Small-time heat decay for stable processes on fractal drums

Small-time heat deca y for stable pro cesses on fractal drums Hyunc hul P ark ∗ and Yimin Xiao † Marc h 17, 2026 Abstract In this pap er, we study the sp ectral heat conten t for isotropic stable pro cesses on fractal drums (namely , op en sets with fractal b oundaries). The sp ectral heat conten t for sub ordinate killed Bro wnian motions by stable sub ordinators w as inv estigated in [21], and the presen t work serv es as a natural extension of [21] for the sp ectral heat con tent for stable pro cesses. Under suitable geometric conditions on the underlying domains, w e sho w that the decay rate of the sp ectral heat conten t for stable pro cesses differs substan tially from that for sub ordinate killed Bro wnian motions when α = d − b , where b is the in terior Mink owski dimension of the b oundary of the underlying op en set. 1 In tro duction The sp ectral heat conten t (SHC) for a giv en sto c hastic pro cess in a domain D measures the to- tal heat remaining in D when the heat particles mov e according to the pro cess with the initial temp erature set at one inside D and main tained at zero outside D . While the SHC for Bro wnian motion has b een well studied since the late 1980s, its analysis for jump-type L´ evy pro cesses b egan only recen tly , from the mid-2010s. The small-time asymptotics for SHC for sub ordinate killed Bro wnian motion by stable sub ordinators was inv estigated in [19] and this was extended for more general sub ordinators in [8]. A closely related pro cess is the sub ordinate Brownian motion, where one observes Brownian motions on a random clock mark ed b y a sub ordinator (a non-decreasing L ´ evy process). F or a non-degenerate sub ordinator, its sample path is discontin uous, and the cor- resp onding sub ordinate Bro wnian motion b ecomes a jump pro cess. The SHC for suc h jump-t yp e L ´ evy pro cesses has b een studied previously; in particular, the small-time asymptotics for SHC for isotropic stable pro cesses were inv estigated in [20], and this was extended to more general L ´ evy pro cesses in [9] and [14]. In [6], the authors studied the SHC for a one-dimensional symmetric L ´ evy pro cess in R with regularly v arying characteristic exp onen t. They sho wed that when the underlying op en set has infinitely man y comp onen ts, the decay rate of SHC is faster than SHC on domains with finitely ∗ Researc h supp orted in part by AMS-Simons Research Enhancement Grants for Primarily Undergraduate Insti- tution (PUI) F aculty 2025-2028 and Research and Creative Pro jects Awards 2025-2026 from SUNY New P altz. † Researc h supp orted in part by the NSF grant DMS-2153846. 1 man y comp onen ts (see [6, Lemma 4.9]). A natural question arises concerning the precise decay rate of the SHC on domains with infinitely man y comp onents. A natural wa y for a b ounde d open set to p ossess infinitely many comp onen ts is to consider sets of the form given in (2.2). Domains of the form (2.2) in R d t ypically hav e fractal b oundaries, and such open sets are referred to as fractal drums in [12, 13]. V arious problems concerning fractal drums ha v e b een in vestigated, including the W eyl–Berry conjecture for the eigenv alues of the Laplacian on b ounded domains in [10] and its connection to the Riemann Hyp othesis in [11]. SHC on fractal drums with resp ect to Brownian motion w as studied in [15]. Subsequen tly , SHC on fractal drums asso ciated with jump pro cesses w as examined. In particular, [21] inv estigated the SHC for sub ordinate killed Bro wnian motions b y stable sub ordinators. The purp ose of this pap er is to study the SHC on fractal drums for isotropic stable processes. A stable pro cess can b e realized as a sub ordinate Brownian motion driven by a stable sub ordinator, where one observ es the underlying Brownian motion on the random clock determined by a stable sub ordinator. The resulting pro cess evolv es through jumps, which is related, but different from the pro cess studied in [21]. Due to the scaling property of stable processes, the SHC for stable pro cesses inherits a corresp onding scaling prop ert y under similitudes. Ho wev er, the additivity prop ert y of the SHC under disjoint unions, crucial in [21], fails for jump pro cesses. This failure arises b ecause the time-c