The exit distribution for iterated Brownian motion in cones

ARTICLE IN PRESS Stochastic Processes and their Applications 116 (2006) 36–69 www.elsevier.com/locate/spa The exit distribution for iterated Brownia...

Download PDF

362KB Sizes 1 Downloads 197 Views

Report

PDF Reader
Full Text

ARTICLE IN PRESS

Stochastic Processes and their Applications 116 (2006) 36–69 www.elsevier.com/locate/spa

The exit distribution for iterated Brownian motion in cones Rodrigo Ban˜uelosa,,1, Dante DeBlassieb a

Department of Mathematics, Purdue University, West Lafayette, IN 47906, USA Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA

b

Received 14 July 2004; received in revised form 11 July 2005; accepted 13 July 2005 Available online 9 August 2005

Abstract We study the distribution of the exit place of iterated Brownian motion in a cone, obtaining information about the chance of the exit place having large magnitude. Along the way, we determine the joint distribution of the exit time and exit place of Brownian motion in a cone. This yields information on large values of the exit place (harmonic measure) for Brownian motion. The harmonic measure for cones has been studied by many authors for many years. Our results are sharper than any previously obtained. r 2005 Elsevier B.V. All rights reserved. Keywords: Iterated Brownian motion; Harmonic measure; Exact constants

1. Introduction Roughly speaking, iterated Brownian motion (IBM) is ‘‘Brownian motion run at an independent one-dimensional Brownian clock.’’ Of course, this is not rigorous because the one-dimensional Brownian motion can take negative values, whereas Brownian motion is deﬁned only for nonnegative times. There are two natural ways to get around this. First, one can use the absolute value of the one-dimensional Corresponding author. Tel.: +1 765 494 1977; fax: +1 765 494 0548. 1

E-mail addresses: [email protected] (R. Ban˜uelos), [email protected] (D. DeBlassie). Supported in part by NSF Grant # 9700585-DMS.

0304-4149/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.spa.2005.07.003

ARTICLE IN PRESS R. Ban˜uelos, D. DeBlassie / Stochastic Processes and their Applications 116 (2006) 36–69

37

Brownian motion. This process is one of the subjects of the papers Allouba and Zheng [2] and Allouba [3], where various connections with the biharmonic operator are presented. Those authors call their process ‘‘Brownian-time Brownian motion’’ (BTBM). The other rigorous deﬁnition of IBM is the one we will use and it is due to [8]. He uses a natural extension of Brownian motion to negative times, called ‘‘two-sided Brownian motion.’’ Formally, let X þ ; X be independent n-dimensional Brownian motions started at z 2 Rn and suppose Y is one-dimensional Brownian motion started at 0, independent of X . Deﬁne two-sided Brownian motion by ( X ðtÞ ¼

X þ ðtÞ;

tX0;

X ðtÞ;

to0:

Then iterated Brownian motion is Z t ¼ X ðY t Þ;

tX0.

Although IBM is not a Markov process, it has many properties analogous to those of Brownian motion; we list a few here. (1) For instance, the process scales. That is, for each c40, cZðc4 tÞ is IBM. (2) The law of the iterated logarithm holds [8] lim sup t!0

ZðtÞ 3=4

t1=4 ðlog logð1=tÞÞ

¼

25=4 33=4

a:s.

There is also a Chung-type LIL [22] and various Kesten-type LIL’s [12] for IBM. Other properties for local times are proved in [28]. (3) The process has 4th order variation [8] lim

jLj!0

n X

½Zðtk Þ Zðtk1 Þ4 ¼ 3ðt sÞ

in Lp ,

k¼1

where L ¼ fs ¼ t0 pt1 p ptn ¼ tg is a partition of ½s; t and jLj ¼ max1pkpn jtk tk1 j. An interesting interpretation of IBM, due to [9], is as a model for diffusion in a crack. See [16] for other references. There is a very interesting connection between IBM (as well as the BTBM process of Allouba and Zheng) and the biharmonic operator D2 . Namely, the function uðt; xÞ ¼ E x ½f ðZt Þ

ARTICLE IN PRESS R. Ban˜uelos, D. DeBlassie / Stochastic Processes and their Applications 116 (2006) 36–69

38

solves the Cauchy problem [3,16] q Df ðxÞ 1 uðt; xÞ ¼ pﬃﬃﬃﬃﬃﬃﬃ þ D2 uðt; xÞ, qt 2pt 2 uð0; xÞ ¼ f ðxÞ. The appearance of the initial function f ðxÞ in the PDE can be viewed as a manifestation of the non-Markovian nature of IBM. This connection suggests the possibility of a relationship between IBM and initialboundary or boundary value problems involving the biharmonic operator. While the results of [16] are not encouraging for connections with initial-boundary value problems, the work of [3] suggests there is some hope for ﬁnding connections between probability and Dirichlet-type boundary value problems for the bilaplacian. Such a connection, if found, would be particularly exciting in its possible applications to the spectral theory (the study of eigenvalues and eigenfunctions) of the bilaplacian where very little seems to be known. An important ﬁrst step in exploring this possibility, as in the case of the Laplacian and Brownian motion, is to gain an understanding of the structure of the distribution of the exit place of IBM from open sets, what one may call, by abuse of terminology, the ‘‘harmonic measure’’ associated with IBM. In contrast with the BTBM process of Allouba and Zheng, this distribution does not coincide with the usual harmonic measure associated with the Laplacian. The goal of this article is to study the exit distribution of IBM from a cone in Rn . We chose this domain because it is unbounded and it contains a boundary singularity. In addition, in this setting we are able to obtain explicit formulas which lead to very sharp results. Our metho ds are easily adapted to bounded domains but in general our formulas will not be as explicit and the result will not be as sharp. Let S n1 be the unit sphere in Rn . If D is a proper open subset of S n1 , then the generalized cone C generated by D is the set of rays emanating from the origin 0 passing through D. Throughout we assume qD is C 2;a . Then the Laplace–Beltrami operator DSn1 on Sn1 with Dirichlet boundary conditions on qD has a complete set of orthonormal eigenfunctions mj with corresponding eigenvalues 0ol1 ol2 p l3 p such that [11] DSn1 mj ¼ lj mj mj ¼ 0

and

on qD.

ð1:1Þ

If B is n-dimensional Brownian motion and tC ðBÞ is its exit time from C, then it is known [13] and [14] Px ðtC ðBÞ4tÞCðxÞtp1 =2

as t ! 1,

(1.2)

where rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ n 2 n 1 1 p1 ¼ l1 þ 2 2

(1.3)

ARTICLE IN PRESS R. Ban˜uelos, D. DeBlassie / Stochastic Processes and their Applications 116 (2006) 36–69

39

and CðxÞ ¼

Z 2 p1 =2 1 Gð2 ðp1 þ nÞÞ jxj x m ðZÞmðdZÞ m1 , 1 jxj 2 G p1 þ n2 qD

(1.4)

m being surface measure on S n1 . Here and in what follows, f ðtÞgðtÞ

as t ! 1,

means f ðtÞ !1 gðtÞ

as t ! 1.

In [16] it is shown that if tC ðZÞ t ! 1, 8 p =2 1 ; > : tðp1 þ1Þ=2 ;

is the ﬁrst exit time of IBM Z from C, then as p1 o2; p1 ¼ 2;

(1.5)

p1 42;

where f ðtÞ gðtÞ means there exist constants C 1 and C 2 such that C1p

f ðtÞ pC 2 ; gðtÞ

t large.

In light of Burkholder’s inequalities [10] and (1.2), E x ðjBðtC Þjp Þo1

iff pop1 .

(1.6)

Hence considering the ‘‘fourth order’’ properties of IBM described above, we expect (1.5) should imply ( p1 p2; po2p1 ; p E x ðjZðtC Þj o1 iff po2ðp1 þ 2Þ; p1 42: Indeed, we have the following theorem. We will always assume the positive xn -axis passes through C. If jðZÞ is the angle between Z 2 S n1 and the positive xn -axis, then in polar coordinates y ¼ rZ, the ðn 1Þ-dimensional surface measure s on qC is given by sðdyÞ ¼ rn2 sin jðZÞmðdZÞ dr. Theorem 1.1. As r ! 1, for z ¼ ry, 8 > r2p1 1 ; > > > > < d 5 Pz ðjZðtC ÞjprÞAðz; p1 Þ r ln r; > dr > > > > : rp1 3 ;

(1.7) p1 o1; 2 p1 ¼ 1; 2 p1 41; 2

ARTICLE IN PRESS 40

R. Ban˜uelos, D. DeBlassie / Stochastic Processes and their Applications 116 (2006) 36–69

where for

p1 2

o1,

G p12þn G 3p12þn 1 Z 2p1 2 Aðz; p1 Þ ¼ r m1 ðyÞ m1 ðZÞmðdZÞ 2 qD G p1 þ n2 Z Z 1 q sin jðZÞ m1 ðZÞmðdZÞ wp1 =2 ð1 þ wÞ2 dw, qnZ qD 0 the integrals over qD are taken with respect to mðdZÞ and qnqZ denotes the inward normal derivative at qD; for p21 ¼ 1, Z Z n1 4 2 q Aðz; p1 Þ ¼ 2 1 þ r m1 ðyÞ m1 ðZÞmðdZÞ sin jðZÞ m1 ðZÞmðdZÞ ; 2 qnZ qD qD and for

p1 2

41, p1

Z

Aðz; p1 Þ ¼ 2r m1 ðyÞ

q sinðZÞj m1 ðZÞmðdZÞ E z ðtBM Þ, qnZ qD

where tBM is the first exit time of Brownian motion from C. Corollary 1.2. (a) As r ! 1, 8 1 2p1 > > r ; > > 2p > 1 > > <1 r4 ln r; Pz ðjZðtC Þj4rÞAðz; p1 Þ 4 > > > > 1 > p1 2 > > ; :p þ 2r 1

p1 o1; 2 p1 ¼ 1; 2 p1 41: 2

(b) We have

E z ½jZðtC Þjp o1

iff

8 > po2p1 ; > > > > < po4; > > > > > : pop1 þ 2;

p1 o1; 2 p1 ¼ 1; 2 p1 41: 2

Remark. Below in (3.3) we get a series expansion of the density valid for rajzj, but it is not all that enlightening.

d dr Pz ðjZðtC ÞjprÞ

Along the way to proving Theorem 1.1, we derive the following result of independent interest. Theorem 1.3. For t being the first exit time of Brownian motion B from the cone C, Px ðBt 2 dy; t 2 dtÞ ¼

1 q p ðt; x; yÞsðdyÞ dt, 2 qny C

where qnqy is the inward normal derivative at qC, pC ðt; x; yÞ is the transition density of Brownian motion killed upon exiting C and s is surface measure on qC.

