Chung's law for homogeneous Brownian functionals

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Preprint submitted on 23 Oct 2007

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Aimé Lachal, Thomas Simon

To cite this version:

hal-00143623, version 2 - 23 Oct 2007

AIM

ELACHALANDTHOMASSIMON

Abstra t. Considerthefirstexittime

T

a,b

fromafiniteinterval

[−a, b]

foranhomogeneous flu tuating fun tional

X

of alinear Brownian motion. We showthe existen e of a finite positive onstant

K

su h that

lim

t→∞

t

−

₁

log P[T

ab

> t] = −K.

FollowingChung'soriginalapproa h[8℄,wededu ea"liminf"lawoftheiteratedlogarithm forthetwo-sided supremumof

X

. Thisextends andgives anewpoint ofviewonaresult ofKhoshnevisanandShi[12℄.

ÏîñâÿùàåòñÿÅíçî Îðñèíãåðó â÷åñòü åãî 60-ëåòèÿ

1. Introdu tion

Let

{B

t

, t ≥ 0}

be a linear Brownian motion starting at 0 and

X = {X

t

, t ≥ 0}

be the

homogeneousflu tuating additivefun tional defined by

X

t

=

Z

t

0

V (B

s

) ds,

t ≥ 0,

where

V (x) = x

α

if

x ≥ 0

and

V (x) = −λ|x|

α

if

x ≤ 0,

forsomefixed

α, λ > 0.

Thepro ess

X

appearsinmathemati alphysi sasthesolutionofageneralizedLangevinequationinvolving aharmoni os illatordriven byawhitenoise, andwe referto[14℄and thereferen es therein formoredetailsonthissubje t. Noti ethat

X

is

(1+α/2)

-self-similar,buthasnostationary in rements. In the ase

α = λ = 1

,it is the integrated Brownianmotion:

X

t

=

Z

t

0

B

s

ds,

t ≥ 0,

andalsoaGaussianpro ess. However, inthe other ases, itisnot Gaussiananylonger. For every

a, b > 0

onsider the bilateralexit time

T

ab

= inf{t > 0, X

t

6∈ (−a, b)}.

As arule, studyingthe law of

T

ab

is adiffi ultissue be ause

X

alone isnot Markov, sothat no spe tral theory is available. We referhowever to [14℄ and [15℄ for several distributional

2000Mathemati sSubje tClassifi ation. 60F99,60G17,60G18,60J55,60J65.

properties of the bivariaterandom variable

(T

ab

, B

T

ab

)

and for the solutionto the two-sided exitproblem,i.e. the omputationofthe probability

P

[X

T

ab

= a] .

In [14℄,itwas alsoshown

thatthevariable

T

ab

hasmomentsofanypower,andanexpli itupperboundwasgivenonthe latter -see Proposition 7.1therein. Before this, the upper tailsof

T

ab

inthe ase

α = λ = 1

had been pre iselyinvestigated in [12℄,with anelegant argument relyingon Chung's law of the iteratedlogarithm. Thisresult was then generalizedin[18℄ toa broad lass of Gaussian and sub-Gaussian pro esses, with a different method relying on wavelet de omposition. In this paper,we aimatextendingthe resultsof[12℄ tothe abovenon-Gaussianfun tionals

X,

with a moreelementary proof:

Theorem. For every

a, b > 0

there exists a finite positive onstant

K

su hthat

(1.1)

lim

t→∞

t

−1

_{log P[T}

ab

> t] = −K.

This exponential tail behaviour is typi al for exit-times from a finite interval for self-similar random pro esses. A tually, in most examples available, it appears that the upper tails of the variable

T

ab

are those of an exponential random variable. Some omments on this somewhat intriguinguniversal behaviourare giveninthe lastse tionof [18℄ inthe ase of a sub-Gaussian symmetri pro ess exiting a symmetri interval. See however Example 3.3 in [20℄, where the tail behaviour is shown to be subexponential. Noti e also that the upper tails of the unilateral exit time

T

a∞

of

X

had been thoroughly studied in [10, 11℄ and exhibit an entirely different, polynomial, behaviour whi h again in the framework of self-similarrandom pro esses is typi al for exit-timesfrom asemi-finite interval.

Taking

a = b = 1

, the estimate (1.1) entails by self-similarity that there exists a finite positive onstant

K

′

su h that (1.2)

lim

ε→0

ε

−2/(α+2)

_{log P[||X||}

∞

< ε] = −K

′

,

standard approa h viewing Chung's LIL asa onsequen e of (1.2). Introdu e the notations

X

_t

∗

= sup{|X

s

|, s ≤ t}

and

f (t) = (t/ log log t)

(α+2)/2

for every

t > e,

and set

K

1

for the onstant appearing in(1.1) when

a = b = 1.