hanged Brownian motion, b eing observed on random clo cks, may hav e w andered back in to the domain after the underlying Brownian motion has exited it. Consequen tly , the direct application of the renewal theorem b ecomes nontrivial. T o address this difficult y , we introduce auxiliary terms D ( t ) and R ( t ) (see (3.7)), which quantify the extent to whic h the SHC for stable pro cesses fails to b e additiv e. W e then show that the error term R ( t ) decays exp onen tially fast, a prop ert y that enables the use of the renewal theorem in our analysis. The main results of this pap er are Theorems 3.7 and 3.8. F or each theorem, w e imp ose differen t assumptions, Conditions 3.1 and 3.2, on the underlying open set G . When α ∈ ( d − b , 2), we imp ose Condition 3.7 that eac h rescaled copies of G and extra comp onent G 0 are con tained in interv als that are disjoint with other parts of G when d = 1, or they are contained in C 1 , 1 op en sets O j and G 0 is C 1 , 1 when d ≥ 2. Under these assumptions, we pro ve in Theorem 3.7 that the deca y rate of SHC is of order t d − b α , where b is the interior Mink o wski dimension of the b oundary ∂ G of the domain and there are tw o regimes dep ending on whether the sequence of logarithms of the co efficients of the similitudes is arithmetic or not. A similar phenomenon was observed for SHC for sub ordinate killed Bro wnian motions in [21, Theorem 3.2]. When α ∈ (0 , d − b ], we impose Condition 3.2, which is completely differen t from Condition 3.1. In this case, we assume that tw o different dimensions, the in terior Minko wski dimension and the Hausdorff dimension, of ∂ G coincide and the Hausdorff measure H b ( ∂ G ) is strictly p ositiv e. In Theorem 3.8, we sho w that the deca y rate of SHC when α ∈ (0 , d − b ] is of order t , whic h is a sharp contrast from results from [21] when α = d − b . In [21, Theorems 3.4 and 3.5], the deca y rates of SHC for sub ordinate killed Brownian motions are of order t ln(1 /t ) or t depending on whether α = d − b or α ∈ (0 , d − b ), respectively . F urthermore, 2 when α = d − b , the small-time asymptotics of SHC for sub ordinate killed Bro wnian motions dep ends on whether the sequence of logarithms of the co efficien ts of the similitudes is arithmetic. Hence, when α = d − b , the decay rates for SHC for killed stable pro cesses and sub ordinate killed Bro wnian motions by stable sub ordinators are not comparable. This is b ecause the b oundary of the domain is p olar for the stable pro cess, and the pro cess never se es the b oundary . Consequen tly , it cannot detect that the domain has a fractal b oundary . W e find this phenomenon very in teresting as the decay rates for SHC for jump pro cesses and the corresp onding sub ordinate killed Brownian motions are kno wn to b e comparable for smo oth domains (see [6, 8, 9, 14, 19]). T o the b est of the authors’ knowledge, this provides the first example in which the decay rates of these tw o sp ectral heat conten ts are not comparable. The organization of this pap er is as follows. In Section 2, w e in tro duce the notations, construct fractal drums, and recall the renewal theorem, Theorem 2.1. In Section 3, we pro ve our main results, Theorems 3.7 and 3.8. While proving these theorems, we disco vered minor inaccuracies in [6, Corollary 3.7] when α ∈ ( d − b , 1). See Remark 3.9 for details. F or comparison purp ose, we also in vestigate the regular heat conten t (without Dirichlet exterior condition) on fractal drums and sho w that the asymptotics of the regular heat con tent is not affected b y the fractal b oundary . In Section 4, we pro vide some concrete examples where our main theorems can b e applied to study the SHC for stable processes on an open set on R whose boundary is the ternary Cantor set. W e also inv estigate the regular heat conten t (without Diric hlet exterior condition) on the same set and sho w that the regular heat conten t for this set is the s ame as an op en in terv al (0 , 1). This is because the boundary , the ternary Can tor set, has the Lebesgue measure zero and cannot be observed at an y given fixed time t ev en though the stable pro cess can observe the b oundary in a time interv al (0 , t ). Finally , we inv estigate the SHC on the complement of the mo dified Sierpi ´ nski gasket. In this pap er, we use c i ( i = 1 , 2 , . . . ) to denote constants whose v alues are unimp ortan t and ma y c hange from one app earance to another. The notation P x stands for the law of the underlying pro cesses started at x ∈ R d . The notation D is used for a generic op en set and G will b e used for a set giv en as in (2.2). W e use δ D ( x ) = dist( x, D c ) to denote the distance from x into D c . All sets will b e assumed to be Borel measurable. F or an y (Borel measurable set) A ⊂ R d , m d ( A ) will b e used for the d -dimensional Lebesgue measure of A and H a ( A ) is for the a -dimensional Hausdorff measure of A . 