SYMMETRIC STABLE PROCESSES IN PARABOLA SHAPED REGIONS

Transcription

1 SYMMETRIC STABLE PROCESSES IN PARABOLA SHAPED REGIONS RODRIGO BAÑUELOS AND KRZYSZTOF BOGDAN Abstract We identify the critical exponent of integrability of the first exit time of rotation invariant stable Lévy process from parabola shaped region 1 Introduction For d =2, 3, and 0 <β<1, we define the parabola shaped region in R d P β = {x =(x 1, x) : x 1 > 0, x R d 1, x <x β 1 } Let 0 <α<2 By {X t } we denote the isotropic α-stable R d -valued Lévy process ([17]) The process is time-homogeneous, has right-continuous trajectories with left limits, α-stable rotation invariant independent increments, and characteristic function (1) E x e iξ(xt x) = e t ξ α, x R d, ξ R d, t 0 Here E x is the expectation with respect to the distribution P x of the process starting from x R d For an open set U R d, we define τ U =inf{t 0; X t / U}, the first exit time of U ([17]) In the case of the parabola-shaped region P β, we simply write τ β for τ Pβ The main result of this note is the following result Theorem 1 Let p 0 Then E x τ β p < for (some, hence for all) x P β if and only if p<p 0,where (d 1)(1 β)+α p 0 = αβ Theorem 1 may be regarded as an addition to the research direction initiated in [3], where it was proved that for β =1/2, d =2and{X t } replaced by the Brownian motion process {B t }, τ β is subexponentially integrable For The first author was supported in part by NSF grant # DMS The second author was supported in part by KBN (2P03A ) and by RTN (HPRN- CT HARP) 2000 Mathematics Subject Classification: Primary 31B05, 60J45 Key words and phrases: symmetric stable process, parabola, exit time, harmonic measure Date: 7/14/2004 1

2 2 R BAÑUELOS AND K BOGDAN {B t }, the generalizations to all the considered domains P β (along with essential strengthening of the result of [3]) were subsequently obtained in [14], [15], [5], [11] and [2] We should note that this direction of research was influenced to a large degree by the result of Burkholder [10] on the critical order of integrability of the exit times of {B t } from cones For more on the many generalizations of Burkholder s result we refer the reader to [4] and references therein Brownian motion is a limiting case of the isotropic α-stable process and B 2t corresponds to α = 2 via the analogue of (1) Extension of some of the above-mentioned results pertaining to cones to the case 0 <α<2 (that is, for {X t }) were given in [12], [13], [16] and [1], see also [7] It should be emphasized that while there are many similarities between {X t } and {B 2t }, there also exist essential differences in their respective properties and their proofs For example the critical exponent of integrability of the exit time of {X t } is less than 1 for every cone, however narrow it may be ([1]), while it is arbitrarily large for {B t } in sufficiently narrow cones ([10]) A similar remark applies to regions P β The critical exponent of integrability of τ β for {X t } givenintheorem1isqualitatively different from that of {B t } ([2]) Finally, we have recently learned that Pedro J Méndez-Hernández, in a paper in preparation, has obtained results similar to those presented His results apply to more general regions defined by other slowly increasing functions The proof of Theorem 1 is based on estimates for harmonic measure of {X t }, following the general idea used in [2] for {B t } The necessity of the condition p<p 0 for finiteness of E x τ p β is proved in Lemma 2 and the sufficiency is proved in Lemma 6 The main technical result of the paper is Lemma 5 where we prove sharp estimates for harmonic measure of cut-offs of the parabola-shaped region At the end of the paper we discuss some remaining open problems 2 Proofs We begin by reviewing the notation By we denote the Euclidean norm in R d Forx R d, r>0, and a set A R d we let B(x, r) ={y R d : x y < r} and dist(a, x) =inf{ x y : y A} A c is the complement of A We always assume Borel measurability of the considered sets and functions In equalities and inequalities, unless stated otherwise, c denotes some unspecified positive real number whose value depends only on d, α and β It is well known that (X t, P x ) is a strong Markov processes with respect to the standard filtration ([17]) For an open set U R d, with exit time τ U,theP x distribution of X τu : ω x U(A) :=P x {X τu A, τ U < + }, A R d,