Corollary (Chung's law of the iteratedlogarithm). Onehas

lim inf

t→+∞

X

∗

t

f (t)

= K

(α+2)/2

1

a.s.

Noti e that if we introdu ethe familyof time-stret hed fun tionals

X

_t

n

=

X

nt

(n/ log log n)

(α+2)/2

,

t ∈ [0, 1]

for every

n ≥ 3,

then by a straightforward monotoni ity argument our Chung's LIL is equivalent to

lim inf

n→+∞

||X

n

_||

∞

= K

(α+2)/2

1

a.s.

From this fa t and in the spirit of Wi hura's fun tional LIL, it is an interesting question to determine the luster set of the family of pro esses

{X

n

_{, n ≥ 1}}

for the weak topology. This was indeed re ently investigated by Lin and Zhang [19℄ for

m−

fold integrated Brow-nian motion, yielding Chung's LIL for these pro esses as a orollary - see Theorem 1.1 and Corollary 1.1therein. However, in our framework the non-linearity of the kernel

x 7→ V (x)

andthenon-Gaussianityof

X

makesthe situationsignifi antlymore ompli atedingeneral, as itwillalready appear inour proof. Settingnow

˜

X

_t

n

=

X

nt

(n log log n)

(α+2)/2

,

t ∈ [0, 1]

for every

n ≥ 3,

our result reads

lim inf

n→+∞

(log log n)

α+2

_{|| ˜}

_X

n

_||

∞

= K

(α+2)/2

1

a.s.

From this fa t and in the spirit of Strassen's fun tional LIL, it is somewhat tantalizing to determine the set of fun tions

f

su h that

(1.3)

lim inf

n→+∞

(log log n)

α+2

_{|| ˜}

_X

n

_{− f ||}

∞

a.s. exists, as anexpli it fun tion of

f

and

K

1

. In the ase of Brownian motion,this (hard) problem had been initiated by Cs
aki [9℄ and De A osta [1℄, hingingupon shifted Brownian small balls. Of ourse, before investigating (1.3) one should first determine the luster set forthe weak topologyofthe familyof pro esses

{ ˜

X

n

_{, n ≥ 1}.}

2. Proof of the theorem

Fix

a, b > 0

on e and for all,and introdu e the notation

T = T

ab

for on ision. For every

x, y ∈ R,

set

P

(x,y)

for the law of the strong Markov pro ess

t 7→ (B

t

, X

t

)

starting at

(x, y).

We keep the notation

P

= P

(0,0)

for brevity. Considering the fun tion

ϕ(t) = sup

P

_(x,y)

_{[T > t], (x, y) ∈ R × (−a, b) ,}

the simple Markov propertyyields for every

t, s ≥ 0

ϕ(t + s) = sup

P

_(x,y)

_{[T > s, T > t + s], (x, y) ∈ R × (−a, b)}

= sup

Z

R

Z

b

a

P

_(x,y)

_[(B

_s

_{, X}

_s

_{) ∈ du dv, T > s]P}

_(u,v)

_{[T > t], (x, y) ∈ R × (−a, b)}



≤ ϕ(t) × sup

Z

R

Z

b

a

P

_(x,y)

_[(B

_s

_{, X}

_s

_{) ∈ du dv, T > s], (x, y) ∈ R × (−a, b)}



≤ ϕ(t)ϕ(s),

so that the fun tion

ψ(t) = log ϕ(t)

is subadditive. Hen e, there exists

K ∈ [0, +∞]

su h that

lim

t→+∞

t

−1

_{ψ(t) = inf}

t>0

t

−1

_ψ(t)

= −K.

Besides from the se ond equality we see that

K > 0,

sin e the fun tion

ψ

is learly not identi allyzero. This entails

(2.1)

lim sup

t→∞

t

−1

log P[T > t] = −K < 0.

The remainder of the proofwill be givenin two steps. First, we will show the finiteness of

K

, whi h is usually the diffi ult part in small deviation problems. In the ase

α = λ = 1

, it had been obtained in [12℄ through an original yet lengthy argument relying on random normalization and Chung's LIL. Here we will provide two proofs whi h are onsiderably simpler. The first one adapts the elementary arguments of Lemma 1 in [5℄ to the two-dimensional Markovpro ess

(B, X)

,while the se ond one is based onthe time-substitution method whi h was used in [10℄ for unilateral passage times - let us stress that its main idea relying on the a.s. ontinuity of the Brownian paths was also impli itly used in [12℄ p. 4258 to obtainChung's LIL. The latter proof is slightlymore involved than the former, nevertheless it allowsto bound the onstant from above - see the Remark 1 below.

breaks down,sothat wehad touse morebare-hand estimates,followingroughly theoutline of Lemma 1in [5℄.

