Local limit theorems for Brownian additive functionals and penalisation of Brownian paths, IX

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Local limit theorems for Brownian additive functionals and penalisation of Brownian paths, IX

Bernard Roynette, Marc Yor

To cite this version:

Bernard Roynette, Marc Yor. Local limit theorems for Brownian additive functionals and penalisation of Brownian paths, IX. 2008. �hal-00257593�

Local limit theorems for Brownian additive functionals and penalisation of Brownian paths, IX

B. Roynette

, M. Yor

12/02/08

(1) Institut Elie Cartan, Universit´e Henri Poincar´e, B.P. 239, 54506 Vandoeuvre les Nancy Cedex

(2) Laboratoire de Probabilit´es et Mod`eles Al´eatoires, Universit´e Paris VI et VII, 4 place Jussieu - Case 188 F - 75252 Paris Cedex 05

(3) Institut Universitaire de France

AbstractWe obtain a local limit theorem for the laws of a class of Brownian additive func- tionals and we apply this result to a penalisation problem. We study precisely the case of the additive functional : !

A⁻_t :="t

01_Xs<0ds, t≥0#

. On the other hand, we describe Feynman- Kac type penalisation results for long Brownian bridges thus completing some similar previous study for standard Brownian motion (see [RVY,I]).

Key wordsLimit theorems for additive functionals, Feynman-Kac functionals, long Brown- ian bridges.

2000 Mathematics Subject Classification 60 F 17, 60 G 44, 60 J 25, 60 J 35, 60 J 55, 60 J 57, 60 J 60, 60 J 65.

0 Notations and introduction

0.1 Notations

• ! Ω,$

X_t,Ft

%

t≥0,F∞=∨t≥0Ft, P_x(x∈R)#

denotes the canonical realisation of the one- dimensional Wiener process. Ω =C$

[0,∞[−→R%

is the space of continuous functions on [0,∞[,$

Xt, t≥0%

the coordinates process on this space,$

F^t, t≥0%

its natural filtration and $

P_x, x ∈ R%

the family of Wiener measures on $ Ω,F∞

%, with P_x(X₀ = x) = 1.

Whenx= 0, we write simply P forP₀.

• We denote by $

L^x_t, t ≥ 0, x ∈ R%

the jointly continuous family of the local times of

$X_t, t≥0%

. We denote $

L_t, t≥0% for$

L⁰_t, t≥0%

, the (continuous) local time process at level 0 and by$

τ_l, l≥0%

its right-continuous inverse : τ_l:= inf&

s >0;L_s> l' $ l≥0%

(0.1)

• To q a positive Radon measure on R, q '= 0, we associate the continuous additive functional :

A^q_t :=

(

R

L^x_tq(dx) (0.2)

Whenqadmits a density with respect to Lebesgue measure, we keep the former notation by still writingq for the density; we have :

A^q_t = ( t

0

q(Xs)ds (0.3)

from the occupation density formula for Brownian motion.

Throughout the following, we shall assume thatqsatisfies one of the three following hypothe- ses :

H1. (The integrable case) (

R

$1 +|x|%

q(dx)<∞. H2. (The left unilateral case)

( 0

−∞

$1 +|x|%

q(dx) < ∞ and there exists α < 1 such that lim

x→∞x^2αq^(a)(x)≥b >0 where q^(a) denotes the absolutely continuous part ofq.

H3. (The right unilateral case) ( _∞

0

$1 +|x|%

q(dx) < ∞ and there exists α < 1 such that lim

x→−∞|x|^2αq^(a)(x)≥b >0.

Of course, if the pair$

(X_t, A^q_t), t≥0%

satisfies H2 (resp. H3), then the pair$

(−X_t, A^q_t), t≥ 0% satisfies H3 (resp. H2).

0.2 Introduction 0.2.1

In [RVY, I], we obtained the following results : i) Under H1, H2 or H3, for anyλ >0 :

tlim→∞

√t Ex

) e

$−^λ₂A^q_t%*

=ϕ_λq(x) (0.4)

(Later, we shall give other presentations ofϕλq).

ii) For any s≥0 and Λ_s∈ Fs :

tlim→∞

E) 1_Λse

$−^λ2A^q_t%*

E) e

$−^λ₂A^q_t%* :=Q^(λq)$ Λs%

(0.5)

where formula (0.5) induces a probabilityQ^(λq)on! Ω,F∞

#

(see [RVY,I] for more details about this penalisation result).

The first part of this work consists in :

• Using the result i) to obtain a limit theorem relative to the law of the additive functional$ A^q_t, t≥0%

. This is the content of Theorem 1.1.

