1. Itrducti Let X fx(t) t 0g be a symmetric stable prcess f idex, with X(0) 0. That is, X has idepedet ad statiary icremets, with characteristic fucti

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1 The mst visited sites f symmetric stable prcesses by Richard F. Bass 1, Nathalie Eisebaum ad Zha Shi Uiversity f Cecticut, Uiversite aris VI ad Uiversite aris VI Summary. Let X be a symmetric stable prcess f idex 2 (1 2] ad let L x t dete the lcal time at time t ad psiti x. Let V (t) be such that L V (t) t x2r L x t. We call V (t) the mst visited site f X up t time t. We prve the trasiece f V, that is, lim t!1 jv (t)j 1 almst surely. A estimate is give ccerig the rate f escape f V. The result exteds a well-kw therem f Bass ad Gri fr Brwia mti. Our apprach is based up Dyki's ismrphism therem, ad relates stable lcal times t fractial Brwia mti ad further t the widig prblem fr plaar Brwia mti. Keywrds. Lcal time, stable prcess, mst visited site, Dyki's ismrphism therem, fractial Brwia mti Mathematics Subject Classicati. 60J55. 1 artially prted by NSF grat DMS

2 1. Itrducti Let X fx(t) t 0g be a symmetric stable prcess f idex, with X(0) 0. That is, X has idepedet ad statiary icremets, with characteristic fucti (11) E(e izx(t) ) exp(?c 0 jzj t) where c 0 > 0 is a cstat. We assume 2 (1 2], s that X admits a jitly ctiuus lcal time prcess fl x t t 0 x 2 Rg (see fr example Byla [2]), which we may rmalize s that fr ay t 0 ad Brel fucti f 0, Z t 0 f(x(s)) ds Clearly, whe 2, X is Brwia mti. We are iterested i the set V(t) Z 1?1 f(x) L x t dx x 2 R L x t y2r Ly t which usually is referred t as the set f the \mst visited sites" f X r the \favrite pits" f X up t time t, see Erd}s ad Revesz [6]. It is kw (Eisebaum [4]) that V(t) is either a siglet, r, fr cutably may t, cmpsed f tw pits, but we will t use this prperty. Let us chse (12) V (t) max x2v(t) x which will be called the (maximal) mst visited site. We meti that the chice f V (t) is irrelevat, i the sese that all the results i this paper remai uchaged if we replace V (t) by ay elemet f V(t). Erd}s ad Revesz [6] were the rst t study the mst visited site, fr simple radm walk. I the case f Brwia mti, we recall the fllwig smewhat surprisig result f Bass ad Gri [1]. Therem A (Bass ad Gri [1]). If X is a Brwia mti, i.e. if 2, the fr ay > 11, lim t!1 (lg t) t 12 jv (t)j 1 a.s

3 I particular, Therem A crms the trasiece f the prcess V, i the sese that lim t!1 jv (t)j 1 almst surely. The prblem f determiig the exact rate f escape f V remais pe t the best f ur kwledge, thugh it is als prved i [1] that almst surely, lim if t!1 t?12 (lg t) jv (t)j 0, fr all < 1. It is atural t ask if the mst visited site is still trasiet fr stable prcesses. The aswer is i the armative. Here is the mai result f this paper. Therem 1.1. Let 1 < 2. Fr > 9(? 1), lim t!1 (lg t) jv (t)j 1 a.s. t1 We say a few wrds abut ur methd. The prf f Therem A by Bass ad Gri relies the Ray{Kight therem ad a path decmpsiti fr the Bessel prcess, tgether with sme martigale prperties related t the Brwia lcal time. I the case f a stable -Brwia prcess, such path decmpsiti r martigale prperty is available. Therefre, we have t adpt a dieret apprach. Our startig pit is a extesi, which has bee btaied i [5], f the classical Ray{Kight therem t symmetric Markv prcesses. I particular, this relates stable lcal times t fractial Brwia mti. It is therefre atural that we shall be usig sme Gaussia techiques. Our methd shws a relatiship betwee fractial Brwia mti ad the widig agle f plaar Brwia mti this may be f idepedet iterest. The rest f the paper is rgaized as fllws. Secti 2 is devted t sme prelimiaries fractial Brwia mti, Ray{Kight therem ad Brwia widig agles. They lead i Secti 3 t ur mai prbability estimate. The prf f Therem 1.1 is cmpleted i Secti 4. Thrughut the paper, the letter c with subscripts detes uimprtat (but ite ad psitive) cstats. 2. relimiaries 2.1. Fractial Brwia mti By fractial Brwia mti f idex (abbreviated FBM() r simply FBM), we mea a cetered Gaussia prcess f(x) x 2 Rg, whse cvariace fucti is give by E((x)(y)) 1 2 jxj + jyj? jx? yj - 3 -

