) + C for some C > 0, independent of n and x. Lemma 3.2 There exist c 3.1 > 0 such that the following holds for all r, j ≥ 0 and x ∈ K \ V * .
Proof. Using Assumption 3.1 in the p.c.f. case, and the fact that l ≥ 2 in the carpet case, we have n r,j (x) ≥ 2 (j−pr(x)) + . Since e < 4 the first inequality is immediate.
Noting that if r ≥ n log t * then Λ r (x) > n for all x ∈ K − V * , the third inequality can be obtained by an easy modification of the proof of Lemma 3.3 in [13]. Now let µ be any Bernoulli measure on K so that µ(V 0 ) = 0, µ(F i (K)) = µ i for all i where µ i ≥ 0 satisfies i µ i = 1. We emphasise that we continue to consider the density of X with respect to µ: the role of µ will be to select subsets of K with different limiting behaviour of p t (x, x).
As in [11], Lemma 6, we have Proof. First note that if x ∈ D k (V 0 ) then any j-walk from x to V 0 requires at least 2 j−k steps. Also, there exists θ < 1 such that µ(D k (V 0 )) ≤ c 0 θ k for k ≥ 1.
We now state our main theorems.

Theorem 3.4
There exists c 3.2 , · · · , c 3.8 > 0 such that the following holds. 1) (Lower estimate) For each x ∈ K \ V * and t ≤ c 3.2 e −n (n = n + c 3.3 p n (x)), 2) (Upper estimate) For each x ∈ K \ V * and for each t which satisfies c 3.5 e −n ≤ t ≤ c 3.6 e −m −c 3.7 log n (n = n + c 3.3 p n (x), n = n + c 3.7 log n ) for some m ≤ n, We note that in general n is not monotone increasing w.r.t. n. As we do not have a good comparison between p t (x, x) and p t/2 (x, x), we need an extra log n in the time interval for the upper estimate. See the proof, which will be given in Section 5, for details.
Remark. For the p.c.f. fractals, we can also obtain the following estimate by the similar (but simpler) argument to the proof of Theorem 3.4. There exist c 3.9 , c 3.10 , c 3.11 > 0 such that for each x ∈ K and for each e −(n+1) ≤ t ≤ e −n , Note that (3.2) always gives some estimate of the kernel for each fixed t whereas Theorem 3.4 does not (unless t is small enough). On the other hand, when t is in the interval where Theorem 3.4 gives the estimate, it is much sharper than that of (3.2).
Concerning the lower bound for the p.c.f. fractals, (3.2) with c 3.10 = 0, which is sharper than (3.2), is proved in Section 5 of [18]. For the p.c.f. fractals with a 'regular harmonic structure', sharper upper estimate of p t (x, x) is given in appendix of [12].
We cannot obtain (3.2) for the carpet cases as D 1 Λn (x) could have a very 'bad shape' in general. When the sizes of two adjacent Λ n -complexes in D 1 Λn (x) are very different, particles could escape from D 1 Λn (x) much faster than e −n .
Using Proposition 3.3, we have the following almost sure result.
Theorem 3.5 Let µ be a Bernoulli measure on K with weights µ i satisfying the hypotheses of Proposition 3.3. There exist c 3.12 , c 3.13 , c 3.14 > 0 and h : The proof is essentially the same as that of Theorem 3.4: we will remark on the necessary modifications in the last part of Section 5.

(3.4)
Corollary 3. 6 The following holds forμ-a.e. x ∈ K: . . t xn ). Since, under µ, x i are independent identically distributed random variables, using the strong law of large numbers . (3.5) Finally, by Theorem 3.5, we have, µ-a.e., log µ x i , and using (3.5) and the law of large numbers completes the proof.
Remarks. 1) The formula for theμ-a.e. spectral dimension (3.4) has the same form as that for the (stationary) homogeneous random Sierpinski gasket studied in [7]. However, in that case, {μ i } i corresponds to the frequency that each random pattern appears.
Note that while d s (µ)/2 ≥ d e s (µ)/2 (recall (2.5)), these two numbers are in general not equal. This lack of correspondence between the asymptotic growth of the eigenvalues and transition density emphasises the lack of uniformity in the behaviour of p t (x, x). 3) Let σ(μ) = ( iμ i log(1/µ i ))/( iμ i log ρ i ) when iμ i log ρ i = 0. We can then write , 0] -this can also be proved directly from (2.2). As d s (μ)/2 is increasing w.r.t. σ(μ), one can calculate the region of d s (μ)/2 asμ varies.
