Some infinite divisibility properties of the reciprocal of planar Brownian motion exit time from a cone

0.18 MB·English

by  Stavros Vakeroudis

Some infinite divisibility properties of the reciprocal
of planar Brownian motion exit time from a cone
S. Vakeroudis M. Yor
∗† ∗‡
January 16, 2012
2
1
0
2 Abstract

n
With the help of the Gauss-Laplace transform for the exit time from a cone
a
J of planar Brownian motion, we obtain some infinite divisibility properties for the

3 reciprocal of this exit time.
1

]
R
Key words: Bougerol’s identity, infinite divisibility, Chebyshev polynomials, Lévy mea-
P
. sure, Thorin measure, generalized Gamma convolution (GGC).
h
t
a
m MSC Classification (2010): 60J65, 60E07.
[

1 1 Introduction
v
8
1 Let (Z = X + iY ,t 0) denote a standard planar Brownian motion , starting from
7 t t t ≥ §
2 x0 + i0,x0 > 0, where (Xt,t 0) and (Yt,t 0) are two independent linear Brownian
≥ ≥
. motions, starting respectively from x and 0.
1 0
0 It is well known [ItMK65] that, since x = 0, (Z ,t 0) does not visit a.s. the point
0 t
2 6 ≥
0 but keeps winding around 0 infinitely often. Hence, the continuous winding process
1
: θ = Im( t dZs),t 0 is well defined. Using a scaling argument, we may assume x = 1,
v t 0 Zs ≥ 0
i without loss of generality, since, with obvious notation:
X R
r
a Z(x0),t 0 (l=aw) x Z(1) ,t 0 . (1)
t ≥ 0 (t/x20) ≥
(cid:16) (cid:17) (cid:16) (cid:17)
∗Laboratoire de Probabilités et Modèles Aléatoires (LPMA) CNRS : UMR7599, Université Pierre et
Marie Curie - Paris VI, Université Paris-Diderot - Paris VII, 4 Place Jussieu, 75252 Paris Cedex 05,
France. E-mail: [email protected]
†Probability and Statistics Group, School of Mathematics, University of Manchester, Alan Turing
Building, Oxford Road, Manchester M13 9PL, United Kingdom.
‡Institut Universitaire de France, Paris, France. E-mail: [email protected]
§When we write: Brownian motion, we always mean real-valued Brownian motion, starting from 0.
For 2-dimensional Brownian motion, we indicate planar or complex BM.
1
From now on, we shall take x = 1.
0
Furthermore, there is the skew product representation:
t dZ
s
log Z +iθ = (β +iγ ) , (2)
t t u u
| | ≡ Z0 Zs (cid:12)u=Ht=R0t |Zdss|2
(cid:12)
where (β +iγ ,u 0) is another planar Brownian m(cid:12)otion starting from log1+i0 = 0
u u
≥
(for further study of the Bessel clock H, see [Yor80]).
We may rewrite (2) as:
log Z = β ; θ = γ . (3)
| t| Ht t Ht
Onenoweasily obtainsthatthetwo σ-fields σ Z ,t 0 andσ β ,u 0 areidentical,
t u
{| | ≥ } { ≥ }
whereas (γ ,u 0) is independent from ( Z ,t 0).
u t
≥ | | ≥
Bougerol’s celebrated identity in law ([Bou83, ADY97] and [Yor01] (p. 200)), which says
that:
(law)
for fixed t, sinh(β ) = δ (4)
t At(β)
u
where (β ,u 0) is 1-dimensional BM, A (β) = dsexp(2β ) and (δ ,v 0) is an-
u ≥ u 0 s v ≥
other BM, independent of (β ,u 0), will also be used. We define the random times
u
≥ R
θ γ
Tc| | inf t : θt = c , and Tc| | inf t : γt = c , (c > 0). From the skew-product
≡ { | | } ≡ { | | }
representation (3) of planar Brownian motion, we obtain [ReY99]:
Tc|γ|
A (β) dsexp(2β ) = H 1 = T θ . (5)
Tc|γ| ≡ Z0 s u− (cid:12)u=Tc|γ| c| |
(cid:12)
Then,Bougerol’sidentity(4)fortherandomtimeTc|θ| y(cid:12)ieldsthefollowing[Vak11,VaY11]:
θ
Proposition 1.1 The distribution of Tc| | is characterized by:
2c2 x 1
E exp = ϕ (x), (6)
"sπTc|θ| (cid:18)−2Tc|θ|(cid:19)# √1+x m
for every x 0, with m = π, and
≥ 2c
2
ϕ (x) = , with G (x) = √1+x √x. (7)
m (G+(x))m +(G (x))m ± ±
−
Comment and Terminology:
If S > 0 a.s. is independent from a Brownian motion (δ ,u 0), we call the density of
u
≥
δ , which is:
S
1 x2
E exp (8)
√2πS −2S
(cid:20) (cid:18) (cid:19)(cid:21)
the Gauss-Laplace transform of S (see e.g. [ChY03] ex.4.16, or [BPY01]). Thus, formula
θ
(6) expresses - up to simple changes -the Gauss-Laplace transform of Tc| |.
2
We also recall several notions which will be used throughout the following text:
a) A stochastic process ζ = (ζ ,t 0) is called a Lévy process if ζ = 0 a.s., it has
t 0
≥
stationary and independent increments and it is almost surely right continuous with left
limits. A Lévy process which is increasing is called a subordinator.
b) Following e.g. [ReY99], a probability measure π on R (resp. a real-valued random
variable with law π) is said to be infinitely divisible if, for any n 1, there is a prob-
≥
ability measure π such that π = π n (resp. if ζ ,...,ζ are n i.i.d. random variables,
n n∗ 1 n
(law)
ζ = ζ +...+ζ ). For instance, Gaussian, Poisson and Cauchy variables are infinitely
1 n
divisible.
It is well-known that (e.g. [Ber96]), π is infinitely divisible if and only if, its Fourier
transform πˆ is equal to exp(ψ), with:
σ2u2 iux
ψ(u) = ibu + eiux 1 ν(dx),
− 2 − − 1+x2
Z (cid:18) (cid:19)
where b R, σ2 0 and ν is a Radon measure on R 0 such that:
∈ ≥ \{ }
x2
ν(dx) < .
1+x2 ∞
Z
This expression of πˆ is known as the Lévy-Khintchine formula and the measure ν as the
Lévy measure.
c) Following [Bon92] (p.29) and [JRY08], a positive random variable Γ is a general-
ized Gamma convolution (GGC) if there exists a positive Radon measure µ on ]0, [
∞
such that:
dx
E e λΓ = exp ∞ 1 e λx ∞e xzµ(dz) (9)
− − −
− − x
(cid:18) Z0 Z0 (cid:19)
(cid:2) (cid:3) (cid:0) λ(cid:1)
∞
= exp log 1+ µ(dz) , (10)
− z
(cid:18) Z0 (cid:18) (cid:19) (cid:19)
with:
µ(dx)
logx µ(dx) and < . (11)
| | x ∞
Z]0,1] Z[1, [
∞
We remark that (10) follows immediately from (9) using the elementary Frullani formula
(see e.g. [Leb72], p.6). The measure µ is called Thorin’s measure associated with Γ.
We return now to the case of planar Brownian motion and the exit times from a cone.
Below, we state and prove the following:
Proposition 1.2 For every integer m, the function x ϕ (x), is the Laplace transform
m
→
of an infinitely divisible random variable K; more specifically, the following decomposi-
tions hold:
3
for m = 2n+1,
•
2 n 1
K = N + a e , a = ; k = 1,2,...,n, (12)
2 k k k sin2 π2k 1
−
k=1 22n+1
X
(cid:0) (cid:1)
for m = 2n,
•
n
1
K = b e , b = ; k = 1,2,...,n, (13)
k k k sin2 π2k 1
−
k=1 2 2n
X
where is a centered, reduced Gaussian va(cid:0)riable(cid:1)and e , k n are n independent
k
N ≤
exponential variables, with expectation 1.
Looking at formula (6), it is also natural to consider:
1
ϕ˜ (x) ϕ (x). (14)
m m
≡ √1+x
We note that:
2
˜
K N +K, (15)
≡ 2
admits the RHS of (6) as its Laplace transform. Hence,
for m = 2n+1,
•
n
K˜ (l=aw) e + a e , (16)
0 k k
k=1
X
for m = 2n,
•
2 n
K˜ (l=aw) N + b e , (17)
k k
2
k=1
X
with obvious notation.
In Section 2 we first illustrate Proposition 1.2 for m = 1 and m = 2 ; we may also verify
θ
equation (6) by using the laws of Tc| |, for c = π/2 and c = π/4, which are well known
[ReY99].
InSection3, weproveProposition1.2,wheretheChebyshev polynomialsplayanessential
role, we calculate the Lévy measure in the Lévy-Khintchine representation of ϕ and we
m
obtain the following asymptotic result:
θ
Proposition 1.3 With c denoting a positive constant, the distribution of Tc|ε|, for every
x 0, follows the asymptotics:
≥
1/ε
2(cε)2 x 1
ε 0
E exp → , (18)
"s πTc|εθ| (cid:18)−2Tc|εθ|(cid:19)#! −→ √x+√1+x π/2c
which, from [JRY08], is the Laplace transform of a s(cid:0)ubordinator Γ(cid:1) G ,t 0 with
t 1/2
≥
Thorin measure that of the arc sine law, taken at t = π/2c.
(cid:0) (cid:0) (cid:1) (cid:1)
Finally, we state a conjecture concerning the case where m is not necessarily an integer.
4
2 Examples
π
2.1 m = 1 c =
⇒ 2
Then:
1
ϕ˜ (x) = , (19)
1
1+x
is the Laplace transform of an exponential variable e .
1
Indeed, with (Z = X + iY = Z exp(iθ ),t 0) a planar BM starting from (1,0),
t t t t t
| | ≥
Tπ|θ/|2 = inf{t : Xt = 0} = inf{t : Xt0 = 1},
with (X0,t 0) denoting another one-dimensional BM starting from 0. Formula (6)
t ≥
states that:
2 x 1
E exp = . (20)
sπT|θ| −2T|θ| ! 1+x
π/2 π/2
 