First proofof the finitenessof the onstant. Fixing

A < 0 < B

and

a < c < 0 < d < b,

introdu e the fun tions

ϕ(t) = inf

˜

P

(x,y)

[T > t], (x, y) ∈ [A, B] × [c, d]

and

Φ(t) = inf

P

_(x,y)

_[(B

_t

_{, X}

_t

_{) ∈ [A, B] × [c, d], T > t] , (x, y) ∈ [A, B] × [c, d] ,}

_{t ≥ 0.}

For every

(x, y) ∈ [A, B] × [c, d]

and

t, s ≥ 0

the simpleMarkov property entails

P

_(x,y)

_{[T > t + s] ≥ P}

_(x,y)

_[(B

_s

_{, X}

_s

_{) ∈ [A, B] × [c, d], T > t + s]}

=

Z

B

A

Z

d

c

P

_(x,y)

_[(B

_s

_{, X}

_s

_{) ∈ du dv, T > s] × P}

_(u,v)

_{[T > t]}

≥ P

(x,y)

[(B

s

, X

s

) ∈ [A, B] × [c, d], T > s] × ˜

ϕ(t)

≥ Φ(s) ˜

ϕ(t),

so that

ϕ(t + s) ≥ ˜

˜

ϕ(s)Φ(t)

for every

t, s ≥ 0.

In parti ular

ϕ(n) ≥ ˜

ϕ(n) ≥ Φ(1) ˜

ϕ(n − 1) ≥ . . . ≥ Φ(1)

n

ϕ(0) = Φ(1)

˜

n

for every

n ∈ N,

whi h entails

t

−1

_{ψ(t) ≥ log Φ(1)}

for every

t > 0,

sin e the fun tion

t 7→ t

−1

_ψ(t)

is de reasing. We finallyget

K ≤ − log Φ(1).

Now the fun tion

(x, y, t) 7→ P

(x,y)

[(B

t

, X

t

) ∈ [A, B] × [c, d], T > t]

is ontinuous on the ompa t

[A, B] × [c, d] × [0, 2]

,sin e it satisfies the heat equation

1

2

∂

2

∂x

2

+ V (y)

∂

∂y

=

∂

∂t

on

R

× (−a, b) × R

+

_.

In parti ular the fun tion

(x, y) 7→ P

(x,y)

[(B

1

, X

1

) ∈ [A, B] × [c, d], T > 1]

is ontinuousonthe ompa t

[A, B] × [c, d]

andsin eitisobviously everywhere positive,one

has

Φ(1) > 0,

whi h ompletes the proof.



Se ond proof of the finiteness of the onstant. Let

L = {L(t, x), t ≥ 0, x ∈ R}

be the lo al-timepro ess asso iated with

B

and

be the inverse lo al time of

B

at zero. It follows easily from the Markov property and a s alingargument that the pro ess

t 7→ (τ

t

, X

τ

t

)

isatwo-dimensional L
evy pro esssu h that

t 7→ τ

t

is a

(1/2)−

stable subordinator and

Y : t 7→ Y

t

= X

τ

t

a

1/(α + 2)

-stable pro ess.

Introdu ing

Θ = inf{t > 0, X

τ

t

6∈ (−a, b)},

the a.s. ontinuityof Brownian traje toriesyieldsthe key-inequality

(2.2)

T ≥ τ

Θ−

a.s.

As inthe proofof Theorem B in [22℄ we now de ompose, for every

c > 0,

P

_{[Θ > t] ≤ P [τ}

_t

_{< ct] + P [Θ > t, τ}

_t

_{≥ ct]}

≤ P

τ

1

< ct

−1



+ P [τ

Θ−

≥ ct]

≤ P

τ

1

< ct

−1



+ P [T ≥ ct]

where we used the 2-self-similarity and the a.s. in reasingness of

τ

in the se ond line, and (2.2)inthe third. ByProposition VIII.3in[4℄and as alingargument,there exists

K

0

finite su h that

lim

t→∞

t

−1

_{log P [Θ > t] = −K}

0

.

By Theorem 5.12.9in [7℄there exists

K

c

→ +∞

as

c → 0

su h that

lim

t→∞

t

−1

_{log P}

_τ

1

< ct

−1



= −K

c

.