ARTICLE IN PRESS R. Ban˜uelos, D. DeBlassie / Stochastic Processes and their Applications 116 (2006) 36–69

41

Hsu [21] has proved this result for bounded C 3 domains. But because the cone C is unbounded with a boundary singularity, there are technicalities not present in the case considered by Hsu. We have the following consequence of Theorem 1.3 that is also of independent interest. Note it gives an improvement of (1.6) above. Theorem 1.4. Let t be the exit time of Brownian motion from C. Then for x ¼ ry and rar, 1 X d 1 n n aj 2 1=2 aj Px ðjBt jprÞ ¼ r22 r12 a1 j g ½1 þ ð1 g Þ dr 2 j¼1 Z q sin jðZÞ mj ðZÞ mðdZÞmj ðyÞ, qnZ qD qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ where aj ¼ lj ðn2 1Þ2 and g ¼ r22rr and the convergence is uniform for gp1 e. þr2

Corollary 1.5. As r ! 1, for x ¼ ry, Z n 2p1 þ22 rp1 q Px ðjBt j4rÞ sin jðZÞ m ðZÞmðdZÞ m1 ðyÞrp1 . 1 qnZ p1 ðp1 þ n2 1Þ qD It follows from the classical estimates for harmonic measure (see [20,17]) that there are constants C 1 and C 2 , depending on x, such that for large r, C 1 rp1 pPx ðjBt j4rÞpC 2 rp1 . However, as far as we know these techniques do not identify the exact limit as Corollary 1.5 above does. It is also interesting to note here that in the case of the parabolic-shaped regions Pa ¼ fðx; Y Þ 2 R Rn1 : x40; jY joAxa g, with 0oao1 and A40, it is proved in [4] that pﬃﬃﬃﬃﬃ l1 r1a , log Pz ðjBt j4rÞ Að1 aÞ

(1.8)

where l1 is the smallest eigenvalue for the Dirichlet Laplacian in the unit ball of Rn1 . In view of Corollary 1.5, it is natural to ask if it is possible to obtain a similar expression for the harmonic measure of the parabolic-shaped regions, and in particular to identify the asymptotics of Pz ðjBt j4rÞ. That is, is it possible to obtain a result similar to that in Ban˜uelos and Carroll but without the logs? At present we do not know the answer to this question. For various results related to the asymptotics of exit times of Brownian motion and heat kernels for parabolic-type regions, we refer the reader to [5,7,15,23,24]. Finally, Allouba and Zheng [3] show the exit distribution of their BTBM process is the same as that of Brownian motion—i.e., harmonic measure (see their [3, Theorem 0.2]). In light of this, Theorem 1.4 above yields the density of the size of the exit place

ARTICLE IN PRESS 42

R. Ban˜uelos, D. DeBlassie / Stochastic Processes and their Applications 116 (2006) 36–69

of BTBM in a cone. Also note for IBM, the exit distribution is not the same as the exit distribution of two-sided Brownian motion in C. The article is organized as follows. In Section 2 we establish various estimates on the terms in the series expansion of the heat kernel of the cone. Then we use them to prove Theorem 1.4 and Corollary 1.5. Using Theorem 1.3, we prove Theorem 1.1 in Section 3. The proof of Theorem 1.3 is given in Section 4, using some results of Pinsky. The proof is independent of the proof of Theorem 1.1.

2. Auxiliary results; proof of Theorem 1.4 and Corollary 1.5 In what follows, we will make repeated use of the following result from Gilbarg and Trudinger [18, Lemma 6.18, p. 111]. Elliptic regularity theorem. Suppose L¼

X i;j

aij ðxÞ

X q2 q þ bj ðxÞ þ cðxÞ qxj qxi qxj j

is a strictly elliptic operator on a domain O Rn . Assume the coefficients of L are in C a ðOÞ, O has a C 2;a boundary portion T and j 2 C 2;a ðOÞ, for some a 2 ð0; 1. If u 2 CðOÞ \ C 2 ðOÞ satisfies Lu ¼ 0 in O and u ¼ j on T, then u 2 C 2;a ðO [ TÞ. The heat kernel for C has a series expansion, due to [6]: For x ¼ ry, y ¼ rZ, 1 rr r2 þr2 X n I aj (2.1) pC ðt; x; yÞ ¼ t1 ðrrÞ12 e 2t mj ðyÞmj ðZÞ, t j¼1 where r; r40, y; Z 2 S n1 , and rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ n 2 1 . aj ¼ lj þ 2

(2.2)

The convergence is uniform for ðt; x; yÞ 2 ðT; 1Þ fx 2 C: jxjoRg C, for any positive constants T and R. The modiﬁed Bessel function I n is given by 1 2k z n X z 1 . I n ðzÞ ¼ 2 k¼0 2 k!Gðn þ k þ 1Þ First we show termwise normal differentiation at the boundary is permitted. Recall q q qny and qnZ denote inward normal differentiation at y 2 qCnf0g and Z 2 qD, respectively. Lemma 2.1. For y ¼ rZ 2 qCnf0g, 1 rr r2 þr2 X n q q pC ðt; x; yÞ ¼ r1 t1 ðrrÞ12 e 2t I aj mj ðZÞ mj ðyÞ qny t qn Z j¼1

ARTICLE IN PRESS R. Ban˜uelos, D. DeBlassie / Stochastic Processes and their Applications 116 (2006) 36–69

43

uniformly for ðt; x; yÞ 2 ðT; 1Þ fx 2 C: jxjoRg fx 2 qC: jxjoRg where T; R40 are arbitrary. Proof. We have 1 * q r ¼ e r þ rSn1 , qr r * where e r is a unit vector in the radial direction and rSn1 is the gradient operator on S n1 . Thus for y 2 qCnf0g q 1 q mj ðZÞ ¼ mj ðZÞ. qny r qnZ Consequently we need only verify the uniform convergence of 1 rr X q I aj mj ðZÞ mj ðyÞ t qn Z j¼1

(2.3)

for ðt; x; yÞ 2 B ¼ ðT; 1Þ fx 2 C: jxjoRg fx 2 qC: jxjoRg. Since kmj k2 ¼ 1, by [11, Theorem 8, p. 102], for some positive cn and bn ¼ bðnÞ depending only on n, bðnÞ=4

sup jmj jpcn lj

.

(2.4)

D

Since ðDSn1 lj Þmj ¼ 0 on D and mj 2 CðDÞ, by the elliptic regularity theorem, mj 2 C 2;a ðDÞ. Hence by the C 2;a nature of qD and the global Schauder estimates [18, Theorem 6.6, p. 98], for some constant K independent of j, sup jrSn1 mj jpK sup jmj j D

D

bðnÞ=4 pKlj .

ð2:5Þ

Here and in what follows, K will be a number whose value might change from line to line, but is independent of j. Hence for y 2 D and Z 2 qD (using (2.2))

mj ðyÞ q mj ðZÞ pKabðnÞ . (2.6) j

qnZ By formula (2.4) [6, p. 303], z n en ez n I n ðzÞpK 2 n þ 12

(2.7)

where K is independent of n and z. Then to show uniform convergence of (2.3) on B, 2 it sufﬁces to show for M ¼ RT , 1 ðMeÞaj abðnÞ X j o1. 1 aj a þ j j¼1 2

(2.8)

By the Weyl asymptotic formula [11, p. 172], there are constants K 1 and K 2 such that K 1 jpan1 pK 2 j; j

jX1.

(2.9)

ARTICLE IN PRESS R. Ban˜uelos, D. DeBlassie / Stochastic Processes and their Applications 116 (2006) 36–69

44

Then for j large, 1 aj aj þ Xðaj Þaj 2 1

1

X expðK 1 j n1 lnðK 1 j n1 ÞÞ. Hence for c ¼ K 1 if Meo1 and c ¼ K 2 if MeX1, the sum in (2.8) is bounded by 1 X

1

1

1

j bðnÞ=ðn1Þ expðcj n1 lnðMeÞ K 1 j n1 lnðK 1 j n1 ÞÞ

j¼1

pK

1 X

j

bðnÞ=ðn1Þ

j¼1

o1

1 1 exp K 1 j ðn1Þ 2

by the integral test:

&

In the sequel, we will use the following bound, which is an immediate consequence of (2.4) and (2.5). Corollary 2.2. For some positive K and bðnÞ independent of j, bðnÞ=2

sup jmj j _ sup jrSn1 mj jpKaj D

.

D

, Lemma 2.3. For some K40, with g ¼ r22rr þr2 R1

r2 þr2

t1 e 2t I a ðrrtÞ dtpKa1 ga 1 2 þr2 R 2 2 a1 aþ1 q 1 1 r 2t (b) jqr I a ðrrtÞ dtjprð1g Þ r2ðgþr2 þg Þ 0 t e (a)

0

for positive r; r and a, with rar. 2

2

þr Proof. Change variables w ¼ r 2t to get Z 1 Z 1 2 þr2 rr 2w 2rrw r2 þ r2 1 r 2t w t e Ia e Ia 2 dw dt ¼ t r 2 þ r2 r þ r2 2w2 0 0 Z 1 ¼ w1 ew I a ðgwÞ dw.