However, we know that: T|θ| (l=aw) 1 , N (0,1).
π/2 N2 ∼ N
The LHS of the previous equality (20) gives:
2 x 1
E N exp N2 = ∞dy y e−x+21y2 = , (21)
π| | −2 1+x
"r (cid:16) (cid:17)# Z0
thus, we have verified directly that (20) holds.
π
2.2 m = 2 c =
⇒ 4
Similarly,
1 1
ϕ˜ (x) = , (22)
2
√1+x1+2x
is the Laplace transform of the variable 2 +2e .
N2 1
Again, this can be shown directly; indeed, with obvious notation:
θ
Tπ|/|4 = inf{t : Xt +Yt = 0, or Xt −Yt = 0}
X0 +Y 1 X0 Y 1
= inf t : t t = , or t − t =
{ √2 √2 √2 √2}
1
= T T˜ (l=aw) T T˜ .
1/√2 ∧ 1/√2 2 ∧
(cid:16) (cid:17)
Hence, formula (6) now writes, in this particular case:
π x 1 1
E exp = . (23)
4(T T˜) −T T˜ √1+x1+2x
(cid:20)r ∧ (cid:18) ∧ (cid:19)(cid:21)
5
This is easily proven, using: T (l=aw) 1 ,T˜ (l=aw) 1 , which yields:
N2 N˜2
E N N˜ exp x N2 N˜2 = 2E N exp xN2 1
| |∨| | − ∨ | | − (N N˜ )
| |≥| |
h(cid:16)= C ∞d(cid:17)u u e(cid:16)−xu2e(cid:16)−u22 udy(cid:17)e(cid:17)−iy22. h (cid:0) (cid:1) i
Z0 Z0
Fubini’s theorem now implies that (23) holds.
Remark 2.1 In a first draft, we continued looking at the cases: m = 3,4,5,6,..., in a
direct manner. But, these studies are now superseded by the general discussion in Section
3.
2.3 A "small" generalization
As we just wrote in Remark 2.1, before finding the proof of Proposition 1.2 (see below,
Subsection 3.1), we kept developing examples for larger values of m, and in particular,
we encountered quantities of the form:
1
, with (x) = 1+ux+vx2. (24)
u,v
(x) P
u,v
P
These quantities turn out to be the Laplace transforms of variables of the form ae +
be, with a,b > 0 constants and e,e two independent exponential variables. In this
′ ′
Subsection, we characterize the polynomials (x) such that this is so.
u,v
P
Lemma 2.2 a) A necessary and sufficient condition for 1/ to be the Laplace trans-
u,v
P
form of the law of ae+be, is:
′
u,v > 0 and ∆ u2 4v 0. (25)
≡ − ≥
b) Then, we obtain:
u √∆ u+√∆
a = − ; b = . (26)
2 2
Proof of Lemma 2.2
i) 1/ is the Laplace transform of ae+be, then:
u,v ′
P
(x) = (1+ax)(1+bx).
u,v
P
Both u = a+b and v = ab are positive.
Moreover, admits two real roots, thus ∆ u2 4v 0; i.e.: (25) is satisfied.
u,v
P ≡ − ≥
ii) Conversely, if the two conditions (25) are satisfied, then the 2 roots of the poly-
nomial are 1/a and 1/b. Hence, (x) = C(1+ax)(1+bx), where C is a constant.
u,v
− − P
However, from the definition of (24), we have: (0) = 1, hence C = 1. Thus,
u,v u,v
1/ is the Laplace transform oPf ae+be. P
u,v ′
P
6
iii) To show b), we note that:
1 1 u √∆ u+√∆
, = − − , − .
−a −b 2v 2v
( )
(cid:26) (cid:27)
as well as: u √∆ u+√∆ = 4v, which finishes the proof of the second part of the
−
Lemma. (cid:16) (cid:17)(cid:16) (cid:17)
3 A discussion of Proposition 1.2 in terms of the Cheby-
shev polynomials
3.1 Proof of Proposition 1.2
a) Assuming, to begin with, the validity of our Proposition 1.2, for any integer m, the
function ϕ should admit the following representation:
m
1
ϕ (x) = , (27)
m
D (x)
m
where
for m = 2n+1, D (x) = √1+xP (x), with P (x) = n (1+a x),
• m n n k=1 k
for m = 2n, D (x) = Q (x), with Q (x) = n (1+Qb x).
• m n n k=1 k
In particular, P and Q are polynomials of degreQe n, each of which has its n zeros, that
n n
is ( 1/a ;k = 1,2,...,n), resp. ( 1/b ;k = 1,2,...,n), on the negative axis R .
k k
It is−not difficult, from the explici−t expression of D (x) = 1 ((G (x))m +(G (x−))m), to
m 2 +
−
find the polynomials P and Q . They are given by the formulas:
n n
P (x) = n C2k+1(1+x)kxn k,
n k=0 2n+1 − (28)
(Qn(x) =Pnk=0C22nk(1+x)kxn−k.
In order to prove Proposition 1.2,Pwe shall make use of Chebyshev’s polynomials of the
first kind (see e.g. [Riv90] ex.1.1.1 p.5 or [KVA02] ex.25, p.195):
m m
y + y2 1 + y y2 1
− − −
T (y)
m ≡ (cid:16) p (cid:17) 2 (cid:16) p (cid:17)
cos(margcos(y)), y [ 1,1]
∈ −
cosh(margcosh(y)), y 1 (29)

