Table Of ContentPenalizations of the Brownian motion by a
fun
tional of its lo
al times
7
Joseph Najnudel
0
0
2 18th January 2007
n
a
J Abstra
tt : RIn∗ this arti
le, we study the family of probability measures (in-
8 dexed by ∈ +), obtained by penalizationof the Brownianmotion by a given
t
1
fun
tional of its lo
al times at time .
t
Weprovethatthis familytendsto alimitmeasurewhen goestoin(cid:28)nity if the
]
R fun
tional satis(cid:28)es some
onditions of domination, and we
he
k these
ondi-
P tions in severalparti
ular
ases.
.
h
t Keywords : penalization, lo
al time, Brownian motion.
a
m
AMS
lassi(cid:28)
ations : 60B10,60J65 (60G17, 60G44, 60J25,60J55).
[
1
v Introdu
tion
6
2 Brownian penalizations have been studied in several arti
les, in parti
ular in
5 [RVY03℄, [RVYW06℄, [RVY05℄. The general prin
i(pRle o,fRt)hes(eXp)enalizations is the
01 following : let(Γt)t≥b0e the Wiener measure on C + , t0t≥<0 tWhe[Γ
ta]n<oni
al
7 pro
ess, and a family of positive weigh(tWs su)
h that W∞;
t t≥0
we
onsider the family of probability measures , obtained from , by
0 Γ
/ (cid:16)penalization(cid:17) with the weight :
h
Γ
t W = t .W
a t W[Γ ]
m t
: (Wt)t≥0
v IWn manytdi(cid:27)erentparti
ular
ases,the family s 0tends to aΛlimit measure
Xi ∞ as → ∞, in=thσe Xfoll,ouwingssense : for all ≥ , and for s measurable
s u
with respe
t to F { ≤ } :
r
a W (Λ ) W (Λ )
t s ∞ s
t→→∞
Up to now, theredoesnotexist ageneraltheoremwhi
h
oversall the di(cid:27)erent
ases forwhi
h
onvergen
eholds. On the other hand, we remark that in many
of these
ases, one has : Γt =F((lty(X))y∈R)
where (lty(X))y∈R is the f(aRm,iRly o)f theRlo
al times of (Xs)s≤t, and F is a mea-
+ +
surable fun
tional from C to .
Γ
These two fa
ts led us to prove that if is of this form, the limit measure
1
W F
∞
exists for a (cid:16)large(cid:17)
lass of fun
tionals .
This proof is the main topi
of our arti
le, whi
h is divided into six se
tions.
In the (cid:28)rst one, we de(cid:28)ne and explain the notations we need to proveourmain
theorem, whi
h is stated at the end of the se
tion.
In Se
tion 2, we prove an equality satis(cid:28)ed by an approximation of a given
fun
tional of lo
al times, and in Se
tion 3, we majorize the error term
orres-
ponding to this approximation.
This allows us to obtain, in Se
tion 4, the asymptoti
behaviour of the ex-
pe
tationoffun
tionalswhi
hsatisfysomeparti
ular
onditions,and(cid:28)nallywe
prove the main theorem in Se
tion 5.
In Se
tion 6, we study the four following examples, for whi
h the Theorem
applies :
1) F((ly)y∈R)R= φ(l0R) (whi
h
orresponds to Γt = φ(lt0(X))), where φ is a
+ +
fun
tion from to , dominated by an integrable and de
reasing fun
tion
ψ
.
F((ly)y∈R)=φ(inf y 0,ly =0 )
Γ2) =φ(sup X ,s t {) ≥ φ } (whi
h
orrespRondsto the wReight
t s + +
{ ≤ } ),wheψre isafun
tion from R∪{∞} to , domina-
ted by a de
reasing fun
tion , whi
h is integrable on .
3)F((ly)y∈R)=exp − −∞∞V(y)lydy ,whereV isapositivemeasurablefun
-
(cid:16) (cid:17) (1+y2)dy
R
tion, not a.e. equal to zero, and integrable with respe
t to .
F((ly)y∈R) = φ(ly1,ly2) y1 < y2 φ(l1,l2) h(l1 l2)
4) , where and ≤ ∧ , for a
h
de
reasing and integrable fun
tion .