Taking

c

smallenough and puttingeverything together yields

lim inf

t→∞

t

−1

_{log P[T > t] ≥ −K}

0

/c > −∞,

whi h entails

K < +∞

as desired.



Remark 1. The positivity parameter

P

[Y

1

> 0]

of the non ompletely asymmetri L
evy

1/(α + 2)

-stable pro ess

Y

had been omputed in [11℄ - see Remark 4 therein. This makes

itpossibletobound fromabovethe onstant

K

0

expli itly: when

λ = 1

i.e.

Y

is symmetri , this an be done in subordinating

Y

to some Brownian motion - see Theorem 4 in [3℄ or Proposition 8 in [21℄ - whereas when

λ 6= 1

, the same method works insubordinating

Y

to some ompletelyasymmetri stablepro esswithinfinitevariation-see Exer iseVIII.1in[4℄ - and using the expli it al ulations of [5℄ inthe ompletely asymmetri ase. On the other hand, the s aling parameter of the stable subordinator

τ

is expli it, so that the onstants

proof allows to exhibit an expli it upper bound on

K

, whi h we will however not in lude here for the sake of brevity. Noti e that in the ase of integrated Brownian motion in a symmetri interval, a lower bound had been given in [12℄, Remark 1.4. Re all also that in the non- ompletely asymmetri framework, the exa t omputation of

K

0

is along-standing and hallenging problem- see [5,3, 2℄ and the referen es therein.

Proof of the existen e of the onstant. Suppose first that

α = λ = 1

and

a = b

. Then by self-similarityand by linearity of the integral one has, for every

x, y ∈ R

and

t > 0

P

_(x,y)

_{[T > t] = P}

_(xt

_−1/2

_,yt

_−3/2

₎

||X||

_∞

_{< at}

−3/2



_{= P}

||X + f

x,y,t

_||

_∞

_{< at}

−3/2



where

||.||

∞

stands for the supremum norm over

[0, 1]

and

f

x,y,t

_{: u 7→ yt}

−3/2

_{+ uxt}

−1/2

_.

Hen e, Anderson's inequality - see e.g. (7.5) in[16℄ - entails

P

_(x,y)

_{[T > t] = P}

_{||X + f}

x,y,t

_||

_∞

_{< at}

−3/2



≤ P

||X||

∞

< at

−3/2



= P[T > t],

so that

ϕ(t) = P[T > t]

for every

t > 0,

and (2.1) is a true limit. Unfortunately, this

simpleGaussianargument annotbeusedingeneral,andwe willhave touse alenghtieryet elementary method, whi h willbe divided into three lemmas. For every

ε > 0,

introdu e

T

ε

= inf {t > 0, X

t

∈ (−a + ε, b − ε)} .

/

Lemma 2. There exist

c

1

, c

2

, K > 0

su h that for every

ε

small enough and every

t

large enough, there exist

x

ε

t

∈ (−K, K)

and

y

ε

t

∈ (−a + ε, b − ε)

su h that

(2.3)

P

(x

ε

t

,y

ε

t

)

[T

ε

> t] ≥ c

2

e

−K(1+c

1

ε)t

_.

Proof. For every

t > 0,

we an hoose

(x

ε

t

, y

t

ε

) ∈ R × (−a + ε, b − ε)

su h that

(2.4)

P

(x

ε

t

,y

ε

t

)

[T

ε

> t + 1] ≥

1

2

sup

P

(x,y)

[T

ε

> t + 1] , (x, y) ∈ R × (−a + ε, b − ε) .

Besides, by s aling and translation we have forevery

(x, y) ∈ R × (−a, b)

P

_(x,y)

_[T

_ε

_{> t + 1] = P}

_(x

ε

,y

ε

)

[T

ab

> t

ε

]

with the notations

x

ε

= x/(1 − 2ε/(a + b))

1/(α+2)

_{, y}

ε

= (y − (b − a)/2)/(1 − 2ε/(a + b)) +

(b − a)/2,

and

t

ε

= (t + 1)/(1 − 2ε/(a + b))

2/(α+2)

_.

Hen e, hoosing some onstant

c

1

> 0

su h that

1 + c

1

ε > (1 − 2ε/(a + b))

−2/(α+2)

sup

P

_(x,y)

_[T

_ε

_{> t + 1] , (x, y) ∈ R × (−a + ε, b − ε)}

= sup

P

_(x,y)

_[T

_ab

_{> t}

_ε

_{] , (x, y) ∈ R × (−a, b)}

≥ sup

P

_(x,y)

_[T

_ab

_{> (1 + c}

₁

_{ε)(t + 1)] , (x, y) ∈ R × (−a, b)}

≥ e

−K(1+c

1

ε)(t+1)

for

t

large enough, so that by (2.4), (2.5)

P

(x

ε

t

,y

ε

t

)

[T

ε

> t + 1] ≥ c

2

e

−K(1+c

1

ε)t

for

t

large enough with

c

2

= e

−K(1+c

1

)

/2.