ð2:10Þ

0

Notice since rar; 0ogo1. (a) Using the expansion of I n ðzÞ given before Lemma 2.1, by monotone convergence Z 1 Z 1 1 aþ2k X g 1 1 w w e I a ðgwÞ dw ¼ waþ2k1 ew dw 2 k!Gða þ 1 þ kÞ 0 0 k¼0 1 aþ2k X g Gða þ 2kÞ ¼ 2 k!Gða þ 1 þ kÞ k¼0 a 1 a 1 1 X aþ2k G 2 þ k G 2 þ k þ 2 ¼ pﬃﬃﬃ , g k!Gða þ 1 þ kÞ 2 p k¼0

ARTICLE IN PRESS R. Ban˜uelos, D. DeBlassie / Stochastic Processes and their Applications 116 (2006) 36–69

where we have used the formula [1, p. 25, 6.1.18] 1 2z1 1 2 GðzÞG z þ Gð2zÞ ¼ pﬃﬃﬃﬃﬃﬃ 2 2 2p

45

(2.11)

for z ¼ a2 þ k. We also make use of the following formulas from [1, formulas 15.1.1 and 15.1.3, p. 556] for the hypergeometric function F: 1 GðcÞ X Gða þ nÞGðb þ nÞ zn , GðaÞGðbÞ n¼0 Gðc þ nÞ n! 1 F a; þ a; 1 þ 2a; z ¼ 22a ½1 þ ð1 zÞ1=2 2a . 2

F ða; b; c; zÞ ¼

Using the ﬁrst one, then the second, yields Z 1 ga G a2 G aþ1 a a 1 1 w 2 2 ; þ ; a þ 1; g F w e I a ðgwÞ dw ¼ pﬃﬃﬃ 2 2 2 2 p Gða þ 1Þ 0 a aþ1 ga G G 2 2a ½1 þ ð1 g2 Þ1=2 a ¼ pﬃﬃﬃ 2 2 p Gða þ 1Þ pﬃﬃﬃﬃﬃﬃ ða1Þ 2 GðaÞ ga 2p 2 2a ½1 þ ð1 g2 Þ1=2 a ¼ pﬃﬃﬃ Gða þ 1Þ 2 p (by (2.11)) ¼ a1 ga ½1 þ ð1 g2 Þ1=2 a .

(2.12)

Then Z

1

w1 ew I a ðgwÞ dwpa1 ga .

0

Thus part (a) is proved. For part (b), we need to differentiate (2.10) under the integral. By looking at difference quotients and using the mean value theorem, differentiation with respect to g under the integral in the right hand side of (2.10) will be allowed if we can show that for ½a; b ð0; 1Þ, Z 1 sup jI 0a ðgwÞjew dwo1. (2.13) 0

g2½a;b

By [19, formula 8.486.2, p. 970] I 0a ðzÞ ¼ 12½I a1 ðzÞ þ I aþ1 ðzÞ. Hence by (2.7), jI 0a ðzÞjpCðaÞez ½za1 þ zaþ1 .

ARTICLE IN PRESS 46

R. Ban˜uelos, D. DeBlassie / Stochastic Processes and their Applications 116 (2006) 36–69

In particular, sup jI 0a ðgwÞjpCðaÞebw ½wa1 þ waþ1 .

g2½a;b

Then since bo1 and a40, (2.13) follows. Thus Z Z 1 d 1 1 w w e I a ðgwÞ dw ¼ ew I 0a ðgwÞ ds dg 0 0 Z 1 1 w ¼ e ½I a1 ðgwÞ þ I aþ1 ðgwÞ dw 2 0 " # 1 ga1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 2 1 g2 ½1 þ 1 g2 a1 " # 1 gaþ1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 1 g2 ½1 þ 1 g2 aþ1 1 p ð1 g2 Þ1=2 ½ga1 þ gaþ1 , 2 where we have used [19, formula 6.611.4, p. 708] for the third equality. We also see that the derivative is nonnegative. To ﬁnish, observe that

Z 1 rr

d Z 1 2 þr2

q r2 r2

1 r 2t 1 w

t e I w e I ðgwÞ dw 2r dt ¼ a a

ðr2 þ r2 Þ2

qr

dg t 0 0 1 prð1 g2 Þ1=2 ðga1 þ gaþ1 Þ 2 , r þ r2 as claimed.

&

To prove Theorem 1.4, we will need the following consequence of (2.10) and (2.12). Corollary 2.4. For g ¼ r22rr o1 þr2 Z 1 r2 þr2 rr t1 e 2t I a dt ¼ a1 ga ½1 þ ð1 g2 Þ1=2 a . t 0 Proof of Theorem 1.4. By Theorem 1.3, (1.7) and Lemma 2.1, for x ¼ ry, Z 1Z d 1 q Px ðjBt jprÞ ¼ pC ðt; x; rZÞrn2 sin jðZÞmðdZÞ dt, dr qD 2 qny 0 Z 1Z 1 rr r2 þr2 X 1 n n ¼ r22 r12 t1 e 2t I aj mj ðyÞ 2 t 0 qD j¼1 sin jðZÞ

q mj ðZÞmðdZÞ dt. qnZ

ARTICLE IN PRESS R. Ban˜uelos, D. DeBlassie / Stochastic Processes and their Applications 116 (2006) 36–69

47

Now for g ¼ r22rr p1 e we have by Corollaries 2.2 and 2.4, þr2

1 Z 1Z rr

X 2 þr2

q 1 r 2t

t e I aj mj ðZÞ

mðdZÞ dt mj ðyÞ

t qnZ qD j¼1 0 pK

1 X

aj bðnÞ a1 j g aj o1,

j¼1

by (2.9) and the integral test. Hence we can exchange summation and integration above to get, uniformly for gp1 e, 1 Z 1 rr r2 þr2 d 1 n2 1n X 2 2 Px ðjBt jprÞ ¼ r r t1 e 2t I aj dt dr 2 t j¼1 0 Z q sin jðZÞ mj ðZÞmðdZÞ mj ðyÞ qnZ qD 1 X 1 n n aj 2 aj ¼ r22 r12 a1 j g ½1 þ ð1 g Þ 2 j¼1 Z q sin jðZÞ mj ðZÞmðdZÞ mj ðyÞ, qnZ qD as claimed.

&

Proof of Corollary 1.5. If r is large, then g ¼ r22rr is small and so by Theorem 1.4, as þr2 r!1 Z d 1 n q n a1 Px ðjBt jprÞ r22 r12 a1 g sin jðZÞ m ðZÞmðdZÞ m1 ðyÞ, 1 1 dr 2 qnZ qD where we have used the fact that since DSn1 m1 ¼ lm1

and

m1 X0,

the Hopf maximum principle [26, Theorem 7, p. 65], implies qnqZ m1 ðZÞ40 on qD. Since ga1 ð2rÞa1 ra1 as r ! 1, we get the desired asymptotic upon integrating and appealing to (1.3). &

3. Proof of Theorem 1.1 Let t be the ﬁrst exit times of X from C and for u; v40, deﬁne Zðu; vÞ ¼ infft40: Y t eðu; vÞg. For typographical simplicity we write t for tC . Then for any A qC, Pz ðZ t 2 AÞ ¼ Pz ðZt 2 A; exit occurs along X pathÞ þ Pz ðZ t 2 A; exit occurs along X pathÞ

ARTICLE IN PRESS 48

R. Ban˜uelos, D. DeBlassie / Stochastic Processes and their Applications 116 (2006) 36–69

¼ Pz ðZt 2 A; Y ðZðt ; tþ ÞÞ ¼ t Þ þ Pz ðZ t 2 A; Y ðZðt ; tþ ÞÞ ¼ tþ Þ ¼ Pz ðX ðt Þ 2 A; Y ðZðt ; tþ ÞÞ ¼ t Þ þ Pz ðX þ ðtþ Þ 2 A; Y ðZðt ; tþ ÞÞ ¼ tþ Þ ¼ 2Pz ðX ðt Þ 2 A; Y ðZðt ; tþ ÞÞ ¼ t Þ by independence and symmetry. Writing f z ðvÞ ¼

d Pz ðt pvÞ dv

for the density of t , by independence of X þ and X , Pz ðZt 2 AÞ Z 1 ¼2 Pz ðX ðt Þ 2 A; Y ðZðt ; vÞ ¼ t Þf z ðvÞ dv Z0 1 Z ¼2 PðY ðZðu; vÞÞ ¼ uÞPz ððX ðt Þ; t Þ 2 dy duÞf z ðvÞ dv 0 Að0;1Þ Z 1Z v Pz ððX ðt Þ; t Þ 2 dy duÞf z ðvÞ dv. ð3:1Þ ¼2 u þ v Að0;1Þ 0 Hence by Theorem 1.3, Lemma 2.1 and (1.7), for y ¼ rZ, z ¼ ry, Z 1Z 1Z d v q Pz ðjZt jprÞ ¼ pC ðu; z; yÞrn2 sin jðZÞmðdZÞ du f z ðvÞ dv dr u þ v qn y qD 0 0 Z 1Z 1Z 1 rr r2 þr2 X v n n e 2u ¼ r22 r12 I aj mj ðyÞ u qD uðu þ vÞ 0 0 j¼1 q mj ðZÞ sin jðZÞmðdZÞ du f z ðvÞ dv. qnZ There is no danger of circular reasoning in using Theorem 1.3 since its proof is selfcontained. Using Corollary 2.2, if we can show for rar, Z 1Z 1Z 1 rr X 2 þr2 v r 2u e abðnÞ I (3.2) mðdZÞ du f z ðvÞ dvo1, a j j u qD uðu þ vÞ 0 0 j¼1 then by monotone convergence and dominated convergence, exchange of summation with integration is allowed and for rar, Z 1 X d q n 2 1n2 2 Pz ðjZ t jprÞ ¼ r r mj ðyÞ sin jðZÞ mj ðZÞmðdZÞ dr qnZ qD j¼1 Z 1Z 1 rr r2 þr2 v e 2u I aj ð3:3Þ f ðvÞ du dv. uðu þ vÞ u z 0 0

ARTICLE IN PRESS R. Ban˜uelos, D. DeBlassie / Stochastic Processes and their Applications 116 (2006) 36–69

49

The work to justify (3.2) has been done in Section 2: The jth term is bounded above by Z 1 1 r2 þr2 rr e 2u I aj du. abðnÞ mðqDÞ j u u 0 Then (3.2) follows from Lemma 2.3a, since rar. d As it stands, the behavior of dr Pz ðjZt jprÞ for large r is not obvious from (3.3). It will turn out that the j ¼ 1 term dominates. In what follows, we write n 1 , (3.4) pj ¼ aj 2 where aj is as in Theorem 1.4. From (1.2) we have Pz ðt 4vÞCðzÞvp1 =2

as v ! 1.