The three (cid:28)rst examples have been already studied by B. Roynette, P. Val-
lois and M. Yor.
As a help to the reader, we mention that Se
tions 2 and 3 are quite te
hni-
al, but it is possible to read the details of these se
tions after Se
tions 4 and
5, whi
h
ontain the prin
ipal steps of the proof of the Theorem.
1 Notations and statement of the main theorem
(B )
t t≥0
I(nLytt)ht≥is0,ayr∈tRi
lteh,e bi
ontindueonuostevserasisotnanodfairtds loon
ea-ldtiimmeenss,iaonndal(Bτlaro)lw≥n0,iaa∈nRmtohteiofna-,
mily of its inverse lo
al times.
T = τa a B
To simplify these notations, we put a 0 ((cid:28)rst hitting time at of )
τ0 =τ
and l l.
Foreveryl∈R+,(Yly,+)y∈R denotesarandompro
essde(cid:28)nedonthewholereal
2
(Yy )
line,su
hthatits(cid:16)positivepart(cid:17) l,+ y≥0isa2-dimensionalsquaredBesselpro-
(Y−y)
ess (BESQ(2)), its (cid:16)negative part(cid:17) l,+ y≥0 is an independent 0-dimensional
Y0 l
squared Bessel pro
ess (BESQ(0)), and its value at zero l,+ is equal to . In
parti
ular,yby
0lassi
al propYeyrtie=s0of ByESQy(0) and BESQ(2) pro
esses, there
exists a.s. 0 ≤ su
h that l,+ i(cid:27) ≤ 0.
Wede(cid:28)nealso(Yly,−)y∈R asapro
esswhi
hhasthesamelawas(Yl−,+y)y∈R,the
pro
ess obtained from (Yly,+)y∈R by (cid:16)reversing the time(cid:17).
In one of the penalization results shown in [RVY05℄, B. Roynette, P. Vallois
(Zl) Zl = B t τ
and M. Yor obtain a limit pro
ess t t≥0, su
h that t t for ≤ l,
(Zl ) B ǫ=sgn(Zl ) u>0
| τl+u| u≥0 is a BES(3) pro
ess indePp(eǫn=den1t)=ofP(,ǫa=nd 1)=1/2τl+u ( )
isanindependentvariablesu
hthat − .Thispro
ess
an be informally
onsidered to be a Brownian motion
onditionned to have a
l
total lo
al time equal to at level zero. By applying Ray-Knight theorems for
(Zl)
Blarwowofnitahnelfoa
maillytiomfeists(stoeeta[lRlYo
9a9lat℄i)mteos istthte≥0h,ailtf-sisumpoosfsitbhleeltaowsshoofw(Ytlhy,+at)yt∈hRe
and (Yly,−)y∈R ((Yly,+)y∈R
orresponds to the paths of (Ztl)t≥0 su
h that ǫ = 1,
and (Yly,−)y∈R
orrespondsto the paths su
h that ǫ=−1).
Thisexplainswhythe pro
esses(Yly,+)y∈R and(Yly,−)y∈R o
urnaturallyinthe
des
ription of the asymptoti
behaviour of Brownian lo
al times.
We also need to de(cid:28)ne some modi(cid:28)
ations of (Yly,+)y∈R and (Yly,−)y∈R : for
l ≥0, a≥0, (Yly,a)y∈R denotes a pro
ess su
h that (Yly,a)y≥0 is markovian with
y a
the in(cid:28)nitesimal genyeratoar o(Yf B−yE)SQ(2) for ≤ and the in(cid:28)nitesimal genera-
tor of BESQ(0) for ≥ , l,a y≥0 is an independent BESQ(0) pro
ess, and
Yl0,a =l. For a≤0, (Yly,a)y∈R has the same law as (Yl−,−ya)y∈R.
F (R,R ) R
+ +
Now, let be a fun
tional from C to , whi
h is measurable with
σ
respe
t to the -(cid:28)eld generated by the topology of uniform
onvergen
e on
inomthpea
atsysmetps.toWtie
s
oonfsEid[Fer(t(hLeyt)fyo∈llRow)]in:gquantities, whi
h will naturallyappear
∞
I+(F)= dlE[F((Yly,+)y∈R)]
Z0
∞
I−(F)= dlE[F((Yly,−)y∈R)]
Z0
I(F)=I (F)+I (F)
+ −
I(F) F σ
WI e obs(eRrv,Re th)at is the integral of with respe
t to the -(cid:28)nite measure
+
on C , de(cid:28)ned by :
∞ ∞
I = dlP + dlP
l,+ l,−
Z0 Z0
where Pl,+ is the law of (Yly,+)y∈R and Pl,− is the law of (Yly,−)y∈R.