2 Preliminaries Let X = { X t } t ≥ 0 b e an isotropic stable pro cess in R d with scaling index α ∈ (0 , 2). The character- istic exp onen t of X t is E [ e iξ X t ] = e − t | ξ | α and the L´ evy density is given by J ( x ) = c ( d,α ) | x | d + α . Note that the isotropic stable pro cess X is a subordinate (time-changed) Bro wnian motion. More precisely , let S ( α/ 2) = { S ( α/ 2) t } t ≥ 0 b e a stable sub ordinator whose Laplace transform is given by E [ e − λS ( α/ 2) t ] = e − tλ α/ 2 , λ > 0 , α ∈ (0 , 2) , 3 and W = { W t } t ≥ 0 is a Browni an motion in R d indep enden t of S . Then, the process W S = { W S t } t ≥ 0 b ecomes an isotropic stable pro cess in R d with index α . Let D ⊂ R d b e an op en set. When ∂ D , the b oundary of D , is fractal (by this, we mean the Hausdorff dimension of ∂ D has a non-in teger v alue), w e will call D a fractal drum as in [12, 13]. Let τ D = inf { t > 0 : X t / ∈ D } b e the first exit time. The sp ectral heat conten t Q ( α ) D ( t ) with resp ect to X on the op en set D is defined by Q ( α ) D ( t ) = Z D P x ( τ D > t ) dx. Note that there is a closely related quantit y called the sp ectral heat con tent for sub or dinate kil le d Br ownian motions (killing the underlying Brownian motion when it first exits the domain, then doing the time-change). The sp ectral heat conten t ˜ Q ( α ) D ( t ) for sub ordinate killed Bro wnian motions with resp ect to the stable sub ordinator S ( α/ 2) on D is defined by ˜ Q ( α ) D ( t ) = Z D P x  τ (2) D > S ( α/ 2) t  dx, where τ (2) D = inf { t > 0 : W t / ∈ D } . The small-time asymptotic b ehavior of ˜ Q ( α ) D ( t ) is inv estigated on C 1 , 1 op en sets in [19] and on fractal drums in [21]. No w we introduce the type of fractal drums that w e consider in this pap er. Recall that a map R : R d → R d is called a similitude with coefficient r > 0 if | R x − R y | = r | x − y | for all x, y ∈ R d . It is w ell kno wn (cf. e.g., [15, p.191]) that any similitude is a composition of a homothet y with co efficien t r , an orthogonal transform, and a translation. Let G 0 b e an op en set in R d and R j ( j = 1 , . . . , N ) b e similitude with co efficien ts r j . F or eac h n ≥ 1, define Υ n = { j = ( j 1 , . . . , j n ) , 1 ≤ j i ≤ N } . Levitin and V assiliev [15] defined the op en set G by G =  ∞ [ n =1 [ j ∈ Υ n R j G 0  ∪ G 0 , (2.1) where, for every j = ( j 1 , . . . , j n ) ∈ Υ n , R j is the similitude defined b y R j = R j 1 ◦ · · · ◦ R j n . W e notice that this definition is a little restrictive, and the op en set G may not ha v e a fractal b oundary if G 0 is an in terv al or an op en set with piecewise smo oth b oundary . F or example, if G 0 = (0 , 1) and R 1 ( x ) = x 3 and R 2 ( x ) = x 3 + 2 3 , then the op en set G defined by (2.1) satisfies G = G 0 = (0 , 1). Ev en though the complement of the ternary Cantor set D = [0 , 1] \ C can b e generated b y taking G 0 = D in (2.1), this G 0 is to o irregular and not helpful for our purp ose. It follows from (2.1) (see also [15, Equation (1.8)]) that G satisfies the follo wing prop ert y G =  N [ j =1 R j G  ∪ G 0 . (2.2) The family of open sets that satisfy (2.2) is m uch larger b ecause it is more flexible to choose the open set G 0 . F or example, the complement of the ternary Can tor set D satisfies (2.2) with G 0 = ( 1 3 , 2 3 ) 4 and the similitudes R 1 and R 2 giv en ab ov e. Hence, in the present pap er, we will use prop erty (2.2) to define the op en sets G . This more general definition allows us to assume that all the sets R j G , 1 ≤ j ≤ N , and G 0 are pairwise disjoint. See Condition 3.1 b elo w. As in [15, Equation (1.7)] we also assume that P N j =1 r d j < 1 < P N j =1 r d − 1 j . Since the expressions