(3.5)

The following lemma will be used to derive asymptotics of the ﬁrst term in (3.3) as well as upper boun ds on the remaining terms. Lemma 3.1. Let aXa1 and set Z 1Z 1 rr r2 þr2 v e 2u I a I¼ f ðvÞ dv du. uðu þ vÞ u z 0 0 (a) For some positive M and 8 > Ka1=2 ð2rÞa r2a ; > > > > < a 2a ln r; Ip Kað2rÞ r > > > > p1 =2 a > : Ka ð2rÞ rap1 ;

K, independent of a, p1 41; 2 p if 1 ¼ 1; 2 p1 o1; if 2 if

for rXM and a41. (b) For p21 41, lim r2þa I ¼ 2ra E x ðt Þ,

r!1

where t is as defined at the beginning of this section. For

p1 2

¼ 1,

lim r2þa ðln rÞ1 I ¼ 4ra CðzÞ,

r!1

and for

p1 2

o1,

lim raþp1 I ¼ CðzÞ2p1 =2 ra

r!1

Proof. Since f z ðvÞ ¼

d Pz ðt 4vÞ, dv

Z 1 G a þ p21 wp1 =2 ð1 þ wÞ2 dw. Gða þ 1Þ 0

ARTICLE IN PRESS R. Ban˜uelos, D. DeBlassie / Stochastic Processes and their Applications 116 (2006) 36–69

50

after an integration by parts, Z 1Z 1 rr r2 þr2 1 I¼ P ðt 4vÞe 2u I a dv du 2 z u ðu þ vÞ 0 0 2

2

þr ) (now change of variables s ¼ r 2u 2 Z 1Z 1 2 r þ r2 2rrs r2 þ r2 þ v Pz ðt 4vÞes I a 2 ds dv ¼ r þ r2 2s 2s2 0 0 Z 1Z 1 2 2sv 2 2rrs s ¼ 2 1þ 2 Pz ðt 4vÞe I a 2 ds dv r þ r2 0 r þ r2 r þ r2 0

Z

1

Z

1

H ds dv

¼ 0

Case 1:

ð3:6Þ

say.

0 p1 2

41. By (3.5), E z ðt Þo1. By (2.7),

a a 2rrs 1 rrs e er2 þr2 s HpK 2 1 Pz ðt 4vÞe . 2 2 2 r þr r þr ða þ 12Þa For ﬁxed r40, choose M 1 independent of a so large that 2rr 1 p ; 2 2 þr

rXM 1 .

r2

(3.7)

Then 2

HpKr Pz ðt 4vÞe

s=2

rrs 2 r þ r2

a

ea ða þ 12Þa ;

rXM 1 .

(3.8)

Here and in what follows, K will be a number whose exact value might change from line to line, but will always be independent of a and r. First observe that by (3.8) a Z 1 Z 1 Z 1Z 1 Kr2 ea rr a 2 H ds dvp Pz ðt 4vÞes=2 sa ds dv r þ r2 a þ 12 0 0 0 0 a Kr2 ea rr a 2 E z ðt Þ2a Gða þ 1Þ. ¼ r þ r2 a þ 12 Now by Stirling’s formula Gða þ 1Þea ea aGðaÞ a ¼ a a þ 12 a þ 12 1

ea aaa2 ea a pK a þ 12 pKa1=2 .

ARTICLE IN PRESS R. Ban˜uelos, D. DeBlassie / Stochastic Processes and their Applications 116 (2006) 36–69

Hence for rXM 1 , Z 1Z 1 2rr a 1=2 H ds dvpKr2 2 a r þ r2 0 0 a 2r pKr2 a1=2 . r

51

ð3:9Þ

Referring back to (3.6), we see that this gives the desired bound in part (a) of the lemma. As for the asymptotic in part (b), notice that Z 1Z 1 Pz ðt 4vÞsa es=2 ds dvo1 0

0

and 2r2 2sv 2 2rrs s a 1 þ P ðt 4vÞe r I z a r!1 r2 þ r2 r2 þ r2 r2 þ r2 a ðrsÞ ¼ 2Pz ðt 4vÞes , Gða þ 1Þ

lim r2þa H ¼ lim

r!1

using the asymptotic z n I n ðzÞ =Gðn þ 1Þ as z ! 0. 2 Hence by (3.8) and the dominated convergence theorem in (3.6), Z 1Z 1 lim r2þa H ds dv lim r2þa I ¼ r!1 r!1 Z0 1 Z0 1 ðrsÞa ds dv 2Pz ðt 4vÞes ¼ Gða þ 1Þ 0 0 ¼ 2ra E z ðt Þ,

(3.10)

as claimed. Case 2: p21 p1. This part is more delicate because this time E z ðt Þ ¼ 1. Let e 2 ð0; 12Þ and use the asymptotics (3.5) and (3.10) to choose M 1 and M 2 such that ð1 eÞCðzÞvp1 =2 pPz ðt 4vÞpð1 þ eÞCðzÞvp1 =2 ;

vXM 1

(3.11)

and a a ð1 eÞ 2z ð1 þ eÞ 2z pI a ðzÞp ; Gða þ 1Þ Gða þ 1Þ

zp2rM 2 .

(3.12)

Notice M 1 is independent of a and M 2 is not. We break up the integral I in (3.6) into three pieces: Z M 1 Z 1 Z 1 Z 1 Z 1 Z M 2 r þ þ I¼ H ds dv 0

0

¼ J1 þ J2 þ J3

M1

say.

M2r

M1

0

ð3:13Þ

ARTICLE IN PRESS 52

R. Ban˜uelos, D. DeBlassie / Stochastic Processes and their Applications 116 (2006) 36–69

It turns out J 3 will dominate as r ! 1. We have [19, 6.611.4, p. 708] Z Z 2 M 1 1 2 s 2rrs 1 1 e Ia 2 J 1p 2 ds dv r 0 r þ r2 0 Z 1 2M 1 2rr es I a 2 s ds ¼ 2 r r þ r2 0 a 2rr 2 2 2M 1 r þr # ¼ 2 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ " rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 a r 2rr 2rr 1 r2 þr2 1 þ 1 r2 þr2

pKr2a ð2rÞa

rXM 3

for

large,

(3.14)

where M 3 is independent of a. As for J 2 , by (2.7) a a 2rrs 2sv 2 s rrs e er2 þr2 a ds dv e K 2 2 r 2 þ r2 a þ 12 M1 M2r r þ r a 2 Z 1 2rrs K rr 2 ea ¼ 2 sa2 es er2 þr2 ds a 2 2 2 2 2 1 r þr r þr r þr aþ2 M2r a Z 1 a 2rrs rr e a2 3s=4 M 2 r=4 r2 þr2 pK 2 ðr2 þ r2 Þ s e e e ds a 2 r þr a þ 12 M2r Z 1 ea a 2a M 2 r=4 a e pKr r sa2 es=2 ds, a þ 12 M2r

2 J 2p 2 r þ r2

Z

1

Z

1

for r large, say rXM 4 , where M 4 is independent of a. Thus ea a eM 2 r=4 2a Gða 1Þ a þ 12 3 ea a eaþ1 ða 1Þa2 pKð2rÞa r2a eM 2 r=4 a þ 12

J 2 pKra r2a

(by Stirling’s formula) pKð2rÞa r2a eM 2 r=4 a3=2 .

(3.15)

Now we examine the dominant piece J 3 . By reversing the order of integration and then changing v into w ¼ r22sv we have þr2 Z

M2r

Z

1

H dv ds (see (3.6))

J3 ¼ 0

M1

ARTICLE IN PRESS R. Ban˜uelos, D. DeBlassie / Stochastic Processes and their Applications 116 (2006) 36–69

53

Z M2r Z 1 2 r2 þ r2 2 w ¼ 2 ð1 þ wÞ Pz t 4 2sM 1 r þ r2 0 2s r2 þr2 2rrs r2 þ r2 es I a 2 dw ds r þ r2 2s Z

M2r

Z

¼ 0

r2 þ r2 2rrs 2 s 1 w e ð1 þ wÞ P t 4 s I dw ds. z a 2sM 1 r2 þ r2 2s 2 2 1

(3.16)

r þr

Write Z

1

ð1 þ wÞ2 wp1 =2 dw

hðuÞ ¼

(3.17)

u

and observe hðuÞphð0Þo1 hðuÞ ln

1 u

for

p1 o1, 2

as u ! 0

1 Now for w4 r2sM 2 þr2 we have

for r2 þr2 2s

p1 ¼ 1. 2

ð3:18Þ

w4M 1 , hence by (3.11), (3.16) becomes

2 p1 =2 r þ r2 2rrs 1 s w ð1 þ wÞ s e Ia 2 J 3 pK dw ds 2sM 1 r þ r2 2s 0 r2 þr2 2 p1 =2 Z M 2 r r þ r2 2sM 1 p1 =21 s 2rrs h 2 e Ia 2 ¼K s ds r þ r2 2 r þ r2 0 a a 2rrs Z M2r 2sM 1 p1 =21 s rrs e er2 þr2 2 2 p1 =2 a ds (by (2.7)) s pKðr þ r Þ h 2 e r2 þ r2 r þ r2 a þ 12 0 Z Kðr2 þ r2 Þap1 =2 ra ra ea M 2 r 2sM 1 aþp1 =21 s r2rrs ¼ h e e 2 þr2 ds s a 2 þ r2 r a þ 12 0 " #a Z M2r re 2sM 1 aþp1 =21 s=2 p1 h 2 e ds ð3:19Þ pKr s r þ r2 r a þ 12 0 Z

M2r

Z

1

2

for r large, independent of a. If p21 o1 then by (3.18), this yields " #a re p1 a p1 2 G a þ J 3 pKr 2 r a þ 12 " #a 2re p p ¼ Krp1 aþ 11 G aþ 11 1 2 2 r aþ2

ARTICLE IN PRESS 54

R. Ban˜uelos, D. DeBlassie / Stochastic Processes and their Applications 116 (2006) 36–69

" pKrp1 pKa

p1 1 2

#a aþp21 12 2re p1 a e aþ 1 2 r a þ 12

(Stirling’s formula)

ð2rÞa rap1 .