F
At the end of this se
tion, we give some
onditions on whi
h turn out to
be su(cid:30)
ient to obtain our penalization result.
Unfortunately, these
onditions are not very simple and we need three more
3
de(cid:28)nitions before stating the main Theorem :
c n R n
+
De(cid:28)nition 1 (a
ondition of domination) : Let hand Rbe in R (generally
+ +
will be an integer). ForFevery de
(rRea,sRing)fun
Rtion from to ,weCs(acy,nt,hha)t
+ +
a measurable fun
tion from C to satis(cid:28)les theR
onditRion
+
i(cid:27) the following holds for every
ontinuous fun
tion from to :
F((ly)y∈R) (ly)y∈[−c,c]
1) depends only on .
sup ly+c n
2) F((ly)y∈R)≤ yy∈∈[[i−−nfcc,,cc]]ly+c! h(cid:18)y∈i[n−fc,c]ly(cid:19)
Intuitively, a fun
tional of the lo
al times satis(cid:28)es the above
ondition if it
depends only on the lo
al times on a
ompa
t set, and if it is small when these
lo
al times are large and don't vary too mu
h.
Now, let us use the notation :
∞
N (h)=ch(0)+ h(y)dy
c
Z0
N (h) < F
c
If ∞, it is possible to prove our main theorem for all fun
tionals
C(c,n,h)
whi
h satis(cid:28)es the
ondition , but this
ondition is restri
tive, sin
e
F [ c,c]
the fun
tional must not depend on the lo
al times outside of − .
In order to relax this restri
tion, we need the following de(cid:28)nition :
n R
+
De(cid:28)Fnition 2 (a less restri
tive
ondition of domi(nRat,iRon)): LRet be in
+
and be a positive and measurable fun
tion from C to .
M 0 F D(n,M)
For all ≥ , let us say that satis(cid:28)es the
ondition i(cid:27) there exists
(c ) [1, [ (h )
k k≥1 k k≥1
Ra sequeRn
e in ∞(,Fa)sequen
e of de
reasing fun
(tRion,sRfr)om
+ + k k≥0 +
R to , and a sequen
e of measurable fun
tions from C to
+
, su
h that :
F =0 (F ) F
0 k k≥1
1) and tends to pointwise.
k 1 F F C(c ,n,h )
k k−1 k k
2) For all ≥ , | − | satis(cid:28)es the
ondition .
N (h ) M
3) ck k ≤ .
k≥1
P N(n)(F) M 0 F
We de(cid:28)ne the quantity as the in(cid:28)mum of ≥ su
h that sa-
D(n,M)
tis(cid:28)es the
ondition .
N(n)(F) < F
Intuitively, if ∞, it means that
an be well-approximated by
fun
tionals whi
h satisfy
onditions given in De(cid:28)nition 1.
F C(c,n,h) c 1 N(n)(F)
Inparti
ular,if satis(cid:28)esthe
ondition for ≥ ,onehas: ≤
N (h) F D(n,N (h))
c c
(one
acn p=rocvehth=ath1satisF(cid:28)es=th0e
ondFiti=onF k 1 , by taking in
k k k=1 0 k
De(cid:28)nition 2 : , , and if ≥ ).
4
F
Now, for a given fun
tional , we need to de(cid:28)ne some other fon
tionals, in-
F
formally obtained from by (cid:16)shifting(cid:17) the spa
e and adding a given fun
tion
to the lo
al time family.
More pre
isely, let us
onsider the following de(cid:28)nition :
x F
aDem(cid:28)enasituiroanble3fRun(l
ot
ioaRnl+atlimfreomanCd(Rsp+a,
Re F)sh(tilofy0t))Ry∈:+RL,,xeatnd ibfe(al0y)rye∈aRl niusmabe
or.n(RtIifn,uRo+uiss)
funR
tion from to , we denote by the fun
tional from C
+
to whi
h satis(cid:28)es :
F(ly0)y∈R,x((ly)y∈R)=F((l0y+ly−x)y∈R)
(ly)y∈R
for every fun
tion .