4 (I particular, FBM(1) is Brwia mti). Uless stated therwise, we always assume the FBM t start frm 0, i.e., (0) 0. The self-similarity f FBM will be frequetly used withut further meti if is a FBM(), the fr ay a > 0, where \ law " detes idetity i law. () law a?2 (a ) We shall als eed the fllwig law f the iterated lgarithm (LIL) if is a FBM() with 2 (0 1], (t) (21) lim p t!0 + 2t lg j lg tj 1 a.s. This ca be fud i Marcus [12]. We meti that, i this paper, we shall ly eed (2.1) i the case 2 (0 1], thugh the latter cditi is t ecessary Ray{Kight therem fr stable prcesses Our basic tl is a extesi f the Ray{Kight therem fr symmetric strg Markv prcesses, which bears a relatively simple frm i the case f stable prcesses. It was prved i [5] by meas f Dyki's ismrphism therem. The latter, which relates the lcal time f Markv prcesses t Gaussia prcesses, turs ut t be a pwerful tl i the study f lcal times ad additive fuctials. See Marcus ad Rse [13] ad [14] fr a deep study f this subject. As befre, let X be a symmetric prcess f idex 2 (1 2], with lcal time deted by L. Let (22) (r) if t > 0 L 0 t > r r 0 which is the (right-ctiuus versi f the) iverse lcal time at 0. Therem B ([5]). Let 1 < 2, ad let be a FBM(? 1) idepedet f X. It is pssible t chse a value fr the rmalizig cstat c 0 i (11), such that (23) L (1) law 1 2 ( + p 2 ) 2 Remark. I the case 2, Therem B takes the frm f the additivity prperty f squared Bessel prcesses, which is a equivalet frm f the usual Ray{Kight therem - 4 -

5 fr Brwia lcal time at (1) see Ray [16], Kight [11]. We als meti that, ulike Dyki's ismrphism therem, (2.3) des t ivlve siged measures. This will eable us t btai sme sharp iequalities fr varius rema f L (1) Scalig fr stable lcal times There are sme easy scalig prperties fr the lcal time prcess f X. Thugh they are very simple, we preset a list f these prperties here i rder t facilitate applicatis. As befre, X is symmetric stable f idex, with lcal time L, ad iverse lcal time at 0 as i (2.2). The we have the fllwig idetities i law fr ay c > 0, L x law ct t 0 x 2 R c (?1) L xc1 t t 0 x 2 R (cr) r 0 law c (?1) (r) r 0 L x(cr) t 0 x 2 R law c L xc1(?1) (r) t 0 x 2 R 2.4. Brwia widigs Let fz(t) t 0g be a plaar Brwia mti, startig frm (1 0). It is kw that every pit is plar fr Z. I particular, with prbability e, Z ever hits the rigi. S there exists a ctiuus determiati f (t), the ttal agle wud by Z arud the rigi up t time t (with, say, (0) 0). Thus recrds the agle ad keeps track f the umber f times the Brwia path has wud arud the rigi, cutig clckwise lps (?2) ad cuterclckwise lps (2). A imprtat feature f the widig agle prcess is that it ca be represeted as a Brwia time chage, amely, (24) (t) B(H(t)) t 0 where B is a stadard e-dimesial Brwia mti startig frm 0, idepedet f the radm clck H. This actually is a particular case f the s-called \skew-prduct represetati" fr plaar Brwia mti see It^ ad McKea [9, p. 270] fr mre details. It is pssible t cmpletely determie the law f H(t) fr each t. Ideed, writig R(t) kz(t)k (the radial part f the plaar Brwia mti), the (25) H(t) Z t ds R 2 (s)