in (2.2) are pairwise disjoin t, the Lesb esgue measure of G can b e written as | G | = N X j =1 r d j | G | + | G 0 | . (2.3) The condition P N j =1 r d j < 1 < P N j =1 r d − 1 j ensures that G has a finite volume and there exists a unique num b er b ∈ ( d − 1 , d ) such that N X j =1 r b j = 1 . It follo ws from [15, Theorem A] that the n umber b is equal to the interior Minko wski dimension of ∂ G . F or more details, see [15] (pages 193–194). W e recall the following renewal theorem from [15]. Consider the follo wing r enewal e quation f = Lf + ϕ, (2.4) where f : R → R , L γ f ( z ) = f ( z − γ ), γ ∈ R , and Lf ( z ) = P N j =1 c j L γ j f ( z ) = P N j =1 c j f ( z − γ j ) with c j > 0, γ j are distinct p oin ts in R , and P N j =1 c j = 1. The follo wing theorem is the renewal theorem on the solution of the renew al equation (2.4). A set of finite real n umbers { γ 1 , · · · , γ N } is arithmetic if γ i γ j ∈ Q for all indices. The maximal num ber γ such that γ i γ ∈ Z is called the span of { γ 1 , · · · , γ N } . If the set is not arithmetic, it is called non-arithmetic . Theorem 2.1 (Renewal Theorem [15]) Supp ose that a map f : R → R satisfies the r enewal e quation (2.4) with lim z →−∞ f ( z ) = 0 , and | ϕ ( z ) | ≤ c 1 e − c 2 | z | , z ∈ R , for some c onstants c 1 , c 2 > 0 . Then, the solution of the r enewal e quation (2.4) is given by f ( z ) = ∞ X n =0 L n ϕ ( z ) = ϕ ( z ) + ∞ X n =1 X c i 1 , ··· ,c i n c i 1 · · · c i n L γ i 1 · · · L γ i n ϕ ( z ) . F urthermor e, if { γ j } is non-arithmetic, then f ( z ) = 1 P N j =1 c j γ j Z ∞ −∞ ϕ ( x ) dx + o (1) as z → ∞ . If { γ j } is arithmetic with sp an γ , then f ( z ) = γ P N j =1 c j γ j ∞ X k = −∞ ϕ ( z − k γ ) + o (1) as z → ∞ . 5 3 Main Results In this section, we prov e the main theorems of this pap er, Theorems 3.7 and 3.8, on the spectral heat con tent for isotropic stable pro cesses on a fractal drum G that satisfies (2.2) for some op en set G 0 . F or e ac h theorem, w e need to imp ose the following conditions dep ending on whether α ∈ ( d − b , 2) and α ∈ (0 , d − b ]. Condition 3.1 When α ∈ ( d − b , 2), the following conditions hold. When d = 1: (a) There exist disjoint op en interv als ( a 0 , b 0 ) and ( a j , b j ), j ∈ { 1 , · · · , N } , such that G 0 ⊂ ( a 0 , b 0 ) and R j G ⊂ ( a j , b j ) for 1 ≤ j ≤ N . (b) ( a 0 , b 0 ) ∩ ( G \ G 0 ) = ∅ and ( a j , b j ) ∩ ( G \ R j G ) = ∅ . When d ≥ 2: (a) G 0 is a b ounded C 1 , 1 op en set. (b) F or each j ∈ { 1 , 2 , · · · , N } , there exist b ounded C 1 , 1 op en sets O j suc h that R j G ⊂ O j , O i ∩ O j = ∅ for i  = j , and O j ∩ G 0 = ∅ for all j ∈ { 1 , 2 , · · · , N } . Condition 3.2 When α ∈ (0 , d − b ] , we assume the fol lowing c onditions hold. (a) The interior Minkowski dimension b of ∂ G and the Hausdorff dimension dim H ( ∂ G ) of ∂ G c oincide and H b ( ∂ G ) > 0 . Here is a remark ab out Condition 3.2. It follo ws from (2.2) that ∂ G =  S N j =1 R j ∂ G  S ∂ G 0 . Th us ∂ ( G \ G 0 ) = N [ j =1 R j ∂ ( G \ G 0 ) , i.e., ∂ ( G \ G 0 ) is a self-similar set. Hence, the Minko wski dimension (denoted by dim M ) and Haus- dorff dimension of ∂ ( G \ G 0 ) alwa ys coincide (cf. [4, Example 2]). If the similitudes R 1 , . . . , R N satisfy the open set condition (cf. e.g., [5, p.139]), one has dim H ∂ ( G \ G 0 ) = dim M ∂ ( G \ G 0 ) = b and H b ( ∂ ( G \ G 0 )) > 0, see [5, Theorem 9.3]. Therefore, if G 0 is a b ounded C 1 , 1 op en set or, more generally , dim H ∂ G 0 < b , then Condition 3.2 holds. W e start with the followin g lemma on the scaling prop ert y of the sp ectral heat conten t Q ( α ) D ( t ), whic h is analogous to [21, Lemma 2.7]. Since the pro of follows from the scaling prop ert y of X and is almost identical to that of [21, Lemma 2.7], it will b e omitted. Lemma 3.3 L et R b e a similitude with c o efficient r and D is any op en set in R d . Then, we have Q ( α ) RD ( t ) = r d Q ( α ) D ( t/r α ) , t > 0 . 