• Obtaining a penalisation result, which is more general than (0.5) i.e, by replacing the exponential function x −→ e^−λx2 by a more general function. This is the content of Theorem 1.5.

0.2.2

In Section 2 of this work, we study in detail the situation whereq= 1_]−∞,0] i.e, A^q_t := A⁻_t := "_t

0 1_Xs<0ds. In particular, we prove a penalisation theorem for long Brownian bridges : this is the content of Theorem 2.1

0.2.3

Section 3 of this work is devoted to the study of Feynman-Kac penalisation for long Brownian bridges, which generalizes what we have done in [RVY,I] for standard Brownian motion. This is the content of Theorem 3.1.

To summarize, this work extends, in the above directions, our preceding work [RVY,I].

1 A local limit theorem for the laws of some Brownian additive functionals and a penalisation result

1.1 A local limit theorem

Theorem 1.1. Let q satisfy one of the hypotheses H1, H2 or H3, and let (A^q_t, t ≥ 0) be defined by$

0.2) $

or (0.3)%

. Then, for everyx∈R, there exists a positive, σ− finite measure νx, carried by R+, such that :

√t P_x(A^q_t ∈dz)−→

t→∞ν_x(dz) (1.1)

The convergence in (1.1) is understood in the following sense : for any functionf :R+−→R+

Borel, and sub-exponential i.e. : there exist two positive constants C1 and C2 such that : 0≤f(x)≤C₁e^−C2x

then

√t E_x+ f(A^q_t),

t−→→∞

(

R+

f(z)ν_x(dz) The measure ν_x is characterized by :

( _∞

0

e^−λ2yν_x(dy) =ϕ_λq(x) (1.2)

1.2 Proof of Theorem 1.1

We first begin with some precisions, taken from [RVY, I],$

see also S. Kotani, [K]%

aboutϕλq, which was defined from (0.3) but admits at least another characterization, namely :

ϕ_λq is the unique solution of the Sturm-Liouville equation :

ϕ^′′(dx) =λϕ(x)q(dx) (1.3)

This equation is taken in the sense of Schwartz distributions, and subject to the following boundary conditions :

Under H1. : ϕ^′(+∞) =−ϕ^′(−∞) = -2

π (1.4)

Under H2. : ϕ^′(−∞) =− -2

π andϕ(+∞) = 0 (1.5)

Under H3. : ϕ^′(+∞) = -2

π andϕ(−∞) = 0 (1.6)

Theorem 1.1. is now an immediate consequence of the next lemma.

Lemma 1.2. Under either of the hypotheses H1, H2, or H3, the function : λ−→ϕ_λq(x) (λ > 0) is, for any real x, completely monotone, i.e., it satisfies :

(−1)ⁿ ∂ⁿ

∂λⁿϕ_λq(x)≥0 (1.7)

Consequently, there exists a positive, σ-finite measureν_x, carried by R+, such that :

ϕ_λq(x) = ( _∞

0

e^−λ2zν_x(dz) (1.8)

We shall give two proofs for Lemma 1.2.

1.3 A first proof of Lemma 1.2

We define, for everyf :R+−→R and every real h'= 0 : D_hf(λ) := f(λ+h)−f(λ)

h (1.9)

Forf(λ) := exp−λ

2A^q_t, we get : (D_h)ⁿ(f)(λ) =e^−λA

q t 2

!e^−A

q t h

2 −1

h

#n

and, hence for allh'= 0 :

(−1)ⁿ(D_h)ⁿ(f)(λ)≥0 (1.10)

Consequently, taking the expectation of the LHS in (1.10), we obtain :

√t(−1)ⁿE_x+

(D_h)ⁿ!

exp−• 2A^q_t#*

≥0 (1.11)

Hence, from (0.4) :

√t(−1)ⁿE_x)

(D_h)ⁿ!

exp−• 2A^q_t#*

t−→→∞(−1)ⁿ(D_h)ⁿ$

ϕ_•q(x)% Thus :

(−1)ⁿ(D_h)ⁿ$

ϕ_•q(x)%

(λ)≥0 (1.12)

Lettingh→0 in (1.12), and using the fact that : D_hf−→

h→0f^′, we get : (−1)ⁿ ∂ⁿ

∂λⁿ

$ϕ_λq(x)%

≥0 (1.13)

1.4 A second proof of Lemma 1.2

We shall only give this second proof under the hypothesis H1 and for x = 0. In [RVY, I], Proposition 4.13, formula (4.43), we have obtained the following explicit formula forϕ_λq(0) :