Set now

K = 2(1 ∨ λ

−1/α

_{)(a + b)}

1/α

_,

fix

ε > 0

and

t

large enough. If

|x

ε

t

| < K,

then by (2.5)

P

_(x

ε

t

,y

t

ε

)

[T

ε

> t] ≥ P

(x

t

ε

,y

ε

t

)

[T

ε

> t + 1] ≥ c

2

e

−K(1+c

1

ε)t

and(2.3)holdssin ene essarily

y

ε

t

∈ (−a+ε, b−ε).

If

x

ε

t

≥ K,

thenintrodu ingthestopping time

S = inf {s > 0, B

s

= K/2} ,

the definition of

K

and the strong Markov property at

S

entail

P

_(x

ε

t

,y

ε

t

)

[T

ε

> t + 1] = P

(x

ε

t

,y

ε

t

)

[S ≤ 1, T

ε

> t + 1] .

Indeed, if

S > 1

then

B

s

≥ K/2

forevery

s ≤ 1,

sothat

X

1

> −a + ε + (K/2)

α

_{> b − ε}

and

T

ε

< 1.

Hen e,

P

_(x

ε

t

,y

ε

t

)

[T

ε

> t + 1] ≤ E

(x

ε

t

,y

t

ε

)

1

{S≤1,X

S

∈(−a+ε,b−ε)}

P

(K/2,X

S

)

[T

ε

> t]



≤ P

(x

ε

t

,y

ε

t

)

[S ≤ 1] sup

P

(K/2,y)

[T

ε

> t] , y ∈ (−a + ε, b − ε)

≤ sup

P

_(K/2,y)

_[T

_ε

_{> t] , y ∈ (−a + ε, b − ε) .}

In parti ular, setting

c

′

2

= e

−K(1+c

1

)

/4

and

x

˜

ε

t

= K/2,

we see by (2.5) that there exists

˜

y

ε

t

∈ (−a + ε, b − ε)

su h that

P

_(˜

_x

ε

t

,˜

y

ε

t

)

[T

ε

> t] ≥ c

′

2

e

−K(1+c

1

ε)t

.

The ase

x

ε

t

≤ −K

an be handled similarly,and the proofof Lemma 2is omplete.



Wenow needtoshowthat the estimate(2.3)remainstrueina suitableneighbourhood of

(x

ε

t

, y

t

ε

)

. Fixing

ε > 0

and

(x

ε

t

, y

ε

t

) ∈ (−K, K) × (−a + ε, b − ε)

as above for

t

large enough,

The key-feature of this neighbourhood is that its volume does not depend on

t

and for this reason, the proofof the following lemma isa bitte hni al:

Lemma 3. There exists

c

3

> 0

su h that for every

ε > 0

inf

P

_(x,y)

_{[T > t] , (x, y) ∈ V}

ε

t

> c

3

e

−K(1+c

1

ε)t

,

t → +∞.

Proof. First, bytranslation invarian e,one has

(2.6)

inf

P

(x

ε

t

,y)

[T > t] , y ∈ [y

ε

t

− ε, y

t

ε

+ ε]

≥ P

(x

ε

t

,y

ε

t

)

[T

ε

> t] ≥ c

2

e

−K(1+c

1

ε)t

as

t → +∞,

where

c

2

is the onstant in (2.3). Suppose now

x

ε

t

≥ 0

and introdu e the stoppingtime

σ

ε

_t

= inf {s > 0, B

s

= x

ε

t

} .

For every

(x, y) ∈ V

ε

t

one gets fromthe Markov property

P

_(x,y)

_{[T > t] ≥ P}

_(x,y)

_{[T > t > σ}

_t

ε

_]

=

Z

t

0

Z

b

a

P

_(x,y)

σ

ε

t

∈ ds, X

σ

ε

t

∈ dv

 P

(x

ε

t

,v)

[T > t − s]

≥

Z

t

0

Z

b

a

P

_(x,y)

σ

ε

t

∈ ds, X

σ

ε

t

∈ dv

 P

(x

ε

t

,v)

[T > t]

≥

Z

t

0

Z

y

ε

_t

+ε

y

ε

t

−ε

P

_(x,y)

