THICK POINTS FOR TRANSIENT SYMMETRIC STABLE PROCESSES

Let T ( x; r ) denote the total occupation measure of the ball of radius r centered at x for a transient symmetric stable processes of index (cid:12) < d in IR d and (cid:3) (cid:12);d denote the norm of the convolution with its 0-potential density, considered as an operator on L 2 ( B (0 ; 1) ; dx ). We prove that sup j x j(cid:20) 1 T ( x; r ) = ( r (cid:12) j log r j ) ! (cid:12) (cid:3) (cid:12);d a.s. as r ! 0. Furthermore, for any a 2 (0 ; (cid:12) (cid:3) (cid:12);d ), the Hausdor(cid:11) dimension of the set of \thick points" x for which lim sup r ! 0 T ( x; r ) = ( r (cid:12) j log r j ) = a , is almost surely (cid:12) − a= (cid:3) (cid:12);d ; this is the correct scaling to obtain a nondegenerate \multifractal spectrum" for transient stable occupation measure. We also show that the lim inf scaling of T ( x; r ) is quite di(cid:11)erent: we exhibit positive, (cid:12)nite, non-random c 0 (cid:12);d ; C (cid:12);d , such that c 0 (cid:12);d < sup x lim inf r ! 0 T ( x; r ) =r (cid:12) < C (cid:12);d a.s. Abstract Let T ( x; r ) denote the total occupation measure of the ball of radius r centered at x for a transient symmetric stable processes of index (cid:12) < d in IR d and (cid:3) (cid:12);d denote the norm of the convolution with its 0-potential density, considered as an operator on L 2 ( B (0 ; 1) ; dx ). We prove that sup j x j(cid:20) 1 T ( x; r ) = ( r (cid:12) j log r j ) ! (cid:12) (cid:3) (cid:12);d a.s. as r ! 0. Furthermore, for any a 2 (0 ; (cid:12) (cid:3) (cid:12);d ), the Hausdor(cid:11) dimension of the set of \thick points" x for which lim sup r ! 0 T ( x; r ) = ( r (cid:12) j log r j ) = a , is almost surely (cid:12) − a= (cid:3) (cid:12);d ; this is the correct scaling to obtain a nondegenerate \multifractal spectrum" for transient stable occupation measure. We also show that the lim inf scaling of T ( x; r ) is quite di(cid:11)erent: we exhibit positive, (cid:12)nite, non-random c 0 (cid:12);d ; C (cid:12);d , such that c 0 (cid:12);d < sup x lim inf r ! 0 T ( x; r ) =r (cid:12) < C (cid:12);d a.s.


Introduction
The symmetric stable process {X t } of index β < d in IR d does not hit fixed points, hence does not have local times. Nevertheless, the path will be "thick" at certain points in the sense of having larger than "usual" occupation measure in the neighborhood of such points. Our first result tells us just how "thick" a point can be.
Recall that the occupation measure µ X T is defined as for all Borel sets A ⊆ IR d . As usual, we let Theorem 1.1 Let {X t } be a symmetric stable process of index β < d in IR d . Then, for any R ∈ (0, ∞) and any T ∈ (0, ∞], lim →0 sup |x|≤R µ X T (B(x, )) β | log | = βΛ β,d a.s. (1.2) Remarks: • Our proof shows that for any T ∈ (0, ∞], • The scaling behavior of stable occupation measure around any fixed time t is governed by the stable analogue of the LIL of Ciesielski-Taylor, see [11]; for any T ∈ (0, ∞] and t ≤ T , lim sup Our next result is related to (1.4) in the same way that the formula of Orey and Taylor [5] for the dimension of Brownian fast points is related to the usual LIL. It describes the multifractal nature, in a fine scale, of "thick points" for the occupation measure of {X t } (We call a point x ∈ IR d a thick point if x is in the set considered in (1.5) for some a > 0; similiarly, t > 0 is called a thick time if it is in the set Thick a considered in (1.6) for some a > 0 and T > 0.) Theorem 1.2 Let {X t } be a symmetric stable process of index β < d in IR d . Then, for any T ∈ (0, ∞] and all a ∈ (0, βΛ β,d ], Equivalently, for any T ∈ (0, ∞] and all a ∈ (0, βΛ β,d ], Denote the set in (1.6) by Thick a . Then Thick a = ∅ at the critical value a = βΛ β,d .
• For any x / ∈ {X t 0 ≤ t ≤ T } and small enough, µ X T (B(x, )) = 0. Hence, the equivalence of (1.5) and (1.6) is a direct consequence of the equivalence between spatial and temporal Hausdorff dimensions for stable motion due to Perkins-Taylor (see [7,Eqn. 0 Our next theorem gives a precise estimate of the total duration in [0, 1] that the stable motion spends in balls of radius that have unusually high occupation measure. Such an estimate (which is an analogue in our setting of the "coarse multifractal spectrum", cf. Riedi [10]), cannot be inferred from Theorem 1.2. Theorem 1.3 Let {X t } be a symmetric stable process of index β < d in IR d , and denote Lebesgue measure on IR by Leb. Then, for any a ∈ (0, βΛ β,d ), a.s. Theorem 1.3 will be derived as a corollary of the following theorem which provides a pathwise asymptotic formula for the moment generating function of the ratio µ X 1 (B(X t , )) / β . It follows from the proof that both sides of (1.8) are infinite if θ ≥ Λ −1 β,d . The thick points considered thus far are centers of balls B(x, ) with unusually large occupation measure for infinitely many radii, but these radii might be quite rare. The next theorem shows that for "consistently thick points" where the balls B(x, ) have unusually large occupation measure for all small radii and the same center x, what constitutes "unusually large" must be interpreted more modestly. Theorem 1.5 Let {X t } be a symmetric stable process of index β < d in IR d . Then for some non-random 0 < c β,d < C β,d < ∞ we have, (1.9) Remarks: • In particular, replacing the lim sup by lim inf in (1.5) and (1.6) results with an a.s. empty set for all a > 0.
• The new assertion in (1.9) is the right hand inequality; the inequality on the left is an immediate consequence of the existence of "stable slow points", see [6,Theorem 22].
It is an open problem to determine the precise asymptotics in (1.9), even in the special case of Brownian motion.
This paper generalizes the results of our paper [3] on thick points of spatial Brownian motion to all transient symmetric stable processes. There are several sources of difficulty in this extension: • Ciesielski-Taylor [2] provide precise estimates for the tail of the Brownian occupation measure of a ball in R d ; the existing estimates for stable occupation measure are not as precise.
• The Lévy modulus of continuity for Brownian motion was used in [3] to obtain long time intervals where the process does not exit certain balls.
We overcome these difficulties by • A spectral analysis of the convolution operator defined by the potential density.
• Conditioning on the absence of large jumps in certain time intervals; this creates strong dependence between different scales, and our general results on "random fractals of limsup type" are designed to handle that dependence.
Sections 2 and 3 state and prove the two new ingredients which are needed in order to establish the results in the generality stated here, that is the Localization Lemma 2.2 and the Exponential Integrability Lemma 3.1. Applying these lemmas, Section 4 goes over the adaptations of the proofs of [3] which allows us to establish the results stated above for all transient stable occupation measures.