ϕ_λq(0) = 1

√2π ( _∞

0

)Q⁽⁰⁾_l (exp−λ < Y, q⁻>)·Q⁽²⁾_l (exp−λ < Y, q⁺>)

+Q⁽²⁾_l (exp−λ < Y, q⁻>)Q⁽⁰⁾_l (exp−λ < Y, q⁺ >)*

dl (1.14)

where, in this formula (1.14), the process (Yx, x ≥0) is, under Q⁽⁰⁾_l , $

resp. under Q⁽²⁾_l % , a squared Bessel process with dimension 0, (resp. 2), starting froml, and we denote :

< Y, q⁺>=

( _∞

0

Y_x q(dx) ; < Y, q⁻>=

( ₀

−∞

Y_−x q(dx) (1.15)

It is then clear from (1.14) that : λ−→ϕλq(0) is the Laplace transform of a positive measure, as an integral, with respect to the parameter l of the product of two Laplace transforms of positive measures (indexed byl).

We shall now give some examples for which the measureν_x may be computed explicitly. We recall thatν_x is characterized by :

( _∞

0

e^−λ2zν_x(dz) =ϕ_λq(x) where ϕ_λq(x) is given by (1.3) ... (1.6).

1.5 Computation of ν_x for q(dy) =δ₀(dy)

In this case, the hypothesis H1 is verified and A^q_t =L_t, is the local time at level 0 ϕ_λq(x) =

-2 π

!2

λ+|x|# $

cf [RVY, I], Ex. 4.8, p. 199-200%

= ( _∞

0

e^−λ2z!-2

π1_z≥0dz+ -2

π|x|δ₀(dz)#

(1.16) Thus :

ν_x(dz) = -2

π1_[0,∞](z)dz+ -2

π|x|δ₀(dz) (1.17)

1.6 Computation of ν_x for q(dy) =δ_a(dy) +δ_b(dy) with (a < b)

In this case, the hypothesis H1 is satisfied and A^q_t = L^a_t +L^b_t where (L^a_t, t ≥ 0) resp.

(L^b_t, t ≥0) denotes the local time at level a, resp. at level b. We know $

see [RVY, I], Ex.

4.8, p. 199-200% that

ϕ_λq(x) =





















 -2

π

!1

λ+x−b#

ifx > b -2

π 1

λ ifx∈[a, b]

-2 π

!1

λ+a−x#

ifx < a

(1.18)

= -2

π ( _∞

0

e^−λ2z21

2dz+ (x−b)1_x>bδ₀(dz) + (a−x)1_x<aδ₀(dz)3 Hence :

νx(dz) = -2

π 21

21_[0,∞[(z)dz+ (x−b)⁺δ0(dz) + (a−x)⁺δ0(dz)3

(1.19)

1.7 Computation of ν_x, for q(y) = e^2y

In this case, the hypothesis H2 is satisfied andA^q_t = ( t

0

e^2Xsds.

To begin with, we show : ϕ_λq(x) =

-2 πK₀(√

λ e^x) (1.20)

whereK₀ denotes the Bessel-Mc Donald function with index 0$

see [Leb], p. 108% . Letψ(x) :=

-2 πK0(√

λ e^x). To check (1.20), it suffices to see that : ψ^′′(x) =λe^2xψ(x), ψ(x)_x−→

→∞0, ψ^′(x)_x−→

→−∞− -2

π (1.21)

Now (1.21) follows from$

see [Leb], p. 110% : K₀^′ =−K₁,−K₁^′(z) = 1

zK₁(z) +K₀(z) and

ψ(x)_x∼

→∞

-2 π

! π

2√ λ e^x

#

e^{−√λ ex}_x−→

→∞0 $

[Leb], p.123% ψ^′(x) =−

-2 π

√λ e^xK₁(√

λ e^x) ∼

x→−∞− -2

π

√λ e^x1 2

√2

λ e^x −→

x→−∞− -2

π

$[Leb], p.111% This proves (1.20). But, we also have :

K₀(√

λ e^x) = 1 2

( _∞

0

e^{−t−λe4t2x}dt t

$cf [Leb], p. 119%

= 1

2 ( _∞

0

e^{−λu2 −e}

2x 2u du

u

Hence :

ν_x(dz) = 1

√2π e^−e

2x

2z 1_[0,∞[(z)dz

z (1.22)

1.8 Computation of ν_x for q₀(dx) = 1]−∞,0](x)dx Here, it is the hypothesis H3 which is satisfied, and

A^q_t⁰ = ( t

0

1_]−∞,0](X_s)ds

By scaling, one has, underP₀ :A^q_t^0(law)= t A₁^q0, and it is well known that underP₀, A^q₁⁰ follows the arc sine law, i.e., the beta !1