ð3:20Þ

When p21 ¼ 1, observe that for spM 2 r we have independent of a. Hence by (3.18) and (3.19),

2sM 1 r2 þr2

p 2M r1 M 2 o1 for r large,

" #a Z M2r K re r2 þ r2 a s=2 ln ds J3p 2 s e r r a þ 12 2sM 1 0 " #a Z M2r K re p 2 ½K ln r ln ssa es=2 ds (r large) 1 r r aþ2 0 " #a Z 1 K re a a s=2 p 2 Kðln rÞ2 Gða þ 1Þ ðln sÞs e ds r r a þ 12 0 " #a Z 1 K re a p 2 Kðln rÞ2 Gða þ 1Þ ðln sÞ ds r r a þ 12 0 " #a K 2re Gða þ 1Þ ln r p 2 r r a þ 12 " #a K 2re aGðaÞ ln r ¼ 2 r r a þ 12 pKað2rÞa ra2 ln r

by Stirling’s formula.

ð3:21Þ

Combining (3.14), (3.15), (3.20) and (3.21), we get the upper bound of part (a) in the lemma. For part (b), ﬁrst assume p21 o1. For spM 2 r we have r2rrs 2 þr2 p2rM 2 and for 2

2

r þr 1 w4 r2sM 2 þr2 we have 2s w4M 1 . Hence by (3.11) and (3.12) applied to the t and I a factors in (3.16), we get the that integrand in (3.16) is bounded above by

C a ð1 þ wÞ2

2 p1 =2 a r þ r2 rrs w s1 es 2 r þ r2 2s

pC a rp1 a wp1 =2 ð1 þ wÞ2 saþp1 =21 es . Moreover, after writing (3.16) as Z

1

Z

1

F dw ds,

J3 ¼ 0

0

ARTICLE IN PRESS R. Ban˜uelos, D. DeBlassie / Stochastic Processes and their Applications 116 (2006) 36–69

55

we see from the asymptotics (3.5) and (3.10) that a rrs 2 p1 =2 2 2 þr2 r þ r r w es s1 lim rp1 þa F ¼ lim rp1 þa ð1 þ wÞ2 CðzÞ r!1 r!1 2s Gða þ 1Þ 1 . ¼ ð1 þ wÞ2 wp1 =2 CðzÞ2p1 =2 sp1 =2þa1 es ra Gða þ 1Þ Since wp1 =2 ð1 þ wÞ2 saþp1 =21 es is integrable on s; w40, we can apply the dominated convergence theorem to get Z 1Z 1 p1 þa lim r J3 ¼ lim rp1 þa F dw ds r!1 r!1 0 0 Z 1 CðzÞ2p1 =2 ra G a þ p21 wp1 =2 ð1 þ wÞ2 dw. ¼ Gða þ 1Þ 0 Combining this with (3.14) and (3.15) and using that lim rp1 þa I ¼ lim rp1 þa J 3 , r!1

p1 2

o1, we get

r!1

which is the claimed value in part (b). Finally, assume p21 ¼ 1. Consider the integral 2 1 a Z M2r Z 1 2 rrs 2 r þ r s 1 w J4 ¼ ð1 þ wÞ e s dw ds 2sM 1 r 2 þ r2 2s 0 2 2 r þr

2

2

þr which is just J 3 in (3.16) with the factors involving t and I a replaced by ðr 2s wÞ1 rrs a and ðr2 þr2 Þ , respectively. These are more or less the asymptotics from (3.5) and (3.10). Then Z M2r 2sM 1 J4 ¼ h 2 2ðr2 þ r2 Þa1 ðrrÞa sa es ds. r þ r2 0 2M 1 M 2 1 o1 for large r, hence by (3.18), for such r the integrand of For spM 2 r, r2sM 2 þr2 p r

ðr2þa Þ=ðln rÞJ 4 is bounded above by r2þa r2 þ r2 a2 a s 1 ½K ln r ln ssa es ln s e pC a Ca r ln r ln r 2sM 1 1 ½K ln r þ ð ln sÞ _ 0sa es pC a ln r pC a ½K þ ð ln sÞ _ 0sa es , which is integrable on s40. Moreover, the limit of the integrand of r2þa ðln rÞ1 J 4 is 2sM 1 1 2þa lim r ðln rÞ h 2 2ðr2 þ r2 Þa1 ðrrÞa sa es r!1 r þ r2 r2 þ r2 ¼ lim ðln rÞ1 ln 2ra sa es r!1 2sM 1 ¼ 4ra sa es .

ARTICLE IN PRESS 56

R. Ban˜uelos, D. DeBlassie / Stochastic Processes and their Applications 116 (2006) 36–69

Hence by dominated convergence again, lim r2þa ðln rÞ1 J 4 ¼ 4ra Gða þ 1Þ.

r!1

By (3.11)–(3.12) ð1 eÞ2 CðzÞ ð1 þ eÞ2 CðzÞ J 4 pJ 3 p J4. Gða þ 1Þ Gða þ 1Þ Multiply by r2þa ðln rÞ1 , let r ! 1 then let e ! 0 to end up with lim r2þa ðln rÞ1 J 3 ¼ 4ra CðzÞ.

r!1

By (3.14) and (3.15), we get lim r2þa ðln rÞ1 I ¼ 4ra CðzÞ,

r!1

as desired. Now we can prove Theorem 1.1. Write the sum in (3.3) as show that 1 X

bj ðrÞb1 ðrÞ as

r ! 1,

P1

j¼1 bj ðrÞ.

If we can

(3.22)

j¼1

then by Lemma 3.1 b, the conclusion of Theorem 1.1 will hold. To this end, write " # 1 1 X X bj ðrÞ bj ðrÞ ¼ b1 ðrÞ 1 þ . b ðrÞ j¼1 j¼2 1 It sufﬁces to show 1 X bj ðrÞ ! 0, b ðrÞ j¼2 1

as r ! 1. There is no danger in dividing by b1 ðrÞ because as in the proof of Corollary 1.5, the factor Z q sin jðZÞ m1 ðZÞmðdZÞ m1 ðyÞ qn Z qD in b1 is positive. By Lemma 3.1(a) and (2.6), for some constants K and M independent of j, for rXM 8 p1 bðnÞþ1=2 > 41; Kaj ð2rÞaj r2aj ; > > > 2 > < p1 n n bðnÞþ1 ¼ 1; ð2rÞaj r2aj ln r; jbj ðrÞjpr22 r12 Kaj 2 > > > p bðnÞþp1 =2 > 1 > o1: ð2rÞaj raj p1 ; : Kaj 2

ARTICLE IN PRESS R. Ban˜uelos, D. DeBlassie / Stochastic Processes and their Applications 116 (2006) 36–69

57

Then by Lemma 3.1 b, jbj ðrÞj pKabðnÞþ1 ð2rÞaj ra1 aj . j b1 ðrÞ By the integral test and (2.9), for any 0oeo1, the series aj 1 X 2r ajbðnÞþ1 r j¼2 2r converges uniformly on 1e or. Thus, since aj 4a1 for jX2,

aj a1

X 1 1 X bj ðrÞ

bðnÞþ1 2r a1 lim

aj ¼ 0,

p lim Kð2rÞ r!1

r b ðrÞ r!1 j¼2 1 j¼2

and (3.22) follows as desired. &

4. Proof of Theorem 1.3 The unbounded, nonsmooth nature of C leads to technicalities not encountered in the bounded C 3 case considered by Hsu [21]. We now state the following result used to prove Theorem 1.3. Before giving its proof, we show how it yields Theorem 1.3. We follow Hsu’s idea of ﬁnding the Laplace transform in t of the density. Here and in what follows we will write Be ðyÞ ¼ fz 2 Rn : jz yjoeg. Theorem 4.1. (a) Let x 2 C, y 2 qCnf0g with jxjajyj. Then for l40, Z 1 Z 1 q q elt pC ðt; x; yÞ dt ¼ elt pC ðt; x; yÞ dt. qny 0 qn y 0 (b) For x 2 C and y 2 qCnf0g,

q qny

pC ðt; x; yÞ40.