≡ ≥
( 1)mcosh(margcosh( y)), y 1.
− − ≤

b) We now start the proof of Proposition 1.2 in earnest. First, we remark that:
1
ϕ (x) = , (30)
m
T √1+x
m
(cid:0) (cid:1)
7
hence:
D (x) = T √1+x , (31)
m m
(cid:16) (cid:17)
with x 1, thus we are interested only in the positive zeros of T , and we study sepa-
m
≥ −
rately the cases m odd and m even.
m = 2n+1
D (y) 1+yP (y) = T 1+y
2n+1 n 2n+1
≡
p (cid:16)p (cid:17)
and the zeros of T are: x = cos π2k 1 , k = 1,2,...,(2n + 1). However, x is
2n+1 k 22n−+1 k
positive if and only if k = 1,2,...,n, thus:
(cid:0) (cid:1)
π2k 1 π2k 1
y = x2 1 = cos2 − 1 = sin2 − ;k = 1,2,...,n.
k k − 2 2n+1 − − 2 2n+1
(cid:18) (cid:19) (cid:18) (cid:19)
Finally:
1
a = ; k = 1,2,...,n, (32)
k sin2 π2k 1
−
22n+1
and (cid:0) (cid:1)
n
x
P (x) = 1+ . (33)
n sin2 π2k−1 !
k=1 22n+1
Y
(cid:0) (cid:1)
m = 2n Similarly, we obtain:
1
b = ; k = 1,2,...,n, (34)
k sin2 π2k 1
−
2 2n
and (cid:0) (cid:1)
n
x
Q (x) = 1+ . (35)
n sin2 π2k−1 !
k=1 2 2n
Y
(cid:0) (cid:1)
3.2 Search for the Lévy measure of ϕ and proof of Proposition
m
1.3
We have proved that ϕ is infinitely divisible. In this Subsection, we shall calculate its
m
Lévy measure. For this purpose, we shall make use of the following (recall that e , k n
k
≤
are n independent exponential variables, with expectation 1):
Lemma 3.1 With (c , k = 1,2,...,n) denotinga sequenceof positiveconstants, n 1
k k=1 (1+ckx)
is the Laplace transform of n c e , which is an infinitely divisible random variable with
k=1 k k Q
Lévy measure:
P
n
dz
e−z/ck .
z
k=1
X
8
Proof of Lemma 3.1
Using the elementary Frullani formula (see e.g. [Leb72], p.6), we have:
n n n
1 dy
= exp log(1+ckx) = exp ∞ e−y 1 e−ckxy
(1+c x) − − y −
k=1 k ( k=1 ) ( k=1Z0 )
Y X X (cid:0) (cid:1)
n
dz
z==cky exp ∞ e−z/ck 1 e−xz ,
− z −
( k=1Z0 )
X (cid:0) (cid:1)
which finishes the proof.
We return now to the proof of Proposition 1.3 and we study separately the cases m
odd and m even and we apply Lemma 3.1 with c = a and c = b respectively.
k k k k
m = 2n+1 Lemma 3.1 yields that, n 1 isthe Laplacetransformofaninfinitely
k=1 (1+akx)
divisible random variable with Lévy measure:
Q
n
dz