Tohnissidneortatthioen
oannddittihoenfaulne
xtpioen
atalstidoen(cid:28)n:eEd[Fin((tLhiyts)yw∈aRy)a|(pBpue)aur≤nsa],tuforarll0y<whsen<wte,
and apply the Markov property.
We are now able to state the main theorem of the arti
le :
F (R,R ) R I(F) > 0
+ +
Theorem : Let be a fun
tional from C to su
h that
N(n)(F)< n 0
anWd ∞ for some ≥ . (R ,R) (X )
+ t t≥0
pIfro
esdse,naontdes(tlhtye(Xst)a)nt∈dRar+d,yW∈Rietnheer
moneatisnuureouosnfCamily of i,ts lo
al titmhees
a(nWon-ai
.as.l
well-de(cid:28)ned), the probability measure :
F (ly(X))
WF = t y∈R .W
t W F(cid:16) (ly(X)) (cid:17)
t y∈R
h (cid:16) (cid:17)i
t
is well-deW(cid:28)nFed for every whi
h is large enough, and there exists a probability
measure ∞ su
h that :
WF(Λ ) WF (Λ )
t s t→→∞ ∞ s
s 0 Λ =σ X ,u s
s s u
for every ≥ and ∈F { ≤ }.
Moreover, this limit measure satis(cid:28)es the following equality :
WF (Λ )=W 1 .I F(lys(X))y∈R,Xs
∞ s Λs I(F)
(cid:0) (cid:1)!
I(F)>0
NR(enm)(Far)k<1.1 : A
onsenquen0
eoftheTheor(eIm(Fi(sLyst)hy∈eRf,aB
st))tsh≥a0tif and
∞ for some ≥ , the pro
ess I(F) is a martingale.
In three of the four examples studied in Se
tion 6, we
ompute expli
itly this
martingale, and in the two (cid:28)rst ones, we
he
k that this
omputation agrees
5
with the results obtained by B. Roynette, P. Vallois and M. Yor.
ly(X)
Remark 2.1 : We point out that our notatioLny, t , for the lo
al times
given in the Theorem, di(cid:27)ers from the notation t, whi
h is used for the lo
al
(B )
s s≤t
timesof .Thisisbe
ause,in one
ase,we
onsiderthe
anoni
alpro
ess
(X )
t t≥0
onagivenprobabilityspa
e,andintheother
ase,we
onsideraBrow-
nianmotiononaspa
ewhi
hisnotmadepre
ise.Hen
e,thetwomathemati
al
obje
tsdeservedi(cid:27)erentwritings,despitethefa
tthattheyarestronglyrelated.
2 An approximation of the fun
tionals of lo
al
times
InordeFrtoprovetheTheorem(,Rw,eRne)edtoRstudytheexpe
tationofF((Lyt)y∈R),
+ +
where is a fun
tion from C to .
wHiolwl reevpelra,
ienFg(e(nLeytr)ayl∈,Rit)ibsydain(cid:30)
auplptrotxoimdoattiohna.t dire
tly, so in this se
tion, we
For the study of this approximation, we need to
onsider the following quanti-
ties : c c c
c = Yy dy, c = Yy dy, c = Yy dy
Il,+ l,+ Il,− l,− Il,a l,a
Z−c Z−c Z−c
c R c= a R
+
for ∈ or ∞, ∈ ;
1 1 1
c = (Yc +Y−c), c = (Yc +Y−c), c = (Yc +Y−c)
Yl,+ 2 l,+ l,+ Yl,− 2 l,− l,− Yl,a 2 l,a l,a
c R a R
+
for ∈ , ∈ ;
Ic,t,+(F)=Z0∞dlEF((Yly,+)y∈R)e−(Ylc1,+−)2/I2l(c,t+−/Itlc,+)φ(cid:18)Ilct,+(cid:19)
q
Ic,t,−(F)=Z0∞dlEF((Yly,−)y∈R)e−(Ylc1,−−)2/I2lc(,t−−/Itlc,−)φ(cid:18)Ilct,−(cid:19)
q
and
I (F)=I (F)+I (F)
c,t c,t,+ c,t,−
c R t > 0 φ R R
+ + +
for ∈ , , where denotes the fun
tion from to su
h that
φ(x)=1 x 1/3 φ(x)=2 3x 1/3 x 2/3 φ(x)=0 x 2/3
in ≤ , − if ≤ ≤ and if ≥ (in
[0,1[
parti
ular,this fun
tion is
ontinuous with
ompa
t support in
luded in ).