6 Mrever, the cditial Laplace trasfrm f H(t) give R(t) was determied by Yr [20] fr a 0, (26) E e?a H(t) j R(t) Ip (R(t)t) 2a I 0 (R(t)t) where I ( ) is the mdied Bessel fucti f idex. 3. Key estimate Thrughut the secti, X is a symmetric stable prcess f idex 2 (1 2], startig frm 0, with lcal time L. The iverse lcal time at 0 is deted by, as i (2.2). Sice the rmalizig cstat c 0 (deed i (1.1)) has iuece Therem 1.1, we shall frm w chse c 0 t be the e satisfyig Therem B (see Secti 2.2), withut further meti. Here is the mai prbability estimate f the paper. Therem 3.1. There exists a cstat c 1 2 (0 1) such that fr all 0 < 12, (31) jxj1 L x (1) < 1 + c 1 54 j lg j 2 rf. We ly have t treat the situati where is sucietly clse t 0. Let f(x) x 2 Rg be a FBM(? 1), idepedet f X. measurable evets E 1 E 2 E 3 E 4 jxj1 L x (1) < (x) < c 2 fr all jxj 1(?1) 2 (x) < c 2 fr all jxj 1 2 (x) < c 2 jxj?1 j lg j fr all 1(?1) jxj 1 Csider the fllwig (S the prbability term the left had side f (3.1) is (E 1 )). By self-similarity, (E 2 ) 2 (x) < c 2 fr all jxj 1 s we ca chse c 2 sucietly large such that (32) (E 2 \ E 3 ) 2 3 (E 3 ) - 6 -

7 O the ther had, by the LIL fr FBM (see (2.1)) ad symmetry, the radm variable 0<u<13 u 1(?1) <jxj1 2 (x) jxj?1 j lg uj is almst surely ite. (Actually the j lg uj term here ca be replaced by lg j lg uj). Therefre, it is pssible t pick c 2 s large (hw large depedig ly ) that (E 4 ) 23. Jitly csiderig this ad (3.2), we are able t x a chice fr c 2 such that (33) (E 2 \ E 3 \ E 4 ) 1 3 Nw, bserve that E 1 ad E 2 \ E 3 \ E 4 are idepedet, ad that 4\ i1 E i E 5 where E 5 L x (1) (x)? 1 < + c 2 f 0 (x) fr all jxj 1 f 0 (x) if 0 jxj 1(?1), mi(jxj?1 j lg j 1) if 1(?1) < jxj 1. Therefre, \ 4 \ 4 (E 1 ) E i E i i2 i1 (E 5 ) 1 2 ((x) + p 2 ) 2? 1 < + c 2 f 0 (x) fr jxj ((x) + p 2 ) 2? 1 < (1 + c 2 ) f 0 (x) fr jxj 1 where we have used (2.3) i the last equality. It is easily checked that fr a 2 (0 1 + c 2 ), if (y + p 2 ) 2 2? 1 < a, the y < c 3 a, where c 3 is a cstat depedig ly the value f c 2. Takig it accut (3.3), we arrive at (34) (E 1 ) 3 (x) < c 3 (1 + c 2 ) f 0 (x) jxj 1 The ext step is t use Slepia's lemma t estimate the prbability term the right had side f (3.4)