σ

ε

t

∈ ds, X

σ

ε

t

∈ dv × inf P

(x

ε

t

,z)

[T > t] , |z − y

ε

t

| ≤ ε

≥ c

2

P

(x,y)

σ

t

ε

≤ t,





X

_σ

ε

t

− y

ε

t





≤ ε e

−K(1+c

1

ε)t

,

where we used (2.6) in the laststep. Hen e, sin e

[−ε/2, ε/2] ⊂ [y

ε

t

− y − ε, y

t

ε

− y + ε],

it suffi es tobound

P

_(x,y)

_[σ

_t

ε

_{≤ t,}



_X

_σ

ε

t

− y

ε

t





≤ ε] ≥ P

_(x,0)

σ

_t

ε

≤ t,



X

σ

ε

t





≤ ε/2



frombelow. Nowsin e

α ≥ 0,

there exists

M > 0

su h that

(2.7)

|u + v|

α

≤ M(|u|

α

+ |v|

α

)

for every

u, v ∈ R,

so that

P

(x,0)

a.s.





X

_σ

ε

t





≤ Mσ

ε

_t



x

α

+



B

_σ

∗

ε

t



α



,

with the notation

B

∗

t

= max{|β

s

| , s ≤ t}

for every

t ≥ 0

, where

{β

s

, s ≥ 0}

is a Brownian

motion starting at zero. With the notations

δ

ε

θ

z

= inf{s > 0, β

s

= z}

forevery

z ∈ R,

this entails

P

_(x,0)

σ

ε

t

≤ t,





X

_σ

ε

t





≤ ε/2



≥ P

h

ρ

ε

_t

≤ t, ρ

ε

_t



B

_ρ

∗

ε

t



α

+ x

α



≤ ε/2M

i

≥ P

h

ρ

ε

_t

≤ t, ρ

t



B

_ρ

∗

ε

t



α

≤ ε/4M, ρ

ε

_t

x

α

≤ ε/4M

i

≥ P

h

ρ

ε

_t

≤ t ∧ (ε/4Mx

α

), B

_ρ

∗

ε

t

≤ x

i

≥ P [ρ

ε

_t

≤ t ∧ (ε/4Mx

α

) ∧ θ

x

]

where in the fourth line we used the obvious fa t that

ρ

ε

t

≤ θ

−x

a.s. By s aling and sin e

0 ≤ δ

ε

t

≤ x,

we know that

(ρ

ε

_t

, θ

x

)

d

= (δ

ε

_t

)

2

θ

−1

, θ

x/δ

ε

t

and

θ

x/δ

ε

t

≥ θ

1

a.s. By Lemma2 we now that

x ≤ K + 1

and sin e

δ

ε

t

∈ [0, 1],

we finally get

P

_(x,0)

σ

ε

t

≤ t,





X

_σ

ε

t





≤ ε/2



≥ P



θ

−1

≤

t ∧ (ε/4Mx

α

₎

(δ

ε

t

)

2

∧ θ

x/δ

t

ε



≥ P [θ

−1

≤ (ε/4M|K + 1|

α

) ∧ θ

1

] ,

whi h finishes the proofof Lemma 3 be ause the right-hand side does not depend on

t

.



Ourlastlemmaisintuitivelyobvious,butwewillgiveaproofforthesakeof ompleteness.

Lemma 4. For every

ε > 0

, there is a onstant

c

ε

su h that

P

_[(B

₁

_{, X}

₁

_{) ∈ V}

_t

ε

_{, T > 1] > c}

_ε

for every

t

large enough.

Proof. Fix

ε > 0

and define

K

asinLemma2. For every

(x, y) ∈ (−K, K) × (−a + ε, b − ε)

, there exists a pie ewise linear fun tion

f

x,y

_{: [0, 1] → R}

starting at zero su h that

f

x,y

1

=

x + 1/2

if

x ≥ 0

and

f

x,y

1

= x − 1/2

if

x < 0

,

g

x,y

1

= y

and

τ

x,y

_{> 1,}

with the notations

g

t

x,y

=

Z

t

0

V (f

_s

x,y

) ds, t ≥ 0,

and

τ

x,y

_{= inf{t > 0, g}

x,y

t

6∈ (−a, b)}.

Besides, sin e from(2.7)we know that a.s.

||X − g

x,y

_||

∞

≤ M||B − f

x,y

||

α

∞

for every

(x, y)

,

by the definition of

V

ε

t

we have for every

t > 0

||B − f

x

ε

t

,y

ε

t

_||

∞

< (ε/2M)

1/α

⊂ {(B

1

, X

1

) ∈ V

t

ε

, T > 1} .