2,1 2

# law. We shall recall the law of A^q_t⁰ under P_x for any x∈R, (see Subsection 2.1 below), which will allow to obtain the following result :

νx(dz) =x+

-2

π δ0(dz) + 1 π e⁻

x2

−

2z1_[0,∞[(z)dz

√z (1.23)

For the moment, we shall prove (1.23) without using the explicit law ofA^q_t⁰. For this purpose, we already observe that :

ϕ_λq0(x) = -2

π 2

e^x√λ 1

√λ1_x≤0+! x+ 1

√λ

# 1_x>03

(1.24) Indeed we have :

ϕ^′′_λq0(x) =λ1_]−∞,0](x)ϕ_λq(x), ϕ^′_λq0(+∞) = -2

π, ϕ_λq0(−∞) = 0.

Then, it remains to see that : ( _∞

0

e^−λ2zν_x(dz) =ϕ_λq0(x) (1.25)

whereνx is defined via (1.23) andϕ_λq0(x) by (1.24). Now, forx >0, one has : ( _∞

0

e^−λ2zνx(dz) = x+

-2 π + 1

π ( _∞

0

e^−λz2 dz

√z

= x₊ -2

π + 1 π

-2

λ Γ(1/2) =x₊ -2

π + -2

π

√1

λ =ϕ_λq(x) whereas forx <0 :

( _∞

0

e^−λ2zν_x(dz) = 1 π

( _∞

0

e^−λ2z−x

2 2z dz

√z = 2 π K_1/2$

|x|√ λ% !x²

λ

#1/4 $

see [Leb], p. 119% However, one has : K_1/2$

|x|√ λ%

=! π 2|x|√

λ

#1/2

e^−|x|√λ. Hence : ( _∞

0

e^−λ2zν_x(dz) = 2 π

!x² λ

#1/4! π 2|x|√

λ

#1/2

e^−|x|√λ = -2

π

√1

λe^−|x|√λ=ϕ_λq(x)

1.9 Computation of ν_x when q(y) = 1_[a,b](y) (a < b)

The hypothesis H1 is satisfied andA^q_t = ( _t

0

1_[a,b](X_s)ds. We shall prove that :

ν_x^(a,b)(dz) =























 -2

π(x−b)δ₀(dz) + 1 π√

z 1_[0,∞[(z)dz! 1 + 2

4∞ n=1

e^{−n2(b−a)22z} #

ifx > b -2

π(a−x)δ₀(dz) + 1 π√

z1_[0,∞[(z)dz! 1 + 2

4∞ n=1

e^{−n2(b−a)22z} #

ifx < a 1

π√ z

4∞ n=0

! e⁻

$

n(b−a)+b−x%2

2z +e⁻

$

n(b−a)+(x−a)%2 2z

#

1_[0,∞[(z)dz ifx∈[a, b]

(1.26)

Here, the explicit form ofϕ^(a,b)_λq (x) is$

see [RVY, I], Ex. 4.7, p. 199% :

ϕ^(a,b)_λq (x) =

























 -2

π

5 1

√λtanh$√

λ^b−₂^a% +x−b 6

ifx > b -2

π

5 1

√λtanh$√

λ^b−₂^a% +a−x 6

if x < a -2

π

5cosh$√

λ$

x−^a+b₂ )%

√λ$ sinh√

λ^b−₂^a)% 6

if x∈[a, b]

(1.27)

It now remains to prove that : ( _∞

0

e^−λ2zν_x^(a,b)(dz) =ϕ^(a,b)_λq (x) (1.28)

whereν_x^(a,b) is defined via (1.26) andϕ^(a,b)_λq via (1.27). But, (1.28) follows, after some elemen- tary computations from the identities, for every realu andv >0 :

cosh$√

λu%

√λsinh$√

λv% = 4∞ n=0

( _∞

0

dh! e^−√λ

$h+(2n+1)v−u%

+e^−√λ

$h+(2n+1)v+u%#

(1.29)

= 4∞ n=0

( _∞

0

dh ( _∞

0

ds$

H_{h+(2n+1)v−u}(s) +Hh+(2n+1)v+u(s)%

e^−λs(1.30) with

H_a(u) := a 2√

πu³ e^−a2/4u = −1

√πu

∂

∂a(e^−a4u2) (a >0)

Passing from (1.29) to (1.30) is obtained by using the elementary formula : e^−√λa=

( _∞

0

e^−λuH_a(u)du= ( _∞

0

e^−λu a 2√

πu³ e^−a4u2du (1.31)