The case of [0, 1] with µ 1 = p > 1/2, µ 2 = q < 1/2, p + q = 1, and ρ i = 2 was discussed in the introduction; some properties of this diffusion process were studied in [10]. For this case, log 2 (1/p) < σ(μ) < log 2 (1/q) and by Proposition 4.7 in [10] (plus an easy Tauberian argument), the corresponding value for dyadic rational points is log 2 (1/p), the infimum of the interval. Using the continuity of p t (x, x) and Corollary 3.6 one can show there exist points for which σ takes the maximum value log 2 (1/q). 4) In the carpet case the sign of σ(μ) is the same as that of (ρ K − 1), and so, for example, if ρ K < 1 then d s (μ) > 2 for anyμ. 5) If min i log ρ i / log(1/µ i ) = max i log ρ i / log(1/µ i ), using Corollary 3.6 and the continuity of p t (x, y), one can prove by an elementary argument that there are (uncountably many) x such that lim t→0 log p t (x, x)/ log t does not exist. 6) Theorem 1.1 now follows from remarks 3) and 5). 7) Using the methods of [6] and [18], together with the (worse) lower bound in (5.2), which will be used for the proof of Theorem 3.4, and a chaining argument we have

Hitting time estimates
In this section, we will prove some hitting time estimates for the process which will be needed for the transition density estimates.
We first give some notation. For each x ∈ K and m ≥ 0, we fix a m-complex which contains x and denote it as be the length of the word of D * Λr (x), and define D 1, * Λr(x)+l (x) to be the union of (Λ * r (x) + l)-complexes which intersect D * Λr(x)+l (x). In the following, we will treat the case where Note that when (4.1) holds for the carpet case, all (Λ * r (x 0 ) + m)-complexes are same size, so that D 1, * Λr(x 0 )+m (x 0 ) is a cube (or square when d = 2). We will consider the process on D 1, * Λr(x 0 )+m (x 0 ) whose Dirichlet form is the same as that of {X t } but whose measure µ satisfies µ(K wv 1 ···vn ) = µ w (µ * ) m µ v m+1 · · · µ vn for each (Λ * r (x 0 )+n)-complex K wv 1 ···vn ⊂ K w = D * Λr (x 0 ) and all n ≥ m. Let X t be the process associated with E and L 2 (K, µ) killed on ∂D 1, * Λr(x 0 )+m (x 0 ). As before, we define T A = inf{t ≥ 0 : X t ∈ A} for A ⊂ K, and write T A ( X) for the analogous hitting time for X. We then have the following.
2) There exist 0 < c 4.3 , 0 < c 4.4 < 1 (independent of x 0 , r, m) such that for all 0 < t < 1, Proof. For the p.c.f. case, 1) can be proved by a simple modification of Lemma 3.5 in [13] (one can also prove it using Green's density killed at ∂D 1, * Λr(x 0 )+m (x 0 )). For the carpet case, 1) is proved in the same way as Proposition 5.5 in [5].
Note that X t is a time change of X t , so that the trajectory of { X t } is the same as that of {X t }, but that X t moves faster than X t . Therefore, we have Thus, we have from Lemma 4. 1 2) that for each 0 < t < 1 and for each x 0 ∈ K, r, m ≥ 0 which satisfies (4.1).
Now, for a process X on K and for n ≥ 0, we define a sequence of hitting times as follows, We have the following estimate of the crossing time distribution.
Proof. As the result is clear when k = 0, we consider the case k ≥ 1. First, for each n ≥ r ≥ 0, set ξ r,n = sup{i : σ n i ≤ T ∂D Λr (x) }. By the structure of K, there exists c 1 > 0 such that c 1 n r,m ≤ ξ r,Λr(x)+m ∀r, m ≥ 0.
We also note that for each r, m ≥ 0 and k ≤ ξ r,Λr(x)+m − 1, {W holds for k ≤ ξ r,Λr(x)+m − 1 by (4.2) ((4.1) clearly holds in these cases). Using these facts, we have for each x ∈ K \ V * , where we use Lemma 1.1 of [2] and (4.4) in the second inequality, and in the last equality we choose c 4.7 so that c . We thus obtain the result.
Let β ≡ log t * . For each c > 0, set Then, by Lemma 4.2, we have for each 0 < t < 1 and x ∈ K \ V * , We now have the following exponential decay of the distribution of hitting times.