(c) Let x 2 C and l40. If f 2 C 2;a ðCÞ is nonnegative with compact support in Cn½f0g [ qBjxj ð0Þ then Z 1 q l lt f ðyÞ G ðx; yÞsðdyÞ, E x ½e f ðBt Þ ¼ 2 qC qny C where G lC is Green’s function for 12 D l on C with Dirichlet boundary conditions. Remarks. (1) Since we use the series expansion of the heat kernel to prove part a), there will be an exchange of summation and integration. This requires the condition jxjajyj as well as the strange hypothesis about the support of f. (2) As we point out below in (4.4), the function uðzÞ ¼ E z ½elt f ðBt Þ

ARTICLE IN PRESS 58

R. Ban˜uelos, D. DeBlassie / Stochastic Processes and their Applications 116 (2006) 36–69

solves the boundary value problem 1 2D l u ¼ 0 in C, ujqC ¼ f . Since a series expansion is known for pC ðt; x; yÞ, the most natural way to try to prove part (c) of Theorem 4.1 is to show directly that Z 1 q l f ðyÞ G ðx; yÞsðdyÞ 2 qC qny C solves the said boundary value problem. It is easy to show the PDE is satisﬁed, but direct veriﬁcation of the boundary condition eludes us. Hence we are forced to take the approach we present below. Proof of Theorem 1.3. Let x 2 C and consider any nonnegative f 2 C 2;a ðCÞ with compact support in Cnðf0g [ qBjxj ð0ÞÞ. Then since Z 1 G lC ðx; yÞ ¼ elt pC ðt; x; yÞ dt, (4.1) 0

we have Z 1Z 0

elt f ðyÞPx ðBt 2 dy; t 2 dtÞ ¼ E x ½elt f ðBt Þ Z 1 Z 1 q lt ¼ f ðyÞ e pC ðt; x; yÞ dt sðdyÞ (Theorem 4.1 c) 2 qC qny 0 Z Z 1 1 q f ðyÞ elt p ðt; x; yÞ dt sðdyÞ (Theorem 4.1 a) ¼ 2 qC qny C 0 Z 1Z 1 q elt f ðyÞ p ðt; x; yÞsðdyÞ dt, ¼ 2 qny C 0 qC qC

by Theorem 4.1 b and Fubini’s Theorem. Inverting the Laplace transform, we get for any 0paob, Z bZ 1 q E x ½f ðBt ÞI ½a;b ðtÞ ¼ f ðyÞ p ðt; x; yÞsðdyÞ dt. 2 qny C a qC Varying f appropriately, this yields Z bZ 1 q pC ðt; x; yÞsðdyÞ dt, Px ðBt 2 A; aptpbÞ ¼ A 2 qny a

(4.2)

where A is any open subset of qCnðf0g [ qBjxj ð0ÞÞ. Since qC \ qBjxj ð0Þ is polar with s measure 0, and since qnqy pC ðt; x; yÞ is continuous as a function of y 2 qCnf0g (by Lemma 2.1), we see (4.2) hol ds for arbitrary Borel A qCnf0g. This yields Theorem 1.3. & Proof of Theorem 4.1(a). Let x 2 C, y 2 qCnf0g with jxjajyj. Write x ¼ ry, y ¼ rZ and g ¼ r22rr . Note since rar, go1. Then by Corollary 2.2 and Lemma 2.3(a) the þr2

ARTICLE IN PRESS R. Ban˜uelos, D. DeBlassie / Stochastic Processes and their Applications 116 (2006) 36–69

59

following interchanges of integration, differentiation and summation are justiﬁed: Z 1 q elt pC ðt; x; yÞ dt qny 0 Z 1 1 q elt pC ðt; x; yÞ dt ¼ r qnZ 0 Z 1 1 rr r2 þr2 X n 1 q ¼ elt t1 ðrrÞ12 e 2t I aj mj ðyÞmj ðZÞ dt r qnZ 0 t j¼1 1 Z 1 rr r2 þr2 q X n n ¼ r12 r2 elt t1 e 2t I aj dt mj ðyÞmj ðZÞ qnZ j¼1 0 t (also using monotone and dominated convergence) 1 Z 1 rr X r2 þr2 q n n ¼ r12 r2 elt t1 e 2t I aj mj ðZÞ dt mj ðyÞ t qn Z j¼1 0 Z 1 1 rr r2 þr2 X q 1n2 n2 ¼r r elt t1 e 2t I aj mj ðZÞ dt mj ðyÞ t qn Z 0 j¼1 Z 1 q elt pC ðt; x; yÞ dt, ¼ qn y 0 by Lemma 2.1.

&

Proof of Theorem 4.1(b). This is an immediate consequence of the Hopf maximum principle for parabolic operators [26, Theorem 6, p. 174]. Proof of Theorem 4.1(c). Part (c) is the hard part. Fix x 2 C and write uðzÞ ¼ E z ½elt f ðBt Þ;

z 2 C,

(4.3)

2;a

where f 2 C ðCÞ is nonnegative with compact support in Cnðf0g [ qBjxj ð0ÞÞ. Choose x0 2 C such that jx0 jojxj and

x0 esupp f .

From now on, x; f and x0 are ﬁxed. Since qC is Lipschitz, Proposition 8.1.9 and Theorem 8.1.10 in [25, pp. 345–346] imply u 2 C 2;a ðCÞ, 1 2D l u ¼ 0

in C,

ujqC ¼ f ,

ð4:4Þ

and in fact Z uðzÞ ¼

f ðyÞkðx; yÞnx0 ðdyÞ, qC

(4.5)

ARTICLE IN PRESS R. Ban˜uelos, D. DeBlassie / Stochastic Processes and their Applications 116 (2006) 36–69

60

where kðx; yÞ is the Martin kernel with pole at y 2 qC normalized by kðx0 ; yÞ ¼ 1 and for any Borel set A qC, nx0 ðAÞ ¼ E x0 ½elt IðBt 2 AÞ. Note since f 2 C 2;a ðCÞ, by the elliptic regularity theorem, u 2 C 2;a ðCnf0gÞ.

(4.6)

Furthermore, since qC is Lipschitz, by Theorem 8.1.4 in [25, p. 337], any sequence yn 2 C with yn ! y 2 qC is a Martin sequence. In particular, if yn ! y along the unit inward normal to qC at y 2 qCnf0g, kðx; yÞ ¼ lim

n!1

¼

G lC ðx; yn Þ GlC ðx0 ; yn Þ

l q qny G C ðx; yÞ . l q qny G C ðx0 ; yÞ

Hence by (4.5) Z q l uðzÞ ¼ f ðyÞ G C ðz; yÞn ðdyÞ, qn y qC

(4.7)

where

q nðdyÞ ¼ G l ðx0 ; yÞ qny C

1 nx0 ðdyÞ.

This representation will allow us to estimate u and its derivatives. It is known that for z 2 C, 1 l 2D l G C ðz; Þ ¼ 0 on Cnfzg

(4.8)

and G lC ðz; Þ is continuous on Cnfzg with boundary value 0. Hence by the elliptic regularity theorem, G lC ðz; Þ 2 C 2;a ðCnf0; zgÞ.

(4.9)

To prove part (c) of Theorem 4.1, choose M so large and e40 so small that eojx0 jojxjoM Be ð0Þ \ supp f ¼ ; c

BM ð0Þ \ supp f ¼ ;. Then choose d40 so small that Bd ðxÞ C. Set c

E ¼ C \ BM ð0Þ \ Bd ðxÞ nBe ð0Þ (see supplementary material).

ARTICLE IN PRESS R. Ban˜uelos, D. DeBlassie / Stochastic Processes and their Applications 116 (2006) 36–69

61

By (4.6) and (4.9) we can apply Green’s second identity: Z ½uðwÞDw GlC ðx; wÞ G lC ðx; wÞDuðwÞ dw E Z q l q l uðyÞ G ðx; yÞ G C ðx; yÞ uðyÞ sðdyÞ, ¼ qny C qny qE where qnqy is the inward normal derivative to qE and sðdyÞ is surface R measure on qE. By (4.4) and (4.8), the left-hand side is 0; then breaking up qE into pieces and solving for the part over qC, Z Z q l q u G C ðx; Þ GlC ðx; Þ u sðdyÞ ¼ qny qny qC qC\BM ð0ÞnBd ð0Þ Z Z Z ¼ þ þ C\qBM ð0Þ

C\qBe ð0Þ

qBd ðxÞ

q qny

(we use the convention that is the unit inward normal to qBM ð0Þ, qBe ð0Þ, qBd ðxÞ, respectively). Below in Theorem 4.7 we will show the ﬁrst two integrals converge to 0 as M ! 1 and e ! 0. In Theorem 4.9 we will show the last integral converges to 2uðxÞ as d ! 0. Thus we will end up with Z 1 q l q uðxÞ ¼ uðyÞ G C ðx; yÞ G lC ðx; yÞ uðyÞ sðdyÞ 2 qC qny qny Z 1 q ¼ f ðyÞ G lC ðx; yÞsðdyÞ, 2 qn y qC using that u ¼ f on qC and G lC ðx; Þ ¼ 0 on qC. Thus gives part (c) of Theorem 4.1. & The representation (4.7) of u will allow us to estimate u and its derivatives. For this we need the next result as well as estimates on G lC and its derivatives. Lemma 4.2. The set qC \ supp f has finite n measure. Proof. By (4.9), G lC ðx0 ; Þ 2 C 2;a ðCnf0; x0 gÞ and by the Hopf maximum principle [26, Theorem 7, p. 65] q Gl ðx0 ; yÞ40; qny C

y 2 qCnf0g.

Since qC \ supp f is compact in qCnf0g, the desired conclusion follows.

&

Lemma 4.3. Suppose r40, z 2 C \ qBr ð0Þ and y 2 qCnf0g with r ¼ jyjar. If g ¼ r22rr is positive and sufficiently small, then þr2

q l

n2 1n2 a1

(4.10)

qn G C ðz; yÞ pKr r g , y

q q l

pKrn2 rn2 ga1 þ rr1n2 ðga1 1 þ ga1 þ1 Þðr2 þ r2 Þ1 ,

G ðz; yÞ

qn qn C z y

(4.11)

ARTICLE IN PRESS 62

R. Ban˜uelos, D. DeBlassie / Stochastic Processes and their Applications 116 (2006) 36–69

q l

1n2 n2 a1 1n2 a1 1

þ ga1 þ1 Þðr2 þ r2 Þ1 ,

qn GC ðz; yÞ pKr ½r g þ rr ðg

(4.12)

z

n

G lC ðz; yÞpKðrrÞ12 ga1 ,

(4.13)

where K40 is independent of z and y. Remark. (1) The proof of the bound in (4.13) really only requires z; y 2 Cnf0g, jzjajyj and 2jzjjyj small. jzj2 þjyj2 (2) A similar remark holds for (4.12). In particular, by the symmetry of G lC , we can replace qnq z G lC ðz; yÞ there by qnq z G lC ðy; zÞ and require only that z; y 2 Cnf0g with jzjajyj and

2jzjjyj jzj2 þjyj2

small.