6 The next lemma concerns the sup er-additivity of the sp ectral heat conten t for stable pro cesses. F or the sp ectral heat conten t of sub ordinate killed Bro wnian motions, an exact additivity prop ert y holds under disjoint unions of domains (see [21, Lemma 2.5]). Ho wev er, in the case of stable pro cesses, the pro cess can jump directly from one comp onent of the domain to another without exiting it. This phenomenon causes the sp ectral heat conten t for stable pro cesses to be sup er- additiv e rather than additiv e. The strict inequality in (3.1) may hold and this is one of the main sources of difficulty in applying the renewal theorem. Lemma 3.4 L et D 1 , D 2 b e op en sets in R d with D 1 ∩ D 2 = ∅ . Then, for α ∈ (0 , 2) Q ( α ) D 1 ∪ D 2 ( t ) ≥ Q ( α ) D 1 ( t ) + Q ( α ) D 2 ( t ) . (3.1) Pro of. The pro of is almost trivial, but we provide the details for the reader’s con venience. Note that Q ( α ) D 1 ∪ D 2 ( t ) = Z D 1 ∪ D 2 P x ( τ D 1 ∪ D 2 > t ) dx = Z D 1 P x ( τ D 1 ∪ D 2 > t ) dx + Z D 2 P x ( τ D 1 ∪ D 2 > t ) dx ≥ Z D 1 P x ( τ D 1 > t ) dx + Z D 2 P x ( τ D 2 > t ) dx = Q ( α ) D 1 ( t ) + Q ( α ) D 2 ( t ) . 2 T o o v ercome the aforementioned difficulty , we estimate the probability that the pro cess jumps from one comp onen t of the domain to another and remains inside the domain up to time t > 0. W e first recall the following estimate on the measure of the set of p oints lo cated at a fixed distance from the b oundary of a C 1 , 1 op en set. Recall that an op en set D in R d is said to b e a C 1 , 1 op en set if there exist p ositive constants R 0 and Λ 0 suc h that, for every z ∈ ∂ D , there exist a C 1 , 1 function ϕ = ϕ z : R d − 1 → R satisfying ϕ (0) = 0, ∇ ϕ (0) = (0 , · · · , 0), ∥∇ ϕ ∥ ∞ ≤ Λ 0 , |∇ ϕ ( x ) − ∇ ϕ ( y ) | ≤ Λ 0 | x − y | and an orthonormal co ordinate system C S z : y = ( e y , y d ) with origin at z such that B ( z , R 0 ) ∩ D = B ( z , R 0 ) ∩ { y = ( e y , y d ) in C S z : y d > ϕ ( e y ) } . The pair ( R 0 , Λ 0 ) is called the C 1 , 1 c haracteristics of the op en set D , and R 0 is called a lo calization radius. Lemma 3.5 [23, L emma 5] L et D b e a b ounde d C 1 , 1 op en set in R d with char acteristic ( R 0 , Λ 0 ) and define for 0 ≤ q < R 0 , D q = { x ∈ D : δ D ( x ) > q } . Then  R 0 − q R 0  d − 1 | ∂ D | ≤ | ∂ D q | ≤  R 0 R 0 − q  d − 1 | ∂ D | , 0 ≤ q < R 0 . 7 Since X is isotropic, the pro jection of X onto any line passing through the origin has the same distribution and is a one-dimensional α -stable pro cess. Let  e 1 , . . . ,  e d b e the standard basis of R d and let X ( i ) t = ⟨ X t ,  e i ⟩ ( i = 1 , . . . , d ) b e the co ordinate pro cesses of X . F or 1 ≤ i ≤ d , let X ( i ) t = sup 0 ≤ s ≤ t X ( i ) s b e the supremum pro cess of X ( i ) . W e recall the follo wing estimate from [18, Equation (3.3)]: There exists a constant c = c ( α ) > 0 suc h that for any L > 0 P ( X ( i ) t > L ) ≤ ct L α . (3.2) No w we are ready to prov e Lemma 3.6. Lemma 3.6 L et α ∈ ( d − b , 2) and assume Condition 3.1 holds. Then, ther e exists t 0 > 0 such that for al l 0 < t ≤ t 0 max Z R j G P x ( τ R j G < t ≤ τ G ) dx, Z G 0 P x ( τ G 0 ≤ t < τ G ) dx ! ≤ cE ( t ) , wher e E ( t ) = t 1 /α , t ln(1 /t ) , or t for α ∈ (1 , 2) , α = 1 , or α ∈ (0 , 1) , r esp e ctively. Pro of. The proof for d = 1 is almost same as d ≥ 2, so we will only provide the pro of for d ≥ 2. F rom Condition 3.1, there exist C 1 , 1 op en sets O j suc h that R j G ⊂ O j for 1 ≤ j ≤ N . Since it is enough to show this only for the case for R 1 G , we use R G and O instead of R 1 G and O 1 for notational simplicity . First, note that X τ RG ∈ G \ RG on the ev ent { τ RG ≤ t < τ G } when the pro cess starts on x ∈ RG . The condition O ∩ O j = ∅ for all O j suc h that O  = O j and O ∩ G 0 = ∅ implies that τ RG = τ O and this implies that when the pro cess starts on RG { τ RG ≤ t < τ G } ⊂ { τ O ≤ t } . Let t 0 b e small enough so that max( t 0 , t 1 /α 0 ) < R 0 / 2, where R 0 is a lo calization radius, one of the C 1 , 1 c haracteristics of O . Assume that t ≤ t 0 . W e write O as O = A ( t ) ∪ B ( t ) , where A ( t ) = { x ∈ O : δ O ( x ) < t 1 /α } , B ( t ) = O \ A ( t ) . Hence, Z RG P x ( τ RG < t ≤ τ G ) dx ≤ Z O P x ( τ RG < t ≤ τ G ) dx = Z A ( t ) P x ( τ RG < t ≤ τ G ) dx + Z B ( t ) P x ( τ RG < t ≤ τ G ) dx. (3.3) F or the first expression ab o v e, we use the trivial upp er b ound P x ( τ RG < t ≤ τ G ) ≤ 1 and obtain Z A ( t ) P x ( τ RG < t ≤ τ G ) dx ≤ m ( A ( t )) . 