In Lemma 5.1 of [13], one of the author proves similar results, but the proof is incomplete as it could be carried out only when k > 0 where k, which appears in the proof, is defined similarly to ours. The proof can be completed following the argument of ours, using Proposition 4.3.

Upper bounds
In this subsection, we will obtain the upper bound of the transition density. For the purpose, we will first obtain the upper estimate of the kernel with Dirichlet boundary condition. For each x ∈ K and n ≥ 0, let p D Λn (x) t (x, y) be the transition density of the process killed at ∂D Λn (x).
Proof. For w ∈ Λ m write f w = f • F w and definē Fw (K) f (x)µ(dx).
Note that for v ∈ F,v = vdµ = w∈Λmv w µ w . In the following, we fix x, m and denote B = D Λm (x). Let u 0 ∈ D(L) with u 0 ≥ 0, Supp u 0 ⊂ B and u 0 1 = 1. Set u t (x) = (P B t u 0 )(x) and g(t) = u t 2 2 . We remark that g is continuous and decreasing. As the semigroup is symmetric and Markov, For each l ≥ 0, Note that C l B ≤ C l+1 B and C l B → ∞ as l → ∞. Let s l = inf{t ≥ 0 : g(t) ≤ eC l B } for l ∈ N. Thus (5.3) holds for 0 < t < s l . Note that s l → 0 as l → ∞. Integrating (5.3) from s l+2 to s l+1 we obtain Thus s l+1 − s l+2 ≤ c 4 e −(m+l) , and iterating this we have This implies that g(c 5 /e m+l ) ≤ g(s l ) = eC l B . Taking l = n − m, it follows that if t/c 5 ≥ e −n , then Using the fact that P t 1→∞ ≤ P t 2 1→2 , we deduce the result.
Proof of Theorem 3.4 2). The main step in the proof is to compare the transition densities of X killed at ∂D Λm (x) with those of the unkilled process. WriteX for X killed at ∂D Λm (x), and let τ = T ∂D Λm (x) . First, by Proposition 4.3, fort ≤ e −m . Set t = st. We first assume e −n ≤t ≤ e −m for some m ≤ n and later determine the right value of s (right interval for t) where the comparison of Dirichlet and Neumann boundaries holds. For k ≥ n define B k = D Λ k (x) ⊂ D Λn (x). Then The second term above equals Since the process X started with measure µ is symmetric, J 2 = P µ (X 0 ∈ B k , X t ∈ B k , t/2 < τ < t) ≤ P µ (X 0 ∈ B k , X t ∈ B k , ∃s ∈ [t/2, t) : X s ∈ ∂D Λm (x)) = P µ (X t ∈ B k , X 0 ∈ B k , ∃s ∈ (0, t/2] : X s ∈ ∂D Λm (x)) = P µ (X 0 ∈ B k , X t ∈ B k , τ ≤ t/2) = J 1 Write a(t/2) = sup{p s (y, y) : y ∈ K, t/2 ≤ s ≤ t}. We have P z (X t ∈ B k , τ ≤ t/2) = E z 1 (τ ≤t/2) P Xτ (X t−τ ∈ B k ) Combining these estimates, letting k → ∞, and using the continuity of p t (·, ·), we deduce that By (2.13) and (5.2), there exists a > 0 such that a(t/2) ≤ c 1 t −a p t (z, z) ∀z ∈ K \ V * , 0 < ∀t < 1.
Thus by Proposition 4.3 Thus the last term in (5.5) is estimated from above by c 2 s −a exp{an −c 3 s −γ }p t (x, x) when e −n ≤ t. Now, by taking s = (hn ) −l < 1 with sufficiently large l, h > 0, we can take c 2 s −a exp{an − c 3 s −γ } ≤ 1/2 for all n ≥ 1. We thus have We thus obtain the result using Proposition 5.1.
We will change the definition of k b so that k b = inf{j ≥ḡ(x) : n r,j (x) e r+βj ≤ bt/c 4.7 , n r,j−1 (x) e r+β(j−1) ≥ bt/c 4.7 }. (5.8) Then (4.3) holds for this k b (with bt instead of t). Noting that p r (x)/2 in Lemma 3.2 corresponds to α log r in this case, it is not hard to modify Proposition 4.3 using k b in (5.8). Then, the lower and upper bounds can be proved in the same way. The extra log n term is taken into C 2 log n.