(Note that (1.31) is nothing else but a translation of : E$

e^−λ22Ta%

= exp(−λa), where T_a denotes the hitting time of level a > 0 by Brownian motion starting from 0, and H_a is the density ofT√^a

2) We now show (1.29).

cosh$√

λu%

√λsinh$√

λv% = 1

√λe^{−√λ(v−u)}1 +e^−2√λu 1−e^−2√λv

= 1

√λe^{−√λ(v−u)}(1 +e^−2√λu)!4^∞

n=0

e^−2n√λv#

= 1

√λ 24^∞

n=0

e^{−√λ{v−u+2nv}}+ 4∞ n=0

e^−√λ

&

2(u+nv)+(v−u)'3

= 1

√λ 24^∞

n=0

! e⁻

√λ$

(2n+1)v−u% +e⁻

√λ$

u+(2n+1)v%#3

= ( _∞

0

e^−√λh24^∞

n=0

e^−√λ

$(2n+1)v−u%

+e^−√λ

$u+(2n+1)v%3 dh

= 4∞ n=0

( _∞

0

$e^−√λ

$h+(2n+1)v−u%

+e^−√λ

$h+(2n+1)v+u%%

dh.

Remark 1.9

(i) If in formula (1.26), we take : b= 0, and we letatend to−∞, we obtain :

a→−∞lim ν^a,0(dz) =







 -2

πx₊δ₀(dz) + 1

π√z1_[0,∞[(z)dz if x >0 1

π√ ze⁻

x2

−

2z 1_[0,∞[(z)dz if x≤0

(1.32)

We note that the RHS of (1.32) is nothing else but the measure ν_x associated with q0(y) = 1_]−∞,0]$

see (1.23)%

. This may be interpreted as ”a continuity property” ofϕ^a,b, asa−→ −∞.

(ii) In the same spirit, but taking up now the computation from subsection 1.9, where we choose forq the function : q^(c)(y) = 1

2c1_[−c,+c](y), we have : ( _∞

0

e^−λz2 ν_x^(c)(dz)−→

c→0

( _∞

0

e^−λz2 ν_x(dz) = -2

π

!2

λ+|x|#

(1.33) whereν_xis the measure associated toq(dz) =δ₀(dz)$

see (1.16)%

. In other terms, since : 1

2c ( t

0

1_[−c,c](Xs)ds−→_c

→0Lt a.s., we witness there also a ”continuity property of νx^(c) as c−→ 0”. Let us show (1.33) forx= 0 ; from (1.27) :

( _∞

0

e^−λz2 ν₀^(c)(dz) = -2

π

7 cosh$8_λ

2c c% 8λ

2c sinh!8

λ 2c·c#

9

−→c→0

-2 π · 2

λ

and for x'= 0, and csmall enough, we obtain from (1.27) that : ( _∞

0

e^−λz2 ν_x^(c)(dz) = -2

π

2 1

8λ

2c tanh!8

λ

2c ·c#+|x−c|3

−→_c

→0

-2 π

!2

λ+|x|#

1.10 Computation of ν_x when q(y) = 1[0,∞[(y)y^α, α >0

The hypothesis H2 is satisfied, and we have : A^q_t = ( t

0

1_(Xs>0)X_s^αds.

We now show the existence of a constant C_α >0 such that : ν₀(dz) = C_α

z^1+α2+α

1_[0,∞[(z)dz (1.34)

Indeed, thanks to the scaling property, we have : E0$

e^{−λ2R0t1Xs>0Xsαds}%

= E0$ e

$−^λ₂t^1+α/2A^q₁%%

= E₀! exp!

−1 2A^q

λ^2+α2 t

##

(1.35) Thus, multiplying (1.35) by√

tand letting ttend to +∞, we obtain : ϕ_λq(0) = 1

λ^2+α1

ϕ_1·q(0) = 1 λ^2+α1

c^′_α =c_α ( _∞

0

e^−λ2z dz z^1+α2+α

The same computations, performed this time with x'= 0, lead to : ϕ_λq(x) = 1

λ^2+α1 ϕ_1q$

xλ^2+α1 %

,i.e. ν_x= 1 λ^2+α1 ν^(λ)

xλ^2+α1 , whereν^(λ) is the image ofν by the applicationz−→λz.