3 STABLE PROCESS IN PARABOLA 3 is a subprobability measure concentrated on U c (probability measure if U is bounded) called the harmonic measure When r>0, x <rand B = B(0,r) R d, the corresponding harmonic measure has the density function P r (x, y) =dωb x /dy (the Poisson kernel for the ball) We have [ ] r 2 x 2 α/2 (2) P r (x, y) =C(d, α) y x d if y >r, y 2 r 2 where C(d, α) =Γ(d/2)π d/2 1 sin(πα/2), and P r (x, y) = 0 otherwise ([6]) The scaling property of X t plays a role in this paper Namely, for every r> 0andx R d the P x distribution of {X t } is the same as the P rx distribution of {r 1 X r α t} (see (1)) In particular, the P x distribution of τ U is the same as the P rx distribution of r α τ ru In short, τ ru = r α τ U in distribution Let P β = P β {x 1 > 1, x <x β 1/2} We claim that if x P β then B(x, x β /5) P β Indeed, let x = (x 1, x) P β and y = (y 1, ỹ) B(x, x β /5) We have x <x β 1/2 <x 1 /2 hence 1 <x 1 x = x x 2 < 5/4x 1 < 5x 1 /4 Then y 1 x 1 x β /5 >x 1 x /5 >x 1 (5/4)x 1 /5=3x 1 /4, and so ỹ x + x β /5 <x β 1 /2+(5x 1/4) β /5 < 3x β 1 /4 < 3(4y 1/3) β /4 <y β 1, which yields y P β, as claimed Lemma 2 If p p 0 then E x τ β p = for every x P β Proof Define τ = inf{t 0 : X t / B(X 0, X 0 β /5)} For y R d and (nonnegative) p by scaling we have E y τ p = E 0 τ p B(0,1) ( y β /5) pα = c y αβp Let x P β, r = dist(x, P c β ), B = B(x, r), and let R be so large that B P β {y 1 R} By strong Markov property we have p E x τ β E x {X τb P β ; E XτB τ p β } E x {X τb P β ; E XτB τ p } = ce x {X τb P β ; X τb αβp } c P r (0,y x) y αβp dy cr α C(d, α) const R R P β {y 1 >R} dy 1 ỹ <y β1 /2 y x d α y αβp dỹ y β(d 1)+αβp d α 1 dy 1 If p p 0,thenβ(d 1)+αβp d α β(d 1)+(d 1)(1 β)+α d α = 1 and the last integral is positive infinity, proving the lemma

4 4 R BAÑUELOS AND K BOGDAN For 0 u<v,weletp u,v β = P β {(x 1, x) : u x 1 <v} Let 0 <s< and define τ s = τ(s) =τ Pβ 0,s Consider the cylinder (3) C = {x =(x 1, x) R d : <x 1 <s, x <s β } Clearly for A P β, (4) P x {X τs A} P x {X τc A}, x R d By Lemma 43 of [9] for A R d {z 1 s} we have [ ] s αβ s αβ/2 (5) P x {X τc A} c z=(z 1, z) A z x d+α (z 1 s) 1 dz, α/2 The following lemma will simplify the use of the estimate (5) x C Lemma 3 Let s 1 and x =(x 1, x) P β with x 1 s/2 Lets u<v and assume that either u s + s β or u = s and v s + s β Then v (6) P x {X τs P u,v β } cs αβ t αβp0 1 dt Proof Denote f(y) = dωx 0,s P β (y) dy Let y =(y 1, ỹ) P s, β From (4) and (5) we can conclude that [ ] s αβ s αβ/2 f(y) c y x d+α (y 1 s) 1 α/2 Since s 1, we have ỹ y 1 and y 2y 1 As y x y 1 x 1 y 1 /2, [ ] f(y) cs αβ y1 d α s αβ/2 (y 1 s) 1 α/2 If u s + s β then v P x {X τs P u,v β } cs αβ y1 d α dỹdy 1 u ỹ <y β 1 v v = cs αβ t d α+β(d 1) dt = cs αβ t αβp0 1 dt If u = s and v s + s β,weconsider and u s+sβ I(s) =s αβ t αβp0 1 dt, II(s) =s αβ s+sβ s s u t αβp 0 1 s αβ/2 dt (t s) α/2 u