On the one hand,by ompa ity, we an learly hoose the fun tions

f

On the otherhand, the Onsager-Ma hlup formula - see e.g. Theorem 7.8in[16℄- entails

P

||B − f

x

ε

t

,y

t

ε

_||

∞

< (ε/2M)

1/α



≥ c

′

_ε

exp



−

1

2

Z

1

0

df

x

ε

t

,y

t

ε

s

ds

!

2

ds





≥ c

′

_ε

e

−M/2

where

c

′

ε

= P

||B||

∞

< (ε/2M)

1/α



. Putting everything together and setting

c

ε

= c

′

ε

e

−M/2

ompletes the proof of Lemma4.



We an now on lude the proof of the existen e of the onstant. Fix

ε > 0

, take

t > 0

large enoughand suppose first that

x

ε

t

≥ 0

. Bythe Markov property attime 1,

P

_{[T > t] ≥ P[(B}

₁

_{, X}

₁

_{) ∈ V}

ε

t

, T > t]

≥ P[(B

1

, X

1

) ∈ V

t

ε

, T > 1] × inf

P

(x,y)

[T > t − 1], (x, y) ∈ V

t

ε

≥ c

ε

inf

P

(x,y)

[T > t], (x, y) ∈ V

t

ε

≥ c

ε

c

3

e

−K(1+c

1

ε)t

,

wherewe usedLemma4inthethird lineand Lemma3inthefourth. The ase

x

ε

t

< 0

being

handled analogously,we finallyobtain, for every

ε > 0,

lim inf

t→+∞

1

t

log P[T > t] ≥ −K(1 + c

1

ε),

whi h ompletes the proofin letting

ε

tendto 0.



Remarks 5. (a) By the self-similarity of

B

, one an a tually extend the definition of the fun tionals

X

toevery

α > −1

withanabsolute onvergen eoftheintegral. Inthesymmetri

ase

λ = 1,

itis even possibleto extend this definitiontoevery

α ∈ (−3/2, 1],

viewing

X

as

a Cau hyprin ipal valuepro ess:

X

t

= lim

ε→0

Z

t

0

1

{|B

s

|>ε}

|B

s

|

α

_sgn(B

s

) ds = lim

ε→0

Z

R

1

{|x|>ε}

|x|

α

sgn(x)(L(t, x) − L(0, x)) dx

where in the se ond equality we used the o upation formula and where the se ond limit exists a.s. sin e the map

x 7→ L(t, x)

is a.s.

η

-Holder for every

η < 1/2

. For

α = −1

the pro ess

X

is then up to a multipli ative onstant the Hilbert transform of

L

while for

Above, the subadditivity argument and the finiteness of the onstant do not rely on the spe ifi valueof

α

,so that one gets with the same notations

−∞ < lim inf

t→∞

t

−1

_{log P[T}

ab

> t] ≤ lim sup

t→∞

t

−1

_{log P[T}

ab

> t] < 0,

whi h is a weaker version of our main result. However, the positivity assumption on

α

is ru ialfor Lemma 2whi h isthe key-step in our proofof the existen eof the onstant. We believe that the limitin(2.1) is alsoa true limitwhen

α

is negative, but the proofrequires probably less bare-hand arguments than ours.

(b) In the ase

α = λ = 1,

the pro ess

(B, X)

is a Gaussian diffusion and in this ase

it is known that the fun tion

f

t

: (x, y) 7→ P

(x,y)

[T > t]

is log- on ave for every

t > 0

-see e.g. Proposition 1.3 in [13℄. Hen e, in the ase of a symmetri interval, its maximum is attained in

(0, 0)

and this gives another proof of the existen e of the onstant. Despite Theorem1.2. in[13℄,ourintuitionisthatthefun tion

f

t

remainslog- on aveingeneral,but we were unable to prove this. If this were true, the existen e of the onstant would follow immediatelyinthe ase

λ = 1

and for a symmetri interval. Letus stress that the fun tion

f

t

already exhibits some on avity properties in the framework of non-Gaussiansymmetri stable pro esses [2℄.

3. Proof of the orollary

We will follow the outline of [12℄ se tions 2.4 and 2.5, whi h are themselves a variation onChung's original argument. First, arguingwith (1.2) and the first Borel-Cantellilemma exa tlyas inse tion 2.4of [12℄, one an show that

(3.1)

lim inf

t→+∞

X

∗

t

f (t)

≥ K

(α+2)/2

1

a.s.

and we leave the verifi ation to the reader (beware the minor orre tion

R → log R

on the last line p. 4258). Moreover, the arguments of se tion 2.3 in [12℄ applied to our L
evy

(1 + α/2)

-stable pro ess

Y : t 7→ X

τ

t

entail withoutmajor modifi ation

(3.2)

lim inf

t→+∞

X

∗

t

f (t)

< ∞

a.s.