Proof. Write z ¼ ry and y ¼ rZ in polar coordinates. Then rar and by Theorem 4.1(a), for g ¼ r22rr , þr2 Z 1 q l q G C ðz; yÞ ¼ elt p ðt; z; yÞ dt qny qny C 0 Z 1 1 rr r2 þr2 X n n pKr2 r12 t1 e 2t I aj dt abðnÞ j t 0 j¼1 (by Lemma 2.1 and Corollary 2.2) 1 Z 1 rr X r2 þr2 n n t1 e 2t I aj pKr2 r12 ajbðnÞ dt t j¼1 0 (Fatou’s Lemma) n

n

pKr2 r12

1 X

abðnÞ1 ga j j

j¼1

(by Lemma 2.3(a)) n

n

pKr2 r12 ga1

ðg smallÞ,

which is (4.10). q , hence Now qnq z ¼ qr

Z q q q 1 lt n 1n 1 r2 þr2 G lC ðz; yÞ ¼ e r 2 r 2 t e 2t qnz qny qr 0 1 rr X q I aj mj ðZÞ dt mj ðyÞ t qn Z j¼1

ARTICLE IN PRESS R. Ban˜uelos, D. DeBlassie / Stochastic Processes and their Applications 116 (2006) 36–69

(by Theorem 4.1 a) and Lemma 2.1) 1 Z 1 rr X r2 þr2 q n q n ¼ r2 r12 t1 elt e 2t I aj mj ðZÞ dt mj ðyÞ qr t qn Z j¼1 0

63

(4.14)

(by Lemma 2.3 and Corollary 2.2). If for g~ ¼ u22ur we can show there is K þr2 independent of a; u and r such that

Z

q 1n 1 1 lt u2 þr2 ur

u 2

2t I t e e dt a

qu

t 0 n

1

n

pK½u2 a1 g~ a þ ru12 ð1 g2 Þ2 ð~ga1 þ g~ aþ1 Þðu2 þ r2 Þ1

ð4:15Þ

for 0o~gpg, then by Corollary 2.2 we can differentiate under the summation in (4.14). Moreover, taking u ¼ r and using Corollary 2.2, for small g we also get the estimate

q q l

qn qn GC ðz; yÞ

z y " # 1 1 X X n n n bðnÞ1 bðnÞ pKr2 r2 aj gaj þ rr12 aj ðgaj 1 þ gaj þ1 Þðr2 þ r2 Þ1 j¼1 n2

j¼1

n2 a1

pKr ½r g þ rr

1n2

ðg

a1 1

þg

a1 þ1

Þðr2 þ r2 Þ1 ,

giving (4.11). To prove (4.15), note by Lemma 2.3

Z

q 1n 1 1 lt u2 þr2 ur

u 2 2t t e e Ia dt

qu t 0 n n n p 1 u2 Ka1 g~ a þ u12 rð1 g~ 2 Þ1=2 ð~ga1 þ g~ aþ1 Þðu2 þ r2 Þ1 2 n n pK½u2 a1 g~ a þ ru12 ð1 g2 Þ1=2 ð~ga1 þ g~ aþ1 Þðu2 þ r2 Þ1 , as desired. To prove (4.12), we repeat the proof of (4.11) almost word for word. The only n n change is in (4.14) where the initial r2 is replaced by r12 and the qnqZ mj ðZÞ is replaced by mj ðZÞ. For the proof of (4.13), note by (4.1), (2.1) and Corollary 2.2, 1 Z 1 rr X r2 þr2 1n2 l G C ðz; yÞpðrrÞ t1 e 2t I aj dt ajbðnÞ t 0 j¼1 n

pKðrrÞ12

1 X

abðnÞ1 ga j j

(by Lemma 2.3 a))

j¼1 n

pKðrrÞ12 ga1

for g small:

&

Corollary 4.4. Given z 2 Cnf0g, for any compact set E qCnf0g with E \ qBjzj ð0Þ ¼ ;, there is a neighborhood N of z in Cnf0g such that

q sup

rw G lC ðw; yÞ

: w 2 N; y 2 E o1. qny

ARTICLE IN PRESS R. Ban˜uelos, D. DeBlassie / Stochastic Processes and their Applications 116 (2006) 36–69

64

Proof. For g ¼ jwj2jwjjyj , we have for some neighborhood N of z in Cnf0g, 2 þjyj2 supfg: w 2 N; y 2 Ego1. Then we can use Lemma 2.3 and Corollary 2.2 as we did in the proof of (4.11) above to get the desired conclusion. & Corollary 4.5. Suppose rajxj and z 2 C \ qBr ð0Þ. Then for some constant K independent of z, ( 1nþa1 r 2 ; r small; G lC ðx; zÞpK n r12a1 ; r large; ( nþa1

r 2 ; r small;

q l

qn G C ðx; zÞ pK rn2a1 ; r large; z where q=qnz is the inward normal derivative on qBr ð0Þ. is small. Hence by the remark after Proof. If r is small or large, g ¼ r22rjxj þjxj2 Lemma 4.3, n

n

G lC ðx; zÞpKjxj12 r12 ga1 and

q l

1n2 n2 a1 1n2 a1 1

þ ga1 þ1 Þðr2 þ jxj2 Þ1 .

qn GC ðx; zÞ pKjxj ½r g þ jxjr ðg z

The desired bounds follow from these inequalities.

&

As another application of Lemma 4.3, we get bounds on the function Z q f ðyÞ G l ðz; yÞnðdyÞ uðzÞ ¼ qny C qC (from (4.7)) and its normal derivatives for large and small z. Lemma 4.6. (a) For some constant K, for z 2 Cnf0g with jzj ¼ M sufficiently large n

juðzÞjpKM 12a1

q

pKM n2a1 uðzÞ

qn

z

where qnq z is the inward normal derivative on qBM ð0Þ. (b) For some constant K, for z 2 Cnf0g with jzj ¼ e sufficiently small, n

juðzÞjpKe12þa1

q

pKen2þa1 . uðzÞ

qn

z

Proof. Note if y 2 qC \ supp f , then for r large or r small, in y. Then by (4.7) and (4.10) with jzj ¼ r ¼ M or e, Z 2rjyj a1 n n juðzÞjpK f ðyÞjyj2 r12 2 nðdyÞ. r þ jyj2 qC

2rjyj r2 þjyj2

is uniformly small

ARTICLE IN PRESS R. Ban˜uelos, D. DeBlassie / Stochastic Processes and their Applications 116 (2006) 36–69

65

Since supp f is compact in Cnðf0g [ qBjxj ð0ÞÞ, we have 0o inffjyj: y 2 supp f gp supfjyj: y 2 supp f go1 and so by Lemma 4.2, when r ¼ e, n

juðzÞjpKe12þa1 ;

e small;

and when r ¼ M, n

juðzÞjpKM 12a1 ;

M large.

The derivative estimates are a bit more delicate. Looking at the last term in (4.11): for y 2 supp f , n

r12 ðga1 1 þ ga1 þ1 Þðr2 þ r2 Þ1 " a1 1 a1 þ1 # 2rr 2rr n ¼ r12 þ ðr2 þ r2 Þ1 r2 þ r2 r2 þ r2 ( 1a1 ½r þ r1a1 r2 ; r large; 1n2 pKr ra1 1 þ ra1 þ1 ; r small; ( a1 n 2; r large; r pK a1 n2 r ; r small: Thus for y 2 supp f and jzj ¼ M large, (4.11) yields

a1

q q

2Mr l a1 n2

pK M n2 G ðz; yÞ þ M

qn qn C

M 2 þ r2 z y n

n

pK½M 2a1 þ M a1 2 n

pKM a1 2 and if jzj ¼ e is small,

q q

2er a1 l a1 n2

pK en2 G ðz; yÞ þ e

qn qn C

e2 þ r2 z y n

n

pK½e2þa1 þ ea1 2 n

¼ Ke2þa1 . Then if we can differentiate under the integral

Z

q

q q l

p

nðdyÞ uðzÞ f ðyÞ G ðz; yÞ

qn

qn qn C

z z y qC ( n KM a1 2 if jzj ¼ M is large; p n Ke2þa1 if jzj ¼ e is small; (by Lemma 4.2), as desired. As for differentiation under the integral, by bounding difference quotients via the mean value theorem, it is easy to see by Corollary 4.4 the exchange is justiﬁed. &

ARTICLE IN PRESS R. Ban˜uelos, D. DeBlassie / Stochastic Processes and their Applications 116 (2006) 36–69

66

Theorem 4.7. We have Z q q uðzÞ G lC ðx; zÞ G lC ðx; zÞ uðzÞ sðdzÞ ! 0, qnz qnz C\qBM ð0Þ as M ! 1 or as M ! 0. Here

q qnz

is the unit inward normal derivative on qBM ð0Þ.