e(Ylc,+)2/2(t−Ilc,+) c t
We observe that the expression √1−Ic /t is not well-de(cid:28)ned if Il,+ ≥ ;
l,+
φ( c /t)=0
but this is not important here, sin
e Il,+ in that
ase.
Now, the main result of this se
tion is the following proposition :
(R ,R) R
+ +
Proposition 2 : For all measurable fun
tionals from C to , su
h
6
F((ly)y∈R) (ly)y∈[−c,c] c 0
that depends only on for some ≥ , the following equa-
lity holds :
1 c
√2πtE F((Lyt)y∈R)1|Bt|≥cφ t Lytdy =Ic,t(F)
(cid:20) (cid:18) Z−c (cid:19)(cid:21)
t>0
for all .
G (R ,R) R R
0 + + +
Proof : (LGet((X )be a,tf)u)n
tional from C × to (,Rsu,
Rh)that the
0 s s≥0 t≥0 +
pro
ess : , de(cid:28)ned on the
anoni
al spa
e C , is pro-
gressively measurable.
ω R R G ((ω ) ,t)
+ 0 s s≥0
For every
ontinuous fun
tion from to , depends only
(ω )
s s≤t
on ; let us take :
G((ω ) )=G ((ω ) ,t)
s s≤t 0 s s≥0
Now,byresultsbyC.Leuridan(see[Leu98℄),P.BianeandM.Yor(see[BY88℄),
one has : ∞ ∞ ∞
dtG((Bs)s≤t)= dl daG((Bs)s≤τa)
l
Z0 Z0 Z−∞
Byusinginvarian
epropertiesofBrownianmotionfortimeandspa
ereversals,
one obtains :
∞ ∞ ∞
dtE[G((B ) )]= dl daE[G((Zl,a) )]
s s≤t s s≤τl+Ta→0
Z0 Z0 Z−∞
(Zl,a) Zl,a = B s τ
where s s≤τl+Ta→0 denotes a pro
ess su
h that s s for ≤ l and
(Zl,a )
τl+u u≤Ta→0 is the time-reversedpro
ess of aBrownianmotion starting from
a B
, independent of , and
onsideredupto its (cid:28)rsthitting timeofzero(denoted
T
a→0
by ).
U R∗ J (F)
Therefore, for all Borel sets of +, if we de(cid:28)ne c,U by :
1 c
Jc,U(F)= dtE F((Lyt)y∈R)1|Bt|≥cφ t Lytdy
ZU (cid:20) (cid:18) Z−c (cid:19)(cid:21)
we have, by taking G0 and G su
h that G((Bs)s≤t)=F((Lyt)y∈R) :
∞ c Lydy
Jc,U(F)=Z0 dtE"F((Lyt)y∈R)1|Bt|≥cφ R−∞−∞c Ltytdy!1R−∞∞Lytdy∈U#
∞ cRLy,l,ady
=Z0 dlZR\[−c,c]daE"F((Ly,l,a)y∈R)φ R−∞−∞c Ly,l,ady!1R−∞∞Ly,l,ady∈U#
(Ly,l,a)y∈R R Zl,a
where is the
ontinuous family of the total lo
al times of .