8 T this ed, let fw 1 (t) t 0g ad fw 2 (t) t 0g be tw idepedet real-valued Brwia mtis, with W 1 (0) W 2 (0) 0. Dee the prcess U fu(x) x 2 [?1 1]g by U(x) 8 < Clearly, fr ay x 2 [?1 1], W 1 (x?1 ) if 0 x 1, 1 2 W 1(jxj?1 ) + p 3 2 W 2(jxj?1 ) if?1 x 0. (35) E(U 2 (x)) jxj?1 E ( 2 (x)) I rder t apply Slepia's lemma, we have t cmpare the cvariace fuctis f U ad. Let (x y) 2 [?1 1] 2. There are tw pssible situatis. the rst case is whe xy 0. We rst assume x 0 ad y 0. The (writig a ^ b fr mi(a b)) E (U(x)U(y)) (x ^ y)?1 1 2 The situati where x 0 ad y 0 is similar. x?1 + y?1? jx? yj?1 E((x)(y)) Ather pssibility is that xy < 0. Withut lss f geerality, we assume x > 0 ad y < 0. I this case, Therefre, we have prved that E(U(x)U(y)) 1 (x ^ jyj)?1 2 1 x?1 + jyj?1? (x + jyj)?1 2 E((x)(y)) (36) E(U(x)U(y)) E ((x)(y)) fr ay?1 x y 1. I view f (3.5) ad (3.6), we ca apply Slepia's lemma (see [19]), t see that fr ay -egative Brel fucti f, (x) < f(x) jxj 1 U(x) < f(x) jxj 1 W 1 (t) < f(t 1(?1) ) W 12 (t) < 2f(?t 1(?1) ) 0 t 1 where we have writte W 12 (t) W 1 (t) + p 3 W 2 (t) fr brevity

9 (37) Takig f(x) c 3 (1 + c 2 ) f 0 (x) c 4 f 0 (x), ad gig back t (3.4), (E 1 ) 3 3 W 1 (t) < c 4 f 0 (t 1(?1) ) W 12 (t) < 2c 4 f 0 (?t 1(?1) ) 0 t 1 W 1 (t) < c 4 W 12 (t) < 2c 4 0 t W 1 (s) < c 4 s j lg j < s 1 3((E 6 ) + (E 7 ) (E 8 )) where E 6 E 7 E 8 W 1 () <? p j lg j 0 t W 1 (t) < c 4 W 12 (t) < 2c 4 W 1 (s)? W 1 () < c 4 s j lg j + p j lg j < s 1 It remais t estimate (E i ) fr i 6, 7, 8. By the well-kw Mill's rati fr Gaussia tails (see fr example Shrack ad Weller [18, p. 850]), (38) (E 6 ) 1 p 2 j lg j exp? 1 2 j lg j2 T estimate (E 7 ), bserve that by scalig, (E 7 ) W 1 (t) < c 4 p W1 (t) + p 3 W 2 (t) < 2c 4 p 0 t 1 (W 1 (t) W 2 (t)) 2 D 0 t 1 where D f(x y) x < c 4 p x + p 3 y < 2c4 p g. I wrds, the evet the righthad side says that, startig frm (0 0), the plaar Brwia mti (W 1 W 2 ) stays i D durig [0 1]. A gemetric bservati (usig traslati ad rtati) reveals that (E 7 ) startig frm ( p c 5 0), the agular part f plaar Brwia mti lies i (?3 3) durig [0 1] with c 5 4(c 4 ) 2 3. Let Z dete a plaar Brwia mti startig frm (1 0), with agular part (see Secti 2.4). By scalig, (E 7 ) 0s1(c 5 ) j(s)j < 3

10 Recall frm (2.4) that (s) B(H(s)), where H is a ctiuus clck, idepedet f the e-dimesial Brwia mti B. It is well-kw (see fr example Csrg} ad Revesz [3, p. 43]) that fr a > 0 ad x > 0, 0sa jb(s)j < x 4 1X k0 (?1) k 2k + 1 exp? (2k + 1)2 2 a 8x 2 4 exp? 2 a 8x 2 Hece, by cditiig H ad the usig (2.6), (E 7 ) 4 h E exp 0uH(1(c 5 ))? 9 8 H? 1 4 E h I32 (R(t)t) I 0 (R(t)t) jb(u)j < 3 c 5 i i where t 1(c 5 ). T cmpute the expectati the right had side, we use sme techiques which derive their ispirati frm It^ ad McKea [9, pp. 270{271]. Recall that R as a tw-dimesial Bessel prcess is Markvia, whse semi-grup is kw, see Revuz ad Yr [17, Chap. XI] (R(t) 2 dy) y t exp? 1 + y2 2t y I 0 dy y > 0 t O the ther had, accrdig t Gradshtey ad Ryzhik [8, p. 967], fr x > 0, r 2 x csh x? sih x I 32 (x) x 32 c 6 x 32 1l f0<x<1g + x?12 e x 1l fx1g Csequetly (recallig that t 1(c 5 )), (39) Z t? y 52? y 2 Z 1? y 12? y (E 7 ) c 7 exp? dy + exp t 2t t t? y2 dy 2t 0 c 8 34 It remais t estimate (E 8 ). By the statiarity f Brwia icremets, (E 8 ) W 1 (t) < c 4 (t + ) j lg j + p j lg j t 0 < t 1? W 1 (t) < c 4 t j lg j + (c 4 + 1) p j lg j 0 < t