Question 1.4 We know $

see [R,Y], chap. X%

that, ifq is an integrable function, then :

√1 t

( t 0

q(x+Xs)ds_t−→^law

→∞

! (

q(x+y)dy#

|N|=! (

q(y)dy#

|N| (1.36)

whereN is a standard Gaussian variable, and on the LHS of (1.36), (X_s, s≥0) is a Brownian motion starting from 0. Letg denote the density of the r.v. ¯q|N|with ¯q=

(

q(y)dythat is :

g(z) = 1

¯ q

-2 πe⁻

z2

2(¯q)21_[0,∞[(z)

Let us now consider the supplementary hypothesis H, which seems reasonable enough in: view of (1.36) that the densityg_t(x,·) of the r.v. 1

√t ( t

0

q(x+X_s)dsconverges, as t−→ ∞, uniformly on every compact, towardsg.

However, this would imply that, for every function h, which is continuous with compact support, one would have :

√t E_x+ h(A^q_t),

= √

t E_x) h!A^q_t

√t·√ t#*

= √

t ( _∞

0

h(z√

t)g_t(x, z)dz

= ( _∞

0

h(y)gt

! x, y

√t

# dy_t−→

→∞

( _∞

0

h(y)g(0)dy

But, from Theorem 1.1., we know that :

√t E_x+ h(A^q_t),

t−→→∞

(

R+

h(x)ν_x(dz)

Thus, this would imply that the measureνx(dz) would be equal to : 1

¯ q

-2

π 1_[0,∞[dz (1.37)

so that, the measure ν_x would not depend on x, and would be proportional to Lebesgue measure onR+. But clearly, this is not the case for either of the examples in subsections 1.4 to 1.9. Consequently, the hypothesisH: is not satisfied for the corresponding q ^′s. It would be of interest to know for whichq^′s, if any, it is satisfied.

1.11 Penalisation by h(A^q_t)

Letq satisfy one of the previous hypotheses H1, H2 or H3, and denote, as before : A^q_t =

(

R

L^x_tq(dx)!

= ( t

0

q(Xs)ds if q admits a density#

. Let now h : R+ −→ R+ such that :

√t E_x+ h(A^q_t),

t−→→∞

( _∞

0

h(z)ν_x(dz) (1.38)

Then, (1.38) is satisfied, from Theorem 1.1., as soon as h is sub-exponential (for example if h is continuous, with compact support). We shall now study the penalisation of Wiener measure by the functionalh(A^q_t), i.e. : we shall study the limit, as t−→ ∞, of :

Ex$

1Λsh(A^q_t)% E_x$

h(A^q_t)% (s≥0, Λ_s∈ Fs) (1.39)

We have already made this study in two situations : 1) q(dy) =δ0(dy) then A^q_t =Lt $

cf [RVY, II]%

; 2) A^q_t =

(

R

L^y_tq(dy) and h(u) = exp!

−λ

2u# $

cf [RVY, I]% .

This time, Theorem 1.1. allows us to obtain : Theorem 1.5. : Let q, A^q and h as above. Then :

1) For every s≥0, and every Λ_s ∈ Fs :

tlim→∞

E_x$

1_Λsh(A^q_t)% E_x$

h(A^q_t)% exists (1.40)

2) This limit equals Ex(1ΛsMs^h,q) :=Q^h,q(Λs), where

M_s^h,q:=

(

R+

ν_Xs(dz)h(z+A^q_s) (

R+

ν_x(dz)h(z)

(1.41)

Furthermore,(Ms^h,q, s≥0)is a positive martingale. In the case whenh(u) :=e^−λ2u(u, λ≥ 0), we then obtain :

M_s^h,q= ϕ_λq(X_s) ϕ_λq(x) exp!

−λ 2A^q_s#

(1.42)

Proof of Theorem 1.5.

We have : Ex$

1Λsh(A^q_t)% E_x$

h(A^q_t)% = Ex$ 1ΛsE_b$

h(a+A^q_t−s)% E_x$

h(A^q_t)%

from the Markov property, where b=Xs and a=A^qs. Thus, from Theorem 1.1. : E_x$

h(A^q_t)%

t→∞∼

√1 t

( _∞

0

ν_x(dz)h(z) and E_b$

h(a+A^q_t−s)%

t→∞∼

√1 t−s

( _∞

0

ν_b(dz)h(a+z) Hence :

E_x$

1_Λsh(A^q_t)% Ex$

h(A^q_t)% ∼

t→∞

√t

√t−s E_x$

1_Λs"

R+ν_Xs(dz)h(z+A^q_s)%

"

R+h(z)ν_x(dz)

t−→→∞ E_x(1_ΛsM_s^h,q)

In the preceding lines, we have been a little careless concerning the exchange of limit and expectation. Likewise, although it is easy to see that (Ms^h,q, s ≥ 0) is a local martingale, some care is needed in order to show that it is a true martingale. However, all this is correct as soon ash is sub-exponential. We leave details to the reader.