Hen
e, by Ray-Knight theorem applied to the independent pro
esses
(B =Z ) (Z )
s s s≤τl and τl+u u≤Ta→0,and
lassi
aladditivitypropertiesofsquared
Bessel pro
esses :
∞ c
Jc,U(F)=Z0 dlZR\[−c,c]daE"F((Yly,a)y∈R)φ IIll∞,,aa!1Il∞,a∈U#
7
∞ c
=Z0 dlZR\[−c,c]daE"F((Yly,a)y∈R)E"φ IIll∞,,aa!1Il∞,a∈U(cid:12)(Yly,a)y∈[−c,c]##
(cid:12)
sin
e F((Yly,a)y∈R) depends only on (Yly,a)y∈[−c,c]. (cid:12)
θ [ c,c] R
+
N∞owY,yifdy is a g−ivcenYy
odnytinuous fun
tion from − to (Y,yth=e iθnyt)egrals :
c l,a and −∞ l,a are independent
onditionally on l,a y∈[−c,c]
∞Yy dy
Rand their
onditRional laws arerespe
tively equalto the laws of 0 θc,(a−c)+
∞Yy dy
and 0 θ−c,(−a−c)+ . R
R
Therefore, by additivity properties of BESQ pro
esses, the
onditional law of :
−c ∞
∞ c = Yy dy+ Yy dy
Il,a−Il,a l,a l,a
Z−∞ Zc
(Yy =θy)
given l,a y∈[−c,c], is equal to the law of :
∞ ∞
Yy dy+ Yy dy
θc+θ−c,0 0,(|a|−c)+
Z0 Z0
(Yy ) (Yy )
where θc+θ−c,0 y≥0 and 0,(|a|−c)+ y≥0 are supposed to be independent.
∞Yy dy
By Ray-Knighttheorem, 0 θc+θ−c,0 has the samelawasthe time spent in
R (B )
+ by s s≤τθc+θ−c, theRrefore :
∞
Yy dy (=d)τ (=d)T
θc+θ−c,0 (θc+θ−c)/2 (θc+θ−c)/2
Z0
Moreover: ∞
Yy dy (=d)T
0,(|a|−c)+ (|a|−c)+
Z0
∞ c (Yy =θy)
Hen
e,the
onditionallawofIl,a−Il,a,given l,a y∈[−c,c],isequaltothe
T
law of (|a|−c)++(θc+θ−c)/2. Consequently :
∞
Jc,U(F)=Z0 dlZR\[−c,c]daEhF((Yly,a)y∈R)ψa(Ilc,a,Ylc,a)i
a >c
where, for | | :
ψ ( ,θ)=E φ I 1
a I +T I+T|a|−c+θ∈U
(cid:20) (cid:18)I |a|−c+θ(cid:19) (cid:21)
u>0 p T
u u
Now, if, for all , denotes the density of the law of , one has :
ψ ( ,θ)= φ( /t)p (t )dt
a |a|−c+θ
I I −I
ZU
and :
∞ c
Jc,U(F)=ZUdtZ0 dlZR\[−c,c]daE(cid:20)F((Yly,a)y∈R)φ(cid:18)Ilt,a(cid:19)p|a|−c+Ylc,a(t−Ilc,a)(cid:21)
8
Byhypothesis,F((Yly,a)y∈R)dependsonlyon(Yly,a)y∈[−c,c].Moreover,fora≥c,
(Yy ) (Yy ) a c (Yy )
l,a y∈[−c,c] has the same law as l,+ y∈[−c,c], and for ≤ − , l,a y∈[−c,c]
(Yy )
has the same law as l,− y∈[−c,c].
Hen
e, we have :
∞ c ∞
Jc,U(F)= dt dlE F((Yly,+)y∈R)φ Ilt,+ pa−c+Ylc,+(t−Ilc,+)da
ZU Z0 (cid:20) (cid:18) (cid:19)Zc (cid:21)
∞ c −c
+ dt dlE F((Yly,−)y∈R)φ Ilt,− p|a|−c+Ylc,−(t−Ilc,−)da
ZU Z0 (cid:20) (cid:18) (cid:19)Z−∞ (cid:21)
θ 0 u>0
Now, for ≥ , :
−c ∞ ∞
p (u)da= p (u)da= p (u)db
|a|−c+θ a−c+θ b
Z−∞ Zc Zθ
∞ b 1
= e−b2/2udb= e−θ2/2u
Zθ √2πu3 √2πu
Therefore : I (F)
c,t
J (F)= dt
c,U
√2πt
ZU
U J (F)
c,U
This equality is satis(cid:28)ed for every Borel set . Hen
e, by de(cid:28)nition of ,
t>0
the equality given in Proposition 2 o
urs for almost every .
t > 0 F
In order to prove it for all , we begin to suppose that is bounded and
ontinuous.