11 Fr ay a > 0 ad b > 0, (310) W 1 (s) < a + bs s > 0 1? e?2ab (see fr example Karatzas ad Shreve [10, p. 197]). It fllws that W 1 (t)? W 1 (12) < (t? 12) j lg j + 1 t 12 Therefre, by the idepedece f Brwia icremets, (E 8 ) 1 c 9 1 c 9 W 1 (t) < c 4 t j lg j + (c 4 + 1) p j lg j 0 < t 12 c 9 W 1 (t) < (t? 12) j lg j c 4 2 j lg j + (c 4 + 1) p j lg j t 12 W 1 (t) < (c 4 + 1)t j lg j + (c 4 + 1) p j lg j t > 0 (311) c 10 p j lg j 2 the last iequality fllwig frm (3.10). Cmbiig (3.7){(3.9) ad (3.11) cmpletes the prf f Therem rf f Therem 1.1 Befre startig the prf f Therem 1.1, we recall sme tati which was already used i the previus sectis X is a symmetric stable prcess f idex 2 (1 2], whse lcal time is L x t. The iverse lcal time at 0 is deted by (r). Als, will dete the FBM(? 1) itrduced i (2.3). As befre, we assume withut lss f geerality that c 0 satises Therem B (see Secti 2.2). Fr brevity, we write L t x2r Lx t t 0 The prf f Therem 1.1 is divided it several small steps. tu Lemma 4.1. Fr ay b > 4, (41) lim r!1 (lg r) b L (r) r? r 1 a.s. rf. Sice b > 4, it is pssible t d a cstat 0 < a < 15 such that b > 4(5a). Dee the sequece r exp( a ). By scalig, L (r ) < r +1 + r +1 (lg r ) b L (1) < r +1 + r r +1 r (lg r ) b L (1) < 1 + c 11 1?a + c 12 ab

12 Applyig Therem 3.1 t c 11?(1?a) + c 12?ab yields that L (r ) < r +1 + r +1 (lg r ) b c (1?a) ab4 (lg ) 2 which is summable fr (recallig that a < 15 ad that b > 4(5a)). A applicati f the Brel{Catelli lemma, tgether with the mticity, gives that (lg r) b lim if L r!1 (r) r? r 1 a.s. Sice b > 4 is arbitrary, this cmpletes the prf f Lemma 4.1. Lemma 4.2. Fr 0 < < 1 ad h > 0, (42) L x (1) > 1 + c 14 exp? 2 9h?1 tu rf. Let be a FBM(? 1), idepedet f the uderlyig stable prcess X. Clearly,? L x (1) > 1 + L x (1) (x) > 1 + By Therem B (see Secti 2.2), we have L x (1) > jxj1? (x) + p 2 2 > 1 + j(x)j > 2 j(x)j > 2h (?1)2 This yields (4.2) by meas f a tail estimate fr geeral Gaussia prcesses due t Marcus ad Shepp [15] if fy (t) t 2 Tg is a buded real-valued cetered Gaussia prcess, the lg t2t jy (t)j > x? x2 2 2 x! 1 with 2 t2t E(Y 2 (t)). tu Lemma 4.3. Fr ay M > 0, there exists c 15 > 0 ad h 2 (0 1) such that, wheever 0 < h < h ad M h (?1)2, (43) jl x (1)? 1j < c