2 A detailed study for q

= 1

, A

:=

(

1

ds

Throughout this section, we choose q0 = 1_]−∞,0]. Thus, the hypothesis H3 is now satisfied.

We shall study this situation in detail, which we are able to do as we know $

see [Y]% the law of A^q_t⁰ =

( _t

0

q₀(X_s)ds under P_x, for every real x $

see (2.5) and (2.7) below%

. We shall, successively :

- compute explicitly the measure ν_x starting from the knowledge of the law of A^q_t⁰ and we shall recover the result of subsection 1.8 above ;

- study the penalisation, not only of the process (Xt, t≥0) byh(A^q_t⁰), but also the penalisa- tion of the ”long bridges” by this functional ;

- describe precisely the behavior of the canonical process under the probability Q^h,q0, where Q^h,q0 is defined via :

Q^h,q0(Λ_s) =E(1_ΛsM_s^h,q0) (s≥0, Λ_s∈ Fs) (2.1)

2.1 The law of A⁻_t and the computation of ν_x To simplify notation, we denote :

A⁻_t = ( t

0

1_(Xs<0)ds= ( t

0

q0(Xs)ds (2.2)

We recall the following result, which is found in [Y]. For any f : [0,1] −→ R+, Borel, sub- exponential (see Theorem 1.1) and anyy >0 :

E₀) f! ( ¹

0

1_(Xs<y)ds#*

= ( ₁

0

du π;

u(1−u)e^−y

2

2uf(u) +f(1) -2

π ( _y

0

e^−α

2

2 dα (2.3)

whereas, for anyy <0, we use : ( ₁

0

1_(Xs<y)ds^law= ( ₁

0

1_(Xs>−y)ds^law= 1− ( ₁

0

1_(Xs<−y)ds (2.4)

and by the scaling property : E_x)

f! ( ^t

0

1_(Xs<0)ds#*

= E_x$

f(A⁻_t )%

=E₀! f! ( ^t

0

1_(Xs<−x)ds##

=E₀! f!

t ( 1

0

1$

Xs<−^√x_t%ds##

hence, from (2.3) and (2.4), ifx≤0 : E_x+

f(A⁻_t),

= ( t

0

dv π;

v(t−v)e^−x2v2f(v) +f(t) -2

π ( ^|x|√

t

0

e^−α22dα (2.5)

t→∞∼ 1 π√

t ( _∞

0

√dv ve^−x

2

2vf(v) (2.6)

whereas, ifx >0 : Ex+

f(A⁻_t),

= f(0) -2

π ( √^x

t

0

e^−α

2 2 dα+

( t 0

dv π;

v(t−v)e^{− x}

2

2(t−v)f(v) (2.7)

t→∞∼ f(0) -2

π ( √^x

t

0

e^−α

2

2 dα+ 1

π√ t

( _∞

0

√dvvf(v) (2.8)

Thus, we obtain, forf :R+−→R+, Borel, sub-exponential !

and then ( _∞

0

√dv

vf(v)<∞#

√t Ex+ f(A⁻_t ,

t−→→∞

( _∞

0

f(z)νx(dz) with

ν_x(dz) =x₊ -2

πδ₀(dz) + 1 πe⁻

x2

−

2z 1_[0,∞[(z)dz

√z

which is precisely (1.23).

2.2 Penalisation by h(A⁻_t). A study of ”long bridges” and of the Q^h−process We recall that, from (2.5), the density ofA⁻_t underP₀, which we denote byp_A−

t , equals : p_A−

t (y) = 1 π

; 1

y(t−y)1_[0,t](y) (: the arc sine law). (2.9)

Throughout the following, hdenotes a function from R+ toR+ such that : ( _∞

0

√dyyh(y)<∞

and we assume, without loss of generality, that : ( _∞

0

√dyyh(y) = 1 (2.10)

Theorem 2.1

1)For every s≥0 and every Λs∈ F^s :

tlim→∞E0(1_Λs|A⁻_t =a) =Q^(a)(Λs) (2.11)

with

Q^(a)(Λ_s) :=

-2 π

1_a<s

√s−a E₀(1_ΛsX_s⁺|A⁻_s =a) + E)

1_Λs

- a

a−A⁻_s 1_(A− s<a)e⁻

(X− s)2 2(a−A− s)

*

(2.12)

!

recall that X_s⁺= 0∨X_s, X_s⁻=−(X_s∧0) andA⁻_s = ( _s

0

1_Xu<0du#

2)For every function h which satisfies (2.10), for every s≥0 and any Λs∈ F^s :

tlim→∞

E₀$

1_Λsh(A⁻_t)% E₀$

h(A⁻_t)% =E(1_ΛsM_s^h) (2.13)

where (M_s^h, s≥0)is the positive martingale given by : M_s^h :=√

2π X_s⁺h(A⁻_s) + ( _∞

0

√dyy e^−(X

s−)2

2y h(A⁻_s +y) (2.14)

(note thatM₀^h = 1).