s t>0
In this
ase, for all , :
1 c 1 c
E F((Lyt)y∈R)1|Xt|≥cφ t Lytdy −E F((Lys)y∈R)1|Xs|≥cφ s Lysdy
(cid:12) (cid:20) (cid:18) Z−c (cid:19)(cid:21) (cid:20) (cid:18) Z−c (cid:19)(cid:21)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) 1 c 1 c (cid:12)
≤E F((Lyt)y∈R)φ t Lytdy −F((Lys)y∈R)φ s Lysdy
(cid:20)(cid:12) (cid:18) Z−c (cid:19) (cid:18) Z−c (cid:19)(cid:12)(cid:21)
(cid:12) (cid:12)
(cid:12) + F P( u [s,t], X =c) (cid:12)
(cid:12) ∞ u (cid:12)
|| || ∃ ∈ | |
t s t
If is (cid:28)xed, the (cid:28)rst term of this sum tends to zero when tends to , by
F φ
ontinuity of , and dominated
onvergen
e.
The se
ond term tends also to :
F P(X =c)=0
∞ t
|| || | |
Therefore, the fun
tion :
1 c
t→E F((Lyt)y∈R)1|Xt|≥cφ t Lytdy
(cid:20) (cid:18) Z−c (cid:19)(cid:21)
is
ontinuous.
I (F) t
c,t
Now, let us prove that is also
ontinuous with respe
t to .
9
t>0
For all :
F((Yly,+)y∈R)e−(Ylc1,+−)2/I2lc(,s+−/Islc,+)φ(cid:18)Ilcs,+(cid:19)s→→tF((Yly,+)y∈R)e−(Ylc1,+−)2/I2l(c,t+−/Itlc,+)φ(cid:18)Ilct,+(cid:19)
q q
φ c < t c t
by
ontinuity of (if Il,+ , it is
lear, and if Il,+ ≥ , the two expressions
s 3t/2
are equal to zero for ≤ ).
s 2t
Moreover, for ≤ :
F((Yly,+)y∈R)e−(Ylc1,+−)2/I2lc(,s+−/Islc,+)φ(cid:18)Ilcs,+(cid:19)≤√3||F||∞e−(Ylc,+)2/4t ≤√3||F||∞e−(Ylc,+)2/16t
q
(Yy )
Re
allingthattheLebesguemeasureisinvariantfortheBESQ(2)pro
ess l,+ y≥0,
we have : ∞ ∞
dlE e−(Ylc,+)2/16t = dle−l2/16t <
∞
Z0 h i Z0
t I (F)
c,t,+
By dominated
onvergen
e, → is
ontinuous.
t I (F) t
c,t,−
Similar
omputations imply the
ontinuity of → , and (cid:28)nally →
I (F)
c,t
is
ontinuous.
F
Consequently, for
ontinuous and bounded, the equality given in Proposi-
t>0 t>0
tion 2, whi
h was proven for a.e. , remains true for every .
Now, by monotone
lass theorem(see [RY99b℄),it is notdi(cid:30)
ult to extend this
equality to every measurable and positive fun
tion, whi
h
ompletes the proo(cid:3)f
of Proposition 2.
This proposition has the following
onsequen
e :
F
Corollary 2 : Let be a fun
tional whi
h satis(cid:28)es the
ondition of Pro-
position 2. The two following properties hold :
t>0
1) For all :
1 c
√2πtE F((Lyt)y∈R)1|Bt|≥cφ t Lytdy ≤√3I(F)
(cid:20) (cid:18) Z−c (cid:19)(cid:21)
t
2) When goes to in(cid:28)nity :
1 c
√2πtE F((Lyt)y∈R)1|Bt|≥cφ t Lytdy →I(F)
(cid:20) (cid:18) Z−c (cid:19)(cid:21)
φ(x)/√1 x √3 x 0
Proof : The (cid:28)rst property is obvious, sin
e − ≤ for all ≥ .
In order to prove the se
ond property, we distinguish two
ases :
I(F)<
1) If ∞, we observe that :
F((Yly,+)y∈R)e−(Ylc1,+−)2/I2l(c,t+−/Itlc,+)φ(cid:18)Ilct,+(cid:19)
q
10