8 It follows from Lemma 3.5, the coarea formula, and the fact R 0 R 0 − s ≤ R 0 R 0 − t 1 /α ≤ R 0 R 0 − R 0 / 2 = 2 m ( A ( t )) = Z t 1 /α 0 | ∂ O s | ds ≤ Z t 1 /α 0  R 0 R 0 − s  d − 1 | ∂ O | ds ≤ 2 d − 1 | ∂ O | t 1 /α . (3.4) No w we handle the second expression in (3.3). W e further divide B ( t ) into B ( t ) = { x ∈ O : δ O ( x ) ≥ t 1 /α } = { x ∈ O : t 1 /α ≤ δ O ( x ) < R 0 / 2 } ∪ { x ∈ O : δ O ( x ) ≥ R 0 / 2 } . When the pro cess starts on { x ∈ O : δ O ( x ) ≥ R 0 / 2 } , we ha ve { τ O ≤ t } ⊂ S d i =1 { X ( i ) t ≥ R 0 2 √ d } . Hence, it follows from (3.2) Z { x ∈O : δ O ( x ) ≥ R 0 / 2 } P x ( τ RG < t ≤ τ G ) dx ≤ Z { x ∈O : δ O ( x ) ≥ R 0 / 2 } P x ( τ O < t ) dx ≤ d Z { x ∈O : δ O ( x ) ≥ R 0 / 2 } P  X (1) t ≥ R 0 2 √ d  ≤ cd |O | t ( R 0 2 √ d ) α . (3.5) Finally , for the set { x ∈ O : t 1 /α ≤ δ O ( x ) < R 0 / 2 } , w e hav e { τ O ≤ t } ⊂ S d j =1 { Y ( i ) t ≥ δ O ( x ) √ d } and this implies Z { x ∈O : t 1 /α ≤ δ O ( x ) δ O ( x ) √ d ) dx = d Z R 0 / 2 t 1 /α | ∂ O u | P ( Y (1) t > u √ d ) du ≤ cd 2 d − 1 d α/ 2 | ∂ O | Z R 0 / 2 t 1 /α t u α du. It is elementary to see that for α ∈ (1 , 2) and t 1 /α < R 0 / 2 Z R 0 / 2 t 1 /α t u α du ≤ Z ∞ t 1 /α t u α du ≤ c 1 t 1 /α . F or α = 1 and t < R 0 / 2 Z R 0 / 2 t t u du ≤ c 2 t ln(1 /t ) , and for α ∈ (0 , 1) and t 1 /α < R 0 / 2 Z R 0 / 2 t 1 /α t u α du ≤ Z R 0 / 2 0 t u α du ≤ c 3 t. This shows that there exists a constant c > 0 such that Z { x ∈O : t 1 /α ≤ δ O ( x ) 0. It remains to prov e that D ( e − z ) e ( d − b ) z α deca ys exp onen- tially as z → ∞ , so that one can apply the Renewal Theorem, Theorem 2.1. By Lemma 3.4, w e clearly hav e D ( t ) ≥ 0. Note that D ( t ) = Q ( α ) G ( t ) − N X j =1 Q ( α ) R j G ( t ) − Q ( α ) G 0 ( t ) = Z S N j =1 R j G ∪ G 0 P x ( τ ( α ) G > t ) dx − N X j =1 Z R j G P x ( τ R j G > t ) dx − Z G 0 P x ( τ G 0 > t ) dx = N X j =1 Z R j G  P x ( τ G > t ) − P x ( τ R j G > t )  dx + Z G 0 ( P x ( τ G > t ) − P x ( τ G 0 > t )) dx = N X j =1 Z R j G P x ( τ R j G ≤ t < τ G ) dx + Z G 0 P x ( τ G 0 ≤ t < τ G ) dx. It follows from Lemma 3.6 that D ( t ) ≤ cE ( t ). This prov es (3.10). Hence, the conclusions (3.8) and (3.9) follow from the Renew al Theorem 2.1 immediately . 2 Next, we inv estigate Q ( α ) D ( t ) when α ∈ (0 , d − b ]. In this case, we imp ose Condition 3.2. 11 Theorem 3.8 L et G b e an op en set given as in (2.2) , α ∈ (0 , d − b ] , and Condition 3.2 is satisfie d. Then, lim t → 0 | G | − Q ( α ) G ( t ) t = Per ( α ) ( G ) = Z G Z G c c ( α, d ) | x − y | d + α dy dx, (3.11) wher e the c onstant c ( d, α ) is given by c ( α, d ) = α Γ  d + α 2  2 1 − α π d/ 2 Γ  1 − α 2  . (3.12) Pro of. By F rostman’s theorem (cf. [7, Page 133] or [5]) and Remark 3 in [7, P age 134], we deriv e that for every α ∈ (0 , d − b ], w e hav e Cap d − α ( ∂ G ) = 0, where Cap d − α denotes the ( d − α )- dimensional Riesz-Bessel capacity . Hence, it follows from [22, Lemma 7] that ∂ G is p olar for X . No w the conclusion follows immediately from [6, Theorem 3.4]. 2 Remark 3.9 W e remark that there is a minor error in [6, Corollary 3.7]. There, the authors claimed that lim t → 0 | D |− Q X D ( t ) t = Per X ( D ) provided the underlying L´ evy pro cess X has b ounded v ariation and D is any op en set with finite Leb esgue measure. An α -stable pro cesses has b ounded v ariation if and only if α ∈ (0 , 1) and P er ( α ) (0 , 1) = Per ( α ) ( D ), where D is the complemen t of the ternary Can tor set since m ((0 , 1) \ D ) = 0. This violates Theorem 3.7 when α ∈ ( d − b , 1). The authors in [6] claimed the b oundary of any op en set in R is p olar for the pro cess, as a single p oin t is p olar. How ev er, the num ber of b oundary p oin ts of an op en set in R can b e uncountable, such as D , and it can b e non-p olar for X . F or comparison purp ose, we now consider the r e gular he at c ontent (RHC) for X on a Borel set D ⊂ R d , which is defined as H ( α ) D ( t ) = Z D P x ( X t ∈ D ) dx. F or any B orel sets D ⊂ Ω ⊂ R d suc h that m d (Ω \ D ) = 0, where m d is the Leb esgue measure in R d , the following lemma sho ws that the regular heat conten t on D is equal to that of Ω. Lemma 3.10 L et D ⊂ Ω b e Bor el sets in R d such that m d (Ω \ D ) = 0 and α ∈ (0 , 2) . Then for al l t > 0 , H ( α ) D ( t ) = H ( α ) Ω ( t ) . Pro of. Note that E [ e iξ X t ] = e − t | ξ | α ∈ L 1 ( R ) and this sho ws that X has a contin uous and b ounded transition density p ( α ) ( t, x ) (heat k ernel). W e write p ( α ) ( t, x, y ) := p ( α ) ( t, y − x ). Then H ( α ) Ω ( t ) = Z Ω P x ( X t ∈ Ω) dx = Z D P x ( X t ∈ Ω) dx (3.13) b ecause m d (Ω \ D ) = 0. On the other hand, this last assumption also implies P x ( X t ∈ Ω) = P x ( X t ∈ D ) + P x ( X t ∈ Ω \ D ) = P x ( X t ∈ D ) . (3.14) 12 Com bining (3.13) and (3.14) yields H ( α ) Ω ( t ) = Z D P x ( X t ∈ D ) dx = H ( α ) D ( t ) . This prov es the lemma. 