3) Formula (2.13) induces a probability Q^h on (Ω,F∞), which admits the following disinte- gration :

Q^h(Λ) = ( _∞

0

Q^(a)(Λ)h(a)

√a da (Λ∈ F∞) (2.15)

where Q^(a) is given by (2.12).

4)Under Q^h, the canonical process $

X_t, t≥0%

satisfies :

i) A⁻_∞is finite a.s., and admits as density h(y)

√y 1_y>0 (2.16)

ii) letg= inf{t; A⁻_t =A⁻_∞}= sup{t; X_t≤0} (2.17) ThenQ^h(g <∞) = 1

iii) the processes(X_t, t≤g) and (X_g+t, t≥0)are independent ;

iv) the process(X_g+t, t≥0)is a 3-dimensional Bessel process starting from 0.

Moreover, while proving Theorem 2.1, we shall give a precise description of the process (Xt; t≤g).

2.3 Proof of Theorem 2.1

2.3.1 Proof of point 1) in Theorem 2.1

For this purpose, we choose a functionh, which is Borel, positive, and satisfies (2.10).

We first write : E₀$

1_Λsh(A⁻_t )%

= ( t

0

E₀(1_Λs|A⁻_t =a)p_A−

t (a)h(a)da (2.18)

Then, conditioning with respect toFs, we obtain : E₀$

1_Λsh(A⁻_t )%

=E₀!

1_ΛsE₀! h!

a+ ( t−s

0

1_(Xu<−x)du###

(2.19)

witha=A⁻_s and x=Xs. Using now (2.5) and (2.6), we obtain : E₀$

1_Λsh(A⁻_t )%

= E₀!

1_Λs1_x<0!! ( ^t−s 0

dv π;

v(t−s−v)e^−x

2

2vh(a+v)#

+h(a+t−s)ψ! |x|

√t−s

###

+E₀!

1_Λs1_x>0)

h(a)ψ! x

√t−s

#+ ( t−s

0

dv π;

v(t−s−v) e^{− x}

2

2(t−s−v)h(a+v)*#

(2.20)

:= (1)_t+ (2)_t (2.21)

whereψ! x

√t

# :=P!

|N| ≤ x

√t

#

= -2

π ( √^x

t

0

e^−α

2

2 dα ∼

t→∞

-2 π

√x t· We now study successively (1)_t and (2)_t. We rewrite (1)_t in the form :

(1)_t

= ( _s

0

p_A−

s(a)da E₀!

1_Λs1_Xs<0! ( ^t−s 0

dv π;

v(t−s−v)e^−X

s2

2vh(a+v) +h(a+t−s)ψ!√|Xs|

t−s

##<<<A⁻_s =a#

=

( _t−s

0

dv π;

v(t−s−v) ( _v+s

v

da p_A−

s(a−v)E₀$

1_Λs1_Xs<0e^−X

2s

2v|A⁻_s =a−v% h(a)

+ ( _t

t−s

p_A−

s (a+s−t)E₀!

1_Λs1_Xs<0ψ! |X_s|

√t−s

#<<<A⁻_s =a+s−t#

h(a)da (2.22) Similarly :

(2)_t

= ( s

0

p_A− s(a)E0

!

1_Λs1_Xs>0ψ!√|Xs| t−s

#<<<A⁻_s =a#

h(a)da (2.23)

+ ( _s

0

da p_A− s(a)

( _t−s

0

dv π;

v(t−s−v)E₀$

1_Λs1_Xs>0e⁻

X2 2(t−s−v)s <

<A⁻_s =a)h(a+v)

= ( _s

0

p_A−

s(a)E₀!

1_Λs1_Xs>0ψ!√|Xs| t−s

#<<A⁻_s =a#

h(a)da (2.24)

+ ( _t−s

0

dv π;

v(t−s−v) ( _v+s

v

p_A−

s(a−v)E₀!

1_Λs1_Xs>0e⁻

X2

2(t−s−v)s <<A⁻_s =a−v# h(a)da

Then, comparing (2.18), (2.22), (2.24) and identifying the ”coefficient of h(a)”, it follows that :

E₀$

1_Λs|A⁻_t =a%

= (˜1)_t+ (˜2)_t