By the 0-1law, we know that the liminf onthe left-hand side is a.s. deterministi , so that Chung's law holds by (3.1) and (3.2), with an unknown finite positive onstant. Noti e in passingthat(3.1)and(3.2)givealsoathirdproofofthefinitenessof

K

inthesymmetri ase

However, toprove that (3.3)

lim inf

t→+∞

X

∗

t

f (t)

≤ K

(α+2)/2

1

a.s.

wewillhavetomodifyslightlytheargumentsofse tion2.5in[12℄,sin ethekernel

x 7→ V (x)

is not linear in general. Fixing a small

ε > 0,

introdu e the numbers

t

n

= n

4n

_{, s}

n

= n

4n+3

and

y

n

= (1 + 2ε)K

(α+2)/2

1

f (t

n

)

for every

n ≥ 1

. Define the sequen e of stoppingtimes

S

0

= 0

and

S

n

= inf{t > t

n

+ S

n−1

, B

t

= 0},

n ≥ 1.

Finally, onsider the events

E

n

=



sup

S

n

≤t≤t

n+1

+S

n









Z

t

S

n

V (B

s

) ds









< y

n+1



and

F

n

= {S

n

< s

n

+ S

n−1

}

for every

n ≥ 1.

On the one hand, setting

r

n

= s

n

− t

n

,

P

x

for the law of

B

starting at

x

, and resuming the notations of Lemma 3, the strong Markov property, the symmetry of

Brownian motionand a s aling argument yield

P

_[F

c

n

] =

Z

R

P

B

_S

n−1

+t

n

∈ dx

 P

x

[θ

0

> r

n

]

=

Z

R

P

_[B

_t

n

∈ dx] P [B

t

< |x|, ∀ t ≤ r

n

]

=

Z

R

P

_[B

₁

_{∈ du] P}

h

B

t

< |u|pt

n

r

n

−1

, ∀ t ≤ 1

i

∼ cpt

n

r

−1

n

∼ cn

−3/2

,

n → ∞

for some positive finite onstant

c

, sothat

X

n≥1

P

_[F

c

n

] < +∞.

By the Borel-Cantellilemma,for almost every

ω

there exists

n

0

(ω)

su h that

S

n

(ω) < S

n

0

(ω)

(ω) + s

n

0

(ω)+1

+ · · · + s

n

for every

n > n

0

(ω).

Hen e, by the definition of

s

n

,

there exists

n

1

(ω) > n

0

(ω)

su h that (3.4)

S

n

(ω) < 2s

n

for every

n ≥ n

1

(ω).

On the other hand, sin e

it followsreadily from the strong Markov property and the definition of

S

n

that the events

E

n

are mutually independent. Besides, using (1.2) and reasoning exa tly as in[12℄ p. 4259

entails

X

n≥1

P

_[E

_n

_{] = +∞.}

By the se ond Borel-Cantelli lemma, an infinity of events

E

n

o ur a.s. and by (3.4), we know that a.s. eventually

[2s

n

, t

n+1

] ⊂ [S

n

, t

n+1

+ S

n

]

. This entails

sup

2s

n

≤t≤t

n+1









Z

t

S

n

V (B

s

) ds









< (1 + 2ε)K

(α+2)/2

1

f (t

n+1

)

i.o.

By Khint hine'sLIL for Brownian motion,

lim inf

n→+∞

1

f (t

n+1

)









Z

2s

n

S

n

V (B

s

) ds









≤ lim inf

n→+∞

2(1 ∨ λ)s

n

f (t

n+1

)

B

∗

_s

_n

α

= 0

a.s.

Putting everythingtogether and letting

ε → 0

yields

(3.5)

lim inf

n→+∞

1

f (t

n

)

sup

2s

n−1

≤t≤t

n









Z

t

2s

n−1

V (B

s

) ds









≤ K

(α+2)/2

₁

a.s.

Finally, we know from (3.2) that

X

∗

2s

n−1

f (t

n

)

→ 0

a.s.

whi h together with (3.5), the usual monotoni ity argument, and the fa t that a.s.

X

_t

∗

_n

≤ X

_2s

∗

_n−1

+

sup

2s

n−1

≤t≤t

n









Z

t

2s

n−1

V (B

s

) ds









,

yields(3.3) as desired.



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