2 The small-time b eha vior of RHC H ( α ) Ω ( t ) for a C 1 , 1 op en set Ω was studied by Acu ˜ na V alv erde [2]. In particular, Theorems 1.1 and 1.2 in [2] show ed that, for d = 1 and Ω = ( a, b ), and for d ≥ 2 and Ω b eing a b ounded C 1 , 1 op en set, the following statements hold: (i) F or 1 < α < 2, lim t → 0 | Ω | − H ( α ) Ω ( t ) t 1 /α = 1 π Γ  1 − 1 α  H d − 1 ( ∂ Ω) (3.15) (ii) F or α = 1, lim t → 0 | Ω | − H (1) Ω ( t ) t log(1 /t ) = 1 π H d − 1 ( ∂ Ω) . (3.16) (iii) F or 0 < α < 1, lim t → 0 | Ω | − H ( α ) Ω ( t ) t = P er ( α ) (Ω) . (3.17) In the ab o ve, H 0 ( ∂ Ω) = #( ∂ Ω) whic h is the cardinalit y of ∂ Ω, and P er ( α ) (Ω) is defined as in (3.11). W e remark that (3.15) and (3.16) follows from [2, Theorems 1.1] directly and (3.17) follo ws from [6, Theorem 3.2]. By com bining (3.15)-(3.17) with Lemma 3.10, we obtain the following corollary of Theorems 1.1 and 1.2 in [2] that is applicable to fractal drums. Corollary 3.11 L et Ω = ( a, b ) if d = 1 and Ω ⊂ R d b e a b ounde d C 1 , 1 op en set if d ≥ 2 . Then for any fr actal drum G ⊂ Ω that satisfies m d (Ω \ G ) = 0 , (3.15) , (3.16) , and (3.17) hold with Ω r eplac e d by G . Theorem 3.7 and Corollary 3.11 sho w that, when α ∈ ( d − b , 2), the rates of decay for the sp ectral heat conten t and regular heat conten t on fractal drums are different. When α ∈ (0 , d − b ), Theorem 3.8 and (3.17) show that the rates of decay for the sp ectral heat conten t and regular heat con tent on fractal drums are the same. 4 Case Studies: the Complemen t of the T ernary Can tor Set and the mo dified Sierpi ´ nski Gask et In this section, we pro vide tw o examples where Theorems 3.7 and 3.8 can b e applied to find the small-time asymptotic b eha vior of the sp ectral heat con tent. Let D b e the complemen t of the ternary Cantor set in [0 , 1] and C b e the ternary Cantor set so that C ∪ D = [0 , 1] and C ∩ D = ∅ . 13 Note that D can b e decomp osed as D = D 0 ∪ R 1 D ∪ R 2 D , where R 1 and R 2 are tw o similitudes on R defined b y R 1 ( x ) = x 3 and R 2 ( x ) = x 3 + 2 3 and D 0 = ( 1 3 , 2 3 ). It is well-kno wn that the interior Mink owski dimension and the Hausdorff dimension of C coincide and are equal to log 2 log 3 . W e find the small-time asymptotic behavior of the regular heat conten t as w ell as the sp ectral heat con tent on D and show that their b eha viors are quite differen t. In the follo wing, the case for α ∈ (1 − ln 2 ln 3 , 2) follo ws from Theorem 3.7 with r 1 = r 2 = 1 3 , b = ln 2 ln 3 , and ρ = 1; and the case for α ∈ (0 , 1 − ln 2 ln 3 ] follows from Theorem 3.8. Example 4.1 L et D b e the c omplement of the ternary Cantor set in [0 , 1] . Then, 1. F or α ∈ (1 − ln 2 ln 3 , 2) , Q ( α ) D ( t ) = | D | − F ( − ln t ) t 1 − ln 2 ln 3 α + o ( t 1 − ln 2 ln 3 α ) as t ↓ 0 , wher e F ( z ) = 1 ln 3 P ∞ n = −∞ ˜ R ( z − nα ) with ˜ R ( t ) =  | D 0 | − Q ( α ) D 0 ( t ) − ˜ A ( t )  t − 1 − ln 2 ln 3 α and ˜ A ( t ) = Q ( α ) D ( t ) − P 2 j =1 Q ( α ) R j D ( t ) − Q ( α ) D 0 ( t ) . 2. F or α ∈ (0 , 1 − ln 2 ln 3 ] , Q ( α ) D ( t ) = | D | − Per ( α ) ( D ) t + o ( t ) , wher e Per ( α ) ( D ) = R D R D c c (1 ,α ) | x − y | 1+ α dy dx . W e turn our attention to the regular heat con tent for X on D . Recall from Lemma 3.6 that E ( t ) = t 1 /α , t ln(1 /t ), or t for α ∈ (1 , 2), α = 1, or α ∈ (0 , 1), resp ectively . F rom (3.15), (3.16), (3.17), and Corollary 3.11, w e obtain the follo wing result. Example 4.2 L et D b e the c omplement of the ternary Cantor set in [0 , 1] . Then, lim t → 0 | D | − H ( α ) D ( t ) E ( t ) = lim t → 0 | (0 , 1) | − H ( α ) (0 , 1) ( t ) E ( t ) =      2 π Γ(1 − 1 α ) if α ∈ (1 , 2) , 2 π if α = 1 , Per ( α ) (0 , 1) , if α ∈ (0 , 1) , wher e Per ( α ) (0 , 1) := R (0 , 1) R (0 , 1) c A α, 1 | x − y | 1+ α dy dx , and A α,d = α 2 α − 1 π − 1 − d/ 2 sin( π α 2 )Γ( d + α 2 )Γ( α 2 ) . Remark 4.3 We r emark th at The or em 3.7 and Example 4.2 show that the sp e ctr al he at c ontent for fr actal drums is very differ ent fr om the SHC for a smo oth domain such as (0 , 1) . In [1, The or em 1.1] the author showe d that for α ∈ (0 , 1] lim t → 0 | (0 , 1) | − H ( α ) (0 , 1) ( t ) E ( t ) = lim t → 0 | (0 , 1) | − Q ( α ) (0 , 1) ( t ) E ( t ) . 