5 STABLE PROCESS IN PARABOLA 5 It is enough to verify that II(s) ci(s) Note that s + s β 2s, hence I(s) s αβ (2s) αβp 0 1 s β = cs αβp 0+αβ+β 1 Since s β II(s) s αβ s αβp0 1 s αβ/2 z α/2 dz = cs αβp0+αβ+β 1, we get II(s) ci(s) The following result is an immediate consequence of Lemma 3 0 Lemma 4 If s 1, x =(x 1, x) R d and x 1 s/2, then (7) P x {X τs P β } cs αβ(p 0 1) Estimate (7) is sharp if, for example, x =(s/2, 0), where 0 =(0,,0) R d 1 Indeed, let B = B(x, cs β ) P β 0,s,wherec>0 is small enough By (2) we have (8) P x {X τs P β } P x {X τb P s, β } c(s β ) α dz 1 (z 1, z) x { z R d α d z d 1 : z <z β1 } s cs αβ t β(d 1) d α dt = cs αβ(p0 1) s We can, however, improve the estimate when x is small relative to s Lemma 5 If s 1, x =(x 1, x) R d and x 1 s/2, then (9) P x {X τs P β } C(x 1 1) αβ s αβp 0 Proof If s S, where0<s< is fixed, then (9) trivially holds with C = S αβp 0, hence from now on we assume s S, wheres =2 k 0 and k 0 2 is such that 4 αβ c ( 2 k 0 ) αβ(p0 1) 1 < αβ(p 0 1) 2 The reason for this choice of k 0 will be made clear later on Here and in what follows c is the constant of Lemma 3 We will prove (9) by induction If s/4 x 1 s/2 then by Lemma 4 P x {X τs P β } c 1 s αβ s αβp 0 4 αβ c 1 x αβ 1 s αβp 0 = c 1 (x 1 1) αβ s αβp 0 Assume that n is a natural number and (9) holds for all x =(x 1, x) P β such that s/2 n+1 x 1 s/2

6 6 R BAÑUELOS AND K BOGDAN Let s/2 n 1andx =(x 1, x) P β be such that s/2 n+2 x 1 <s/2 n+1 Note that 1 s/2 n 4x 1 here We have P x {X τs P β } P x {X τ(s/2 n ) P s/2, β } + s/2 E x {X τ(s/2 n ) P n,s/2 β ; P Xτ(s/2 n ) {X τs P β }} = I + II By Lemma 3, I c(s/2 n ) αβ t αβp0 1 dt = c2 αβp 0 (s/2 n ) αβ s αβp 0 s/2 c2 αβp0+2α x αβ 1 s αβp 0 c 2 (x 1 1) αβ s αβp 0 By Lemma 3 and induction s/2 II c(s/2 n ) αβ t αβp0 1 Ct αβ s αβp 0 dt s/2 n c αβ(p 0 1) (s/2n ) αβ(p0 1) C(s/2 n ) αβ s αβp 0 Thus 4 αβ c αβ(p 0 1) (s/2n ) αβ(p 0 1) C(x 1 1) αβ s αβp 0 I + II R = (x 1 1) αβ s c 4 αβ c αβp αβ(p 0 1) (s/2n ) αβ(p0 1) C For n such that s/2 n 2 k 0 we have R c 2 + C/2 andwecantakec = 2(c 1 c 2 ) in our inductive assumption to the effect that R c 2 + c 1 c 2 C, and so (9) holds for every x =(x 1, x) P β, satisfying s/2 n+2 x 1 s/2 By induction, (9) is true with C =2(c 1 c 2 ) for all x =(x 1, x) P β, satisfying 2 k0 2 x 1 s/2 If x =(x 1, x) P β and 0 <x 1 2 k0 2,then P x {X τs P β } P x {X τ(2 k 0 /2) P β s/2, } + E x {X τ(2 k 0 /2) P β 2 k 0 /2,s/2 ; P Xτ(2 k 0 /2) {X τs P β }} c(2 k 0 /2) αβ t αβp0 1 dt s/2 + c(2 k 0 /2) αβ s/2 2 k 0 /2 t αβp 0 1 2(c 1 c 2 )t αβ s αβp 0 dt c 3 s αβp 0 c 3 (x 1 1) αβ s αβ(p 0 1) We used here our previous estimates The proof is complete