ν+(dz) = e−z/ak. (36)
z
k=1
X
Moreover:
n
1 dz 1 z
= exp ∞ 1 e xz exp , (37)
−
( nk=1(1+akx))1/n (−Z0 z − n k=1 (cid:26)−ak(cid:27))
(cid:0) (cid:1) X
and Q 1 , for n , converges to the Laplace transform of a variable which
(Qnk=1(1+akx))1/n → ∞
is a generalized Gamma convolution (GGC) with Thorin measure density:
n n
1 z 1 π2k 1
µ (z) = lim exp = lim exp z sin2 −
+
n→∞ n k=1 (cid:26)−ak(cid:27) n→∞ n k=1 (cid:26)− (cid:18)2 2n+1(cid:19)(cid:27)
X X
1 π v=πu 2 π/2
= du exp zsin2 u =2 dv exp zsin2(v)
− 2 π −
Z0 n (cid:16) (cid:17)o Z0
h==sin2v 1 1 dh e hz, (cid:8) (cid:9) (38)
−
π h(1 h)
Z0 −
which, following the notatpion in [JRY08], is the Laplace transform of the variable G
1/2
which is arc sine distributed on [0,1].
m = 2n Lemma 3.1 yields that, n 1 is the Laplace transform of an infinitely
k=1 (1+bkx)
divisible random variable with Lévy measure:
Q
n n
dz dz π2k 1
ν (dz) = e−z/bk = exp z sin2 − . (39)
− z z − 2 2n
k=1 k=1 (cid:26) (cid:18) (cid:19)(cid:27)
X X
Moreover 1 , for n , converges to the Laplace transform of a GGC with
(Qnk=1(1+bkx))1/n → ∞
Thorin measure density:
µ (z) = µ (z). (40)
+
−
9
We now express the above results in terms of the Laplace transforms ϕ and ϕ˜ . Using
m m
the following result from [JRY08], p.390, formula (193):
dz
E exp xΓ G = exp t ∞ 1 e xz E exp zG
t 1/2 − 1/2
− − z − −
(cid:26) Z0 (cid:27)
(cid:2) (cid:0) (cid:0) (cid:1)(cid:1)(cid:3) 1 (cid:0) (cid:1) (cid:2) (cid:0) (cid:1)(cid:3)
= (41)
2t
√1+x+√x
with 2t = m = π , with c a positi(cid:0)ve constant, t(cid:1)ogether with (38) and (40), we obtain
2cε
(18).
Remark: The natural question that arises now is whether the results of Proposition
1.2 could be generalized for every m > 0 (not necessarily an integer), in other words
wether ϕ (x) = 2 is the Laplace transform of a generalized Gamma con-
m (G+(x))m+(G−(x))m
volution (GGC, see [Bon92] or [JRY08]), that is:
ϕ (x) = E e xΓm , (42)
m −
(cid:2) (cid:3)
with
(law) ∞
Γ = f (s)dγ , (43)
m m s
Z0
where f : R R and γ is a gamma process.
m + + s
→
This conjecture will be investigated in a sequel of this article in [Vak12].
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