14 However, as one c an se e in The or em 3.7 the or der of de c ay for Q ( α ) D ( t ) is t d − b α when α ∈ ( d − b , 1] . Henc e, for α ∈ ( d − b , 1] we c onclude that H ( α ) D ( t ) and Q ( α ) D ( t ) have differ ent de c ay r ates and one c annot infer the smal l-time asymptotic b ehavior of Q ( α ) D ( t ) fr om H ( α ) D ( t ) . Remark 4.4 More general Cantor-t yp e sets w ere introduced in [16, Chapter 4]. Let E i 1 ,i 2 , ··· ,i k , i j ∈ { 1 , 2 , · · · , N } b e compact sets in R d with the following prop erties: 1. E i 1 ,i 2 , ··· ,i k ,i k +1 ⊂ E i 1 ,i 2 , ··· ,i k for all k ∈ N . 2. max i 1 , ··· ,i k d ( E i 1 ,i 2 , ··· ,i k ) → 0 as k → ∞ , where d ( E ) is the diameter of E . 3. P N j =1 d ( E i 1 , ··· ,i k ,j ) s = d ( E i 1 , ··· ,i k ). 4. F or an y ball B , P B ∩ E i 1 ,i 2 , ··· ,i k  = ∅ d ( E i 1 ,i 2 , ··· ,i k ) s ≤ cd ( B ) s for some constant c > 0. Then, it is shown in [16, Chapter 4] 0 < H s  ∞ \ k =1 [ i 1 , ··· ,i k E i 1 ,i 2 , ··· ,i k  < ∞ . Remark 4.5 F or more general Can tor-type sets, Oh tsuk a [17, P age 151] pro vided necessary and sufficien t conditions for the β -dimensional Riesz-Bessel capacity to b e 0. Hence, we can extend Example 4.1 to these Can tor-type sets as w ell. As another example, w e show that the Sierpi ´ nski gasket b elongs to the ab ov e class. Let T 0 b e an op en triangle whose vertices are at (0 , 0) , (1 , 0), and ( 1 2 , √ 3 2 ). Let R 1 ( x ) = x 2 , R 2 ( x ) = x 2 + ( 1 2 , 0), and R 3 ( x ) = x 2 + ( 1 4 , √ 3 4 ). W e define T i = R i ( T 0 ) and T i 1 , ··· ,i k ,i = R i ( T i 1 ,i 2 , ··· ,i k ) for i ∈ { 1 , 2 , 3 } . The Sierpi ´ nski gasket G is defined by G = ∞ \ k =1 [ i 1 , ··· ,i k T i 1 , ··· ,i k . It is easy to chec k that s = ln 3 ln 2 satisfies all conditions ab o ve and this sho ws that the Hausdorff dimension is ln 3 ln 2 and 0 < H ln 3 ln 2 ( G ) < ∞ . Define G = T 0 \ G . Then, G can b e written as G = R 1 G ∪ R 2 G ∪ R 3 G ∪ V , where V is a triangle whose vertices are (1 / 4 , √ 3 / 4), (1 / 2 , 0) and (3 / 4 , √ 3 / 4). Then, G is a set of the form in (2.1) with P 3 j =1 ( 1 2 ) ln 3 ln 2 = 1 and ∂ G = C . It follows from [15, Theorem A] that the in terior Minko wski dimension of C is also ln 3 ln 2 . 15 Hence, G satisfies Condition 3.1 except that the triangle V is not C 1 , 1 . The sp ectral heat con tent for X on triangles, or more generally domains with p olygonal b oundaries, is not av ailable at this point. Therefore, Theorem 3.7 cannot b e directly applied to G . This can be remedied by mo difying the Sierpi ´ nski gask ets by smo othing out the corners of the initial triangle, so that it b ecomes a C 1 , 1 op en set. W e illustrate the precise construction of the mo dified Sierpi ´ nsi gasket G ′ . Define T ′ 1 = 3 [ j =1 R j T 0 ∪ C , where C is a union of three compact regions near the three corners of the triangle V , so that T 0 \ T ′ 1 is a C 1 , 1 op en set. Define T ′ 1 ,i 2 , ··· ,i k ,i = R i ( T ′ 1 ,i 2 , ··· ,i k ) for i ∈ { 1 , 2 , 3 } and k ≥ 2. Now w e define the mo dified Sierpi ´ nsi gasket G ′ as G ′ = ∞ \ k =2 [ i 2 , ··· ,i k T ′ 1 ,i 2 , ··· ,i k . Finally , we define G ′ = T 0 \ G ′ . Note that G ′ and G ′ satisfy G ′ = 3 [ j =1 R j G ′ ∪ C , (4.1) and G ′ = 3 [ j =1 R j G ′ ∪ ( T 0 \ T ′ 1 ) . Note that sets of the form (4.1) are called the inhomogeneous self-similar set with condensation C in [3]. The set G ′ can b e written as in (2.2) with T 0 \ T ′ 1 b eing C 1 , 1 . Hence, Theorem 3.7 holds for G ′ . W e note that Condition 3.2 holds for G (see the remark immediately follo wing Condition 3.2). Hence, Theorem 3.8 can b e directly applicable to b oth G and G ′ . 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[23] M. v an den Berg: On the asymptotic of the heat equation and b ounds on traces asso ciated with Dirichlet Laplacian. J. F unct. A nal. , 71 , (1987), 279–293. Hyunc h ul P ark Departmen t of Mathematics, State Universit y of New Y ork at New Paltz, NY 12561, USA E-mail: parkh@newpaltz.edu Yimin Xiao Departmen t of Statistics and Probability , Michigan State Univ ersity , East Lansing, MI 48824, USA E-mail: xiaoyimi@stt.msu.edu 18