13 rf. Withut lss f geerality, we ca assume that M 12. Let be a FBM(?1), idepedet f X. Clearly, (44) jl x (1)? 1j < jl x (1) (E 9 )? (E 10 )? 1j < 2 (x) < jl x (1)? (x)j < 2 2 (x) < jl x (1)? (x)j < 2? 2 (x) with bvius tati. By (2.3), (45) (E 9 ) c 16 jxj1 jxj1 j (x) + p 2 (x)j < 2 j(x)j < 4 j(x)j < 4h (?1)2 j(x)j < M 4 O the ther had, (46) (E 10 ) c 17 jxj1 jxj1 jxj1 2 (x) 2 (x) 2 (x) h?1 M h (?1)2 M h (?1)2 We ca chse h sucietly small such that c 17 < c 16. The lemma fllws by jitly csiderig (4.4), (4.5) ad (4.6). tu

14 Lemma 4.4. It is pssible t chse a sucietly small h 2 (0 1) such that, fr all 0 < h < h ad > 0, (47) 1r2 (L x (r)? r) > c 18 exp? h?1 rf. We assume withut lss f geerality that h (?1)2, therwise (4.7) hlds trivially. Dee, Y h (r) T if E 11 L x (r)? L0 (r) r 1 Y h (r) > T 2 jxj(3?t ) 1(?1) h???l x??< (3)? Lx (T )? (3? T ) (3? T ) Sice r 7! Y h (r) is right-ctiuus, we have Y h (T ) whe T < 1. By the triagle iequality, the evet E 11 (thus 1 T 2), L x (3)? L0 (3) Y h (T )? >? (3? T ) 3 3???L?? x (3)? Lx (T )? (3? T ) I ther wrds, E 11 fy h (3) > 3g. Let (F t ) t0 be the atural ltrati f X. Observe that ft < 1g, T L 0 (T ) is measurable with respect t F (T ) ad s is the evet ft 2g. Therefre, give T 2, E 11 is idepedet f F (T ). By the strg Markv ad scalig prperties, (T 2) jl x (1)? 1j < 3 (E 11 ) (Y h (3) > 3) L x (1) > (?1) which, i view f Lemmas 4.2 ad 4.3, yields fr small h ad h (?1)2, (T 2) c 19 exp? h?

15 Sice f 1r2 (L x (r)? r) > g ft 2g, this cmpletes the prf f the lemma. tu Lemma 4.5. Fr ay > 0 ad b > 0 such that (? 1) > 2b, we have (48) lim r!1 (lg r) b r jxjr 1(?1) (lg r)? L x (r)? r 0 a.s. rf. Dee r 2. By scalig ad Lemma 4.4, fr all sucietly large, r (lg r ) b which sums. lemma. r rr +1 1r2? L x (r)? r > jxjr 1(?1) +1 (lg r ) jxj2 1(?1) (lg r )? L x (r)? r > c 18 exp? (lg r ) (?1)?2b (lg r ) b Lemma 4.5 fllws frm a immediate applicati f the Brel{Catelli rf f Therem 1.1. Fix > 9(? 1). It is pssible t chse b > 4 ad " > 0 such that b <? 1 2? 1? 1? " Sice r 7! (r) is a stable subrdiatr f idex (? 1), sme well-kw therems (see fr example, Fristedt [7, Therems 11.2 ad 11.7]) crm that almst surely fr all sucietly large r, tu (49) (410) (r) < r (?1) (lg r) (?1)+" (r) > r (1?")(?1) Let t be very large, say t 2 [(r?) (r)]. By (4.1), (411) L t L (r?) > r + r (lg r) b O the ther had, by (4.9) ad (4.10), ad writig c 20 ((? 1)(1? ")), L x t L x (r) jxjt 1 (lg t) jxj((r)) 1 (lg (r)) jxjc 20 ((r)) 1 (lg r) L x (r) jxjc 20 r 1(?1) (lg r)?1(?1)?" L x (r)