Localization for stable occupation measures
We start by providing a convenient representation of the law of the total occupation measure µ X ∞ (B(0, 1)). This representation is the counterpart of the Ciesielski-Taylor representation for the total occupation measure of spatial Brownian motion in [2, Theorem 1].
Proof of Lemma 2.1: We use p t (x) to denote the transition density of {X t } and for ease of exposition write K, Λ for K β,d , Λ β,d respectively, and let J = µ X ∞ (B(0, 1)). While u 0 is in general only in L 1 (B(0, 1), dx), each application of K lowers the degree of divergence by β (this is easily seen by scaling), so v := K m−1 u 0 , the convolution kernel of K m is bounded and in fact continuous for m large enough. Fix such m and note that for any n ≥ m, Thus, for g(z, u) := ∞ n=m z n−m u n /n!, by the standard Neumann series for the resolvent, for any z ∈ I C such that |z| < Λ −1 := θ * , Taking z ∈ [−3θ * /4, −θ * /4], we find after m integrations by parts that To justify this, we note, on the one hand, that by (2.2) J has all moments, so that P(J > u) ≤ c N /u N for any N , and therefore f j (u) is bounded and goes to 0 as u tends to ∞ for any j. On the other hand, d m g(z, u)/d m u = e zu with d k g(z, u)/d k u = 0 at (z, 0) for k = 0, . . . , m − 1 which controls the boundary terms at u = 0, and writing g(z, u) = z −m (e zu − m−1 n=0 z n u n /n!) and using the fact that z < 0 controls the boundary terms at u = ∞.
Since K is a convolution operator on B(0, 1) with locally L 1 IR d , dx kernel, it follows easily as in [4,Corollary 12.3] that K is a (symmetric) compact operator. Moreover, the Fourier transform relation e i(x·p) u 0 (x) dx = |p| −β > 0 implies that K is strictly positive definite. By the standard theory for symmetric compact operators, K has discrete spectrum (except for a possible accumulation point at 0) with all eigenvalues positive, of finite multiplicity, and the corresponding eigenvectors of K, denoted {φ j } form a complete orthonormal basis of L 2 (B(0, 1), dx) (see [8, Theorems VI.15, VI.16]). Moreover, (f, Kg) B(0,1) > 0 for any non-negative, non-zero, f, g, so by the generalized Perron-Frobenius Theorem, see [9, Theorem XIII.43], the eigenspace corresponding to Λ = λ 1 is one dimensional, and we may and shall choose φ 1 such that φ 1 (y) > 0 for all y ∈ B(0, 1). Noting that φ j are also eigenvectors of (I − zK) −1 with corresponding eigenvalues (1 − zλ j ) −1 , we have by (2.3) and (2.4 where c j := (1, φ j )(v, φ j ) is absolutely summable. Since both integrals in (2.5) are analytic in the strip Re z ∈ [−3θ * /4, −θ * /4], and agree for z real inside the strip, they agree throughout this strip. Considering z = −θ * /2 + it, t ∈ IR we have that a.e. on u ≥ 0 by Fourier inversion and hence for all u > 0 by right continuity. Considering the (m − 1)-st derivative of (2.6), the uniform convergence of j c j λ −k j e −u/λ j for k = 1, . . . , m and u away from 0, shows that Recalling that v, the convolution kernel of K m , is bounded and continuous, we have that λ m j φ j = K m φ j are continuous and bounded functions, and