7 STABLE PROCESS IN PARABOLA 7 Note that (9) is sharp if, for example, x =(x 1, 0) and 1 x 1 s/2 This can be verified as in (8) to the effect that for such x the upper bound in Lemma 5 is of the same order as P x {X τb P s, β },whereb is the largest ball centered at x such that B P β Informally, {X t } goes to P s, β mostly by a direct jump from B This informal rule seems to be related to a thinness of P β at infinity (or its inversion [8] at 0) This is false for cones ([1]) Lemma 6 If p<p 0,thenE x τ β p < for every x P β Proof Let 1/λ 1 be the first eigenvalue of the Green operator G B f( x) = ηb E x f(y 0 t )dt, x R d 1, for our isotropic stable process Y t in R d 1 Here, B is the unit ball in R d 1, η D is the first exit time of Y from D R d 1, and P, E, are, respectively, the distribution and expectation corresponding to Y For r>0, by scaling, 1/(r α λ 1 ) is the eigenvalue for G rb and P x {η rb >t} ce tr α λ 1,0<t< Let C be the cylinder as in (3) For t>0, fixed x =(x 1, x) P β and s 1 2x 1, we have by Lemma 5 that P x {τ β >t} = P x {τ β >t, τ β = τ s } + P x {τ β >t, τ β >τ s } P x {τ C >t} + P x {X τs P β } ce ts αβ λ 1 + cs αβp 0 Let us take s = t (1 ɛ)/(αβ),where0<ɛ 1/2 We get P x {τ β >t} ce tɛ λ 1 + ct (1 ɛ)p 0 ct (1 ɛ)p 0 for large t Thus, letting p =(1 2ɛ)p 0 we get E x τ β p = 0 pt p 1 P x {τ β >t} dt const + const 1 t ɛp 0 1 dt < This finishes the proof Proof of Theorem 1 The result follows from Lemma 2 and Lemma 6 We conclude with three remarks Because of scaling of {X t }, Theorem 1 holds with the same p 0 for the more general parabola-shaped regions of the form {x =(x 1, x) : x 1 > 0, x R d 1, x <ax β 1}, for any 0 <a< If β 0thenP β approaches an infinite cylinder, for which the exit time has exponential moments (compare the proof of Lemma 6) The second endpoint β 1 suggests studying the rate of convergence (to 1) of the critical exponent of integrability of the exit time from the right circular cone as the opening of the cone tends to 0 A partial result in this direction is given in [13] [1] contains more information on our stable processes in cones and a hint how to approach the problem

8 8 R BAÑUELOS AND K BOGDAN Acknowledgment The second author is grateful for the hospitality of the Departments of Statistics and Mathematics at Purdue University, where the paper was prepared in part We are grateful to the referee for pointing out several corrections References [1] R Bañuelos and K Bogdan, Symmetric stable processes in cones, Potential Analysis, 21 (2004), [2] R Bañuelos and T Carroll, Sharp Integrability for Brownian Motion in Parabolashaped Regions (2003), J Funct Anal (to appear) [3] R Bañuelos, R D DeBlassie and R Smits, The first exit time of planar Brownian motion from the interior of a parabola, Ann Prob 29 (2001), [4] R Bañuelos and R Smits, Brownian motion in cones, Prob Th Rel Fields, 108 (1997), [5] M van den Berg, Subexponential behavior of the Dirichlet heat kernel, J Funct Anal, 198, [6] R M Blumenthal, R K Getoor and DB Ray, On the distribution of first hits for the symmetric stable process, Trans Amer Math Soc 99 (1961), [7] K Bogdan and T Jakubowski, Problème de Dirichlet pour les fonctions α- harmoniques sur les domaines coniques (2004), preprint [8] K Bogdan and T Żak, On Kelvin transformation (2004), preprint [9] K Burdzy and T Kulczycki, Stable processes have thorns, Ann Probab 31 (2003) [10] D L Burkholder, Exit times of Brownian motion, harmonic majorization, and Hardy spaces, Advances in Math 26(2) (1977), [11] R D DeBlassie and R Smits Brownian motion in twisted domains, Transactions of the American Mathematical Society (to appear) [12] R DeBlassie, The first exit time of a two dimensional symmetric stable process from awedge, Ann Prob, 18 (1990), [13] T Kulczycki, Exit time and Green function of cone for symmetric stable processes, Probab Math Statist 19(2) (1999), [14] W Li, The first exit time of Brownian motion from unbounded convex domains, Annals of Probability, 31, [15] M Lifshitz and Z Shi, The first exit time of Brownian motion from parabolic domain, Bernoulli 8(6) (2002), [16] P J Méndez-Hernández, Exit times from cones in R n of symmetric stable processes Illinois J Math 46 (2002), [17] K-I Sato, Lévy Processes and Infinitely Divisible Distributions, University Press Springer-Verlag, Cambridge, 1999 Department of Mathematics, Purdue University, West Lafayette, IN Institute of Mathematics, Polish Academy of Sciences, and Institute of Mathematics, Wroc law University of Technology, Wroc law, Poland address: banuelos@mathpurdueedu bogdan@impwrwrocpl