Proof. By Corollary 4.5 and Lemma 4.6, ( 1nþa1 n n n M 2 M 2þa1 þ M 12þa1 M 2þa1 ; integrandpK n n n n M 12a1 M 2a1 þ M 12a1 M 2a1 ; ( 1nþ2a1 M ; M small; ¼K M 1n2a1 ; M large: Since sðqBM ð0ÞÞpKM n1 , we get the desired conclusion. The next order of business is to study this end, introduce the function Z 1 G l ðx; yÞ:¼ elt pðt; x; yÞ dt,

G lC ðx; Þ

M small; M large;

&

in a small neighborhood of x. To

(4.16)

0

where n=2

pðt; x; yÞ ¼ ð2ptÞ

1 2 exp jx yj 2t

(4.17)

is the usual Gaussian kernel. The relevant properties of Gl are stated in the next lemma. Lemma 4.8. (a) For 0ojx yj small, ( nX3; jx yj2n ; l G ðx; yÞpK ln jx yj; n ¼ 2: (b) As jx yj ! 0, n yx ry G l ðx; yÞ pn=2 G jx yj1n . jy xj 2 (c) For some small neighborhood N of x with compact closure in C, supfjry G l ðw; yÞj: w 2 qCnf0g; y 2 Ngo1. Proof. After changing variables u ¼ lt, [19, formula 3.471.12, p. 340] Z 1 1 n=2 l lt 2 G ðx; yÞ ¼ e ð2ptÞ exp jx yj dt 2t 0 Z 1 l jx yj2 n2 n21 n2 u u e exp ¼ ð2pÞ l du 2 u 0 p ﬃﬃﬃﬃﬃ n 1 n n n ¼ 22ð12Þ p2 l1=2ð21Þ jx yj12 K ðn=2Þ1 ð 2l jx yjÞ,

ARTICLE IN PRESS R. Ban˜uelos, D. DeBlassie / Stochastic Processes and their Applications 116 (2006) 36–69

67

where K n is the modiﬁed Bessel function. It is known that K 0 ðzÞ ln z

as z ! 0,

zn 1 K n ðzÞ GðnÞ as z ! 0, 2 2 rﬃﬃﬃﬃﬃ p z 0 as z ! 1, K n ðzÞ e 2z rﬃﬃﬃﬃﬃ p z e K n ðzÞ 2z

(4.18) (4.19)

(4.20)

as z ! 1,

(4.21)

n K 0n ðzÞ ¼ K nþ1 ðzÞ þ K n ðzÞ. z

(4.22)

(4.18)–(4.21) can be found in formulas 9.6.8, 9.6.9, 9.7.4, 9.7.2, respectively, in [1, pp. 375–378]. Formula (4.22) is from [27, formula (4), Section 3.71, p. 79]. Part (a) is an immediate consequence of (4.18)–(4.19). n 1 n n Now for C 1 ¼ 22ð12Þ p2 l1=2ð21Þ , by (4.22) and (4.19), as jx yj ! 0, h pﬃﬃﬃﬃﬃ yx n n ry G l ðx; yÞ ¼ C 1 1 jx yj2 K n21 ð 2ljx yjÞ jy xj 2 i pﬃﬃﬃﬃﬃ 0 pﬃﬃﬃﬃﬃ 1n2 þjx yj 2l K n1 ð 2ljx yjÞ 2 pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ 1n2 ¼ C 1 jx yj 2l K n2 ð 2ljx yjÞ !n2 rﬃﬃﬃ pﬃﬃﬃﬃﬃ 1 n l 1n2 jx yj 2l G C 1 jx yj 2 2 2 n ¼ pn=2 G jx yj1n . 2 This gives part (b). Finally, for some d40, Bd ðxÞ C and so inffjw yj: w 2 qCnf0g, y 2 Bd ðxÞg40. Then by (4.20)–(4.21), part (c) follows. & Theorem 4.9. We have Z q q l l lim uðzÞ G ðx; zÞ G C ðx; zÞ uðzÞ sðdzÞ ¼ 2uðxÞ d!0 qBd ðxÞ qnz C qnz where

q qnz

is the unit inward normal derivative on qBd ð0Þ.

qu Proof. By (4.6) qn is bounded on a neighborhood of x. Since G lC pG l , by Lemma z 4.8(a) for d small, ( )

Z

l

nX3 d2n ; q

G ðx; zÞ uðzÞ

sðdzÞpKsðqBd ð0ÞÞ ! 0,

C qnz ln d; n ¼ 2 qBd ðxÞ

ARTICLE IN PRESS 68

R. Ban˜uelos, D. DeBlassie / Stochastic Processes and their Applications 116 (2006) 36–69

as d ! 0. Thus we just need to show Z q l lim uðzÞ G C ðx; zÞ sðdzÞ ¼ 2uðxÞ. d!0 qBd ð0Þ qnz

(4.23)

It is well-known that for the exit time t of Brownian motion Bt from C, pC ðt; x; zÞ ¼ pðt; x; zÞ E x ½I tot pðt t; Bt ; zÞ. Then G lC ðx; zÞ ¼ G l ðx; zÞ E x ½elt G l ðBt ; zÞ, and as a consequence, for z 2 qBd ðxÞ q l q l q l G C ðx; zÞ ¼ G ðx; zÞ E x elt G ðBt ; zÞ . qnz qnz qnz The exchange of qnq z and E x is justiﬁed as follows. Bound difference quotients via the mean value theorem. Then the exchange is justiﬁed provided supfjrz Gl ðw; zÞj: z 2 B2d ðxÞ; w 2 qCnf0ggo1. This follows from Lemma 4.8, part (c). Furthermore, since u is bounded near x, by Lemma 4.8(c)

Z

l lt q

pKsðqBd ðxÞÞ ! 0 uðzÞE e G ðB ; zÞ sðdzÞ x t

qnz qBd ðxÞ

as d ! 0.

Thus to get (4.23) we need only show Z q lim uðzÞ G l ðx; zÞsðdzÞ ¼ 2uðxÞ. d!0 qBd ðxÞ qnz zx By Lemma 4.8 b, on qBd ðxÞ, qnq z G l ðx; zÞ ¼ jzxj rz G l ðx; zÞpn=2 Gðn=2Þd1n as d ! 0. Since u is continuous and bounded near x, Z n q lim uðzÞ G l ðx; zÞsðdzÞ ¼ uðxÞpn=2 G sðqB1 ðxÞÞ d!0 qBd ðxÞ qnz 2

¼ 2uðxÞ, as desired.

&

Appendix A. Supplementary material Supplementary data associated with this article can be found in the online version 10.1016/j.spa.2005.07.003

ARTICLE IN PRESS R. Ban˜uelos, D. DeBlassie / Stochastic Processes and their Applications 116 (2006) 36–69

69

References [1] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972. [2] H. Allouba, Brownian-time process: the PDE connection II and the corresponding Feynman–Kac formula, Trans. Amer. Math. Soc. 354 (2002) 4627–4637. [3] H. Allouba, W. Zheng, Brownian-time processes: the PDE connection and the half-derivative generator, Ann. Probab. 29 (2001) 1780–1795. [4] R. Ban˜uelos, T. Carroll, Sharp integrability for Brownian motion in parabola-shaped regions, J. Funct. Anal. 218 (2005) 219–253. [5] R. Ban˜uelos, R.D. DeBlassie, R. Smits, The ﬁrst exit time of planar Brownian motion from the interior of a parabola, Ann. Prob. 29 (2001) 882–901. [6] R. Ban˜uelos, R.G. Smits, Brownian motion in cones, Probab. Theory Related Fields 108 (1997) 299–319. [7] M. Vandenberg, Subexponential behaviour of the Dirichlet heat kernel, J. Funct. Anal. 198 (2003) 28–42. [8] K. Burdzy, Some path properties of iterated Brownian motion, in: E. Cinlar, K.L. Chung, M.J. Sharpe (Eds.), Seminar on Stochastic Processes, Birkha¨user, Boston, 1993, pp. 67–87. [9] K. Burdzy, D. Khoshnevisan, Brownian motion in a Brownian crack, Ann. Appl. Probab. 8 (1998) 708–748. [10] D.L. Burkholder, Exit times of Brownian motion, harmonic majorization and Hardy spaces, Adv. in Math. 26 (1977) 182–205. [11] I. Chavel, Eigenvalues in Riemann Geometry, Academic Press, New York, 1984. + A. Fo¨ldes, P. Re´ve´sz, The local time of iterated Brownian motion, J. Theoret. [12] E. Csa´ki, M. Cso¨rgo, Probab. 9 (1996) 717–743. [13] R.D. DeBlassie, Exit times from cones in Rn of Brownian motion, Probab. Theory Related Fields 74 (1987) 1–29. [14] R.D. DeBlassie, Remark on exit times from cones in Rn of Brownian motion, Probab. Theory Related Fields 79 (1988) 95–97. [15] R.D. DeBlassie, R. Smits, Brownian motion in twisted domains, Trans. Amer. Math. Soc. 357 (2005) 1245–1274. [16] R.D. DeBlassie, Iterated Brownian motion in an open set, Ann. Appl. Probab. 14 (2004) 1529–1558. [17] M. Esse´n, K. Haliste, A problem of Burkholder and the existence of harmonic majorants of jxjp in certain domains in Rd , Ann. Acad. Sci. Fenn. Ser. A. I. Math. 9 (1984) 107–116. [18] D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order, second ed., Springer, Berlin, 1983. [19] I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series and Products, Academic, New York, 1980. [20] K. Haliste, Some estimates of harmonic majorants, Ann. Acad. Sci. Fenn. Ser. A. I. Math. 9 (1984) 117–124. [21] P. Hsu, Brownian exit distribution of a ball, in: Seminar in Stochastic Processes, Birkha¨user, Boston, 1986, pp. 108–116. [22] D. Khoshnevisan, T.M. Lewis, Chung’s law of the iterated logarithm for iterated Brownian motion, Ann. Inst. Henri Poincare´ 32 (1996) 349–359. [23] W. Li, The ﬁrst exit time of Brownian motion from an unbounded convex domain, Ann. Probab. 31 (2003) 1078–1096. [24] M. Lifshitz, Z. Shi, The ﬁrst exit time of Brownian motion from parabolic domain, Bernoulli 8 (2000) 745–765. [25] R.G. Pinsky, Positive Harmonic Functions and Diffusion, Cambridge University Press, Cambridge, 1995. [26] M. Protter, H. Weinberger, Maximum Principles in Differential Equations, Springer, New York, 1984. [27] G.N. Watson, A Treatise on the Theory of Bessel Functions, second ed., Cambridge University Press, Cambridge, 1922. [28] Y. Xiao, Local times and related properties of multidimensional iterated Brownian motion, J. Theoret. Probab. 11 (1998) 383–408.

The exit distribution for iterated Brownian motion in cones

The exit distribution for iterated Brownian motion in cones

Recommend Documents