2
With the aid of (2.1) we next provide a localization result for the occupation measure of {X t }.

Lemma 2.2 (The Localization Lemma) Let {X t } be a symmetric stable process of index
, and all u > 0 sufficiently large With h( ) = β | log | and any ρ > d/(d − β), considering u = a| log | in (2.7), by stable scaling we establish that for any a > 0 and > 0 small enough Proof of Lemma 2.2: Let J t := µ X t (B(0, 1)). Recall that φ 1 is a strictly positive function on B(0, 1), hence in (2.1) we have ψ 1 > 0 and θ * = λ −1 out of which the upper bound of (2.7) immediately follows.
Turning to prove the corresponding lower bound, let τ z := inf{s : |X s | > z}, noting that by [1, Proposition VIII.3] for some c > 0 and any u > 0, z > 1, t > 0, (2.10) Let J and J denote two independent copies of J ∞ . Noting that P y (J > u) ≤ P(J > u) for any y ∈ IR d , u > 0, and using the strong Markov property, it is not hard to verify that
Proof of Theorem 1.2, lower bound: In proving the lower bound it suffices to assume that T is finite; by stable scaling, it is enough to consider T = 2 or equivalently, −1 ≤ t ≤ 1 in (1.6). Our proof follows closely the proof of [3, Corollary 4.1] to which the reader is referred for notation and details. Take n = n 3 2 −n/β , n = 1, 2, . . . and b n = 1 − | log n | −2 . With I = [t, t + 2 −n ] ∈ D n , defineĨ = [t − n v 2 −n , t], v = 3β + ρ. Let Z I = 1 if the following two (independent) conditions hold: (4.1) Therefore, if I ∈ D n and Z I = 1, then Ĩ 1 {|Xs−X t |< n } ds ≥ ah( n ) for every t ∈ I. Using stable scaling and [1, Proposition VIII.4] it is easily verified that the second condition in (4.1) has probability at least 1/2 for n sufficiently large. By stable scaling, the lower bound of (2.8) directly implies that for all I ∈ D n and n sufficiently large Noting that the two conditions in (4.1) are independent, we see that for I ∈ D n , and all n large enough, p n := P(Z I = 1) ≥ 2 −aθ * n/β .
where we used the fact that the process {X t } is right-continuous with left limits and has only a countable set of jumps.
Proof of Theorem 1.1: With the above estimates, as well as the lower bound (2.8) of the Localization Lemma 2.2, this follows directly as in the corresponding proof of [3, (1.7)], using now δ = β ρ( ).
Proof of Theorem 1.3: Our proof follows closely the outline of [3,Section 8], to which the reader is referred for notation and details, where we take now h( ) = β | log | and δ n = β n ρ( n ) as needed for applying the lower bound of (2.8). For ρ n = β n | log n | −3β and the i.i.d. random variables we combine [1, Proposition VIII.4] with (2.8) to provide the lower bound on P(Y (n) ≥ a/(1 −δ)) leading to [3, (8.2)] (see the derivation of (4.2) for a similar application).
Proof of Theorem 1.5: We follow the outline of [3, Section 9], relying on Lemma 3.1 to control the exponential moments instead of Girsanov's theorem used in [3, Lemma 9.1].
To deal with the occupation measure of balls not centered at the origin take 0 < α < 1 and fix b = 1 + α > 1. For k ∈ (1, ∞), let Γ k = {x : |x| ∈ [k −1 , k]} and for a > 0, Set η n = 2 −n and δ n = η 1−b −1 n for n = 1, 2, . . .. Let {x j : j = 1, . . . , K n }, denote a maximal collection of points in Γ k such that inf =j |x − x j | ≥ αη n . Let A n be the set of j ; 1 ≤ j ≤ K n such that (4.5) We will shortly prove that for some C β,d < ∞ and all a ≥ C β,d where γ > 0. Assuming this for the moment, let V n,j = B(x j , αη n ). Then, for any x ∈ Γ k there exists j ∈ {1, . . . , K n } such that x ∈ V n,j and B(x, Fixing a ≥ C β,d , if x ∈ D a then a.s. for some m 1 (ω, x, b) < ∞ and all n ≥ m 1 , Therefore, D a ⊆ ∪ n≥m ∪ j∈An V n,j for any m ≥ 1. Therefore Thus, by Borel-Cantelli, it follows that a.s. A n is empty for all n ≥ m 2 (ω), implying that the sets D a are a.s. empty for all T < ∞. By [13, Lemma 5], a.s.
Thus, taking k ↑ ∞ completes the proof of the right side of (1.9), subject only to (4.6).