16 Applyig Lemma 4.5 gives that jxjt 1 (lg t) L x t < r + r (lg r) b which, i view f (4.11), implies that jv (t)j > t 1 (lg t), where V (t) is as befre the mst visited site (see (1.2)). As a csequece, fr > 9(? 1), lim if t!1 (lg t) This cmpletes the prf f Therem 1.1. jv (t)j 1 a.s. t1 tu Ackwledgmets We are grateful t Mike Marcus fr a referece fractial Brwia mti. Refereces [1] Bass, R.F. ad Gri,.S. The mst visited site f Brwia mti ad simple radm walk. Z. Wahrsch. Verw. Gebiete 70 (1985) 417{436. [2] Byla, E.S. Lcal times fr a class f Markv prcesses. Illiis J. Math. 8 (1964) 19{39. [3] Csrg}, M. ad Revesz,. Strg Apprximatis i rbability ad Statistics. Academic ress, New Yrk, [4] Eisebaum, N. O the mst visited sites by a symmetric stable prcess. rbab. Thery Related Fields 107 (1997) 527{535. [5] Eisebaum, N. ad Shi, Z. A Ray{Kight therem fr strg Markv prcesses. J. Theretical rbab. (t appear) [6] Erd}s,. ad Revesz,. O the favurite pits f a radm walk. Mathematical Structures { Cmputatial Mathematics { Mathematical Mdellig 2 pp. 152{157. Sa, [7] Fristedt, B. Sample fuctis f stchastic prcesses with statiary idepedet icremets. I Advaces i rbability ad Related Tpics 3 pp. 241{396. Dekker, New Yrk, [8] Gradshtey, I.S. ad Ryzhik, I.M. Table f Itegrals, Series, ad rducts. Academic ress, New Yrk,

17 [9] It^, K. ad McKea, H.. Diusi rcesses ad Their Sample aths. (2d ritig). Spriger, Berli, [10] Karatzas, I. ad Shreve, S.E. Brwia Mti ad Stchastic Calculus. Secd editi. Spriger, New Yrk, [11] Kight, F.B. Radm walks ad a sjur desity prcess f Brwia mti. Tras. Amer. Math. Sc. 109 (1963) 56{86. [12] Marcus, M.B. Hlder cditis fr Gaussia prcesses with statiary icremets. Tras. Amer. Math. Sc. 134 (1968) 29{52. [13] Marcus, M.B. ad Rse, J. Sample path prperties f the lcal times f strgly symmetric Markv prcesses via Gaussia prcesses. (Special Ivited aper). A. rbab. 20 (1992) 1603{1684. [14] Marcus, M.B. ad Rse, J. Gaussia chas ad sample path prperties f additive fuctials f symmetric Markv prcesses. A. rbab. 24 (1996) 1130{1177. [15] Marcus, M.B. ad Shepp, L.A. Sample behavir f Gaussia prcesses. I rc. Sixth Berkeley Symp. Math. Statist. rbab. 2 pp. 423{441. Uiversity f Califria ress, Berkeley, [16] Ray, D. Sjur times f a diusi prcess. Illiis J. Math. 7 (1963) 615{630. [17] Revuz, D. ad Yr, M. Brwia Mti ad Ctiuus Martigales. Secd editi. Spriger, Berli, [18] Shrack, G.R. ad Weller, J.A. Empirical rcesses with Applicatis t Statistics. Wiley, New Yrk, [19] Slepia, D. The e-sided barrier prblem fr Gaussia ise. Bell System Tech. J. 42 (1962) 463{501. [20] Yr, M. Li de l'idice du lacet brwie, et distributi de Hartma{Wats. Z. Wahrsch. Verw. Gebiete 53 (1980) 71{95. Richard F. Bass Nathalie Eisebaum Zha Shi Departmet f Mathematics Labratire de rbabilites Labratire de rbabilites Uiversity f Cecticut Uiversite aris VI Uiversite aris VI Strrs, CT USA 4 lace Jussieu 4 lace Jussieu aris Cedex 05, Frace aris Cedex 05, Frace bass@math.uc.edu ae@ccr.jussieu.fr shi@ccr.jussieu.fr