Density estimates on a parabolic spde

Publ. Ma . 46 (2002), 77–96
DENSITY ESTIMATES ON A PARABOLIC SPDE
D. M´
a quez-Ca e as1and M. Mellouk2
Abs ac
We conside a gene al class o pa abolic spde’s
∂uε
,x
∂ =∂2uε
,x
∂x2+∂
∂xg(uε
,x)+ (uε
,x)+εσ(uε
,x)˙
W ,x,
wi h ( , x)∈[0,T]×[0,1] and ε˙
W ,x,ε>0, a pe u bed Gaussian
space- ime whi e noise. Fo ( , x)∈(0,T]×(0,1) we p o e he
called Da ies and Va adhan-L´eand e es ima es o he densi y pε
,x
o he solu ion uε
,x.
1. In oduc ion
In his pape we deal wi h he pe u bed pa abolic s ochas ic pa ial
diffe en ial equa ion (spde)
∂uε
,x
∂ =∂2uε
,x
∂x2+∂
∂xg(uε
,x)+ (uε
,x)+εσ(uε
,x)˙
W ,x,(1.1)
( , x)∈[0,T]×[0,1], ε>0, wi h ini ial condi ion uε
0,x =ξ(x) and Di ich-
le ’s bounda y condi ions uε
,0=uε
,1= 0. The p ocess {˙
W ,x,( , x)∈
[0,T]×[0,1]}is a space- ime whi e noise on a comple e p obabili y
space (Ω,F,P); σ, , g :R→Ra e smoo h unc ions and ξis some
eal- alued unc ion defined on [0,1].
I σ= = 0 and g( )= 2/2, he abo e equa ion is called Bu ge s
equa ion. I a ises connec ion wi h he s udy o u bulen fluid mo ion
and he li e a u e a aches g ea impo ance o his ac (see, o in-
s ance, [4]). Recen ly, Bu ge s equa ion pe u bed by space- ime whi e
2000 Ma hema ics Subjec Classifica ion. 60H15, 60H07, 35R60.
Key wo ds. Mallia in Calculus, pa abolic spde’s, Da ies and Va adhan-L´eand e es-
ima es, space ime whi e noise, la ge de ia ions.
1Suppo ed by he g an PB 960088 om he Subdi ecci´on Gene al de Fo maci´on
y P omoci´on del Conocimien o and he g an ERBF MRX CT960075A om he
Eu opean Union.
2Suppo ed by a g an o he INRIA, domaine de Voleceau-Rocquencou , 78153
Chesnay Cedex, F ance.
78 D. M´
a quez-Ca e as, M. Mellouk
noise has been conside ed in se e al pape s ([8], [9], [12] and he e e -
ences he ein). As g= 0, (1.1) is a s ochas ic eac ion-diffusion equa ion,
wha has also been s udied in ensi ely (see, o ins ance, [25], [1]).
The equa ion (1.1) can be igo ously o mula ed as an in eg al e olu-
ion equa ion
(1.2) uε
,x =G (x, ξ)+ε
01
0
G −s(x, y)σ(uε
s,y)W(ds, dy)
−
01
0
∂G −s
∂y (x, y)g(uε
s,y)ds dy +
01
0
G −s(x, y) (uε
s,y)ds dy,
whe e G (x, y) is he undamen al solu ion o he hea equa ion on
[0,T]×[0,1] wi h Di ichle ’s bounda y condi ions and G (x, ξ)=
1
0G (x, y)ξ(y)dy (see Appendix o mo e in o ma ion abou his un-
damen al solu ion). Basic esul s conce ning exis ence and uniqueness
o solu ion o (1.1) a e gi en in [12]. Unde mo e es ic i e assump-
ions on he coefficien s ( ,gand σsmoo h enough), Mo ien [20] has
es ablished ha , o each fixed ( , x)∈(0,T]×(0,1) and ε∈(0,1], uε
,x
is an infini ely diffe en iable unc ional in he sense o Mallia in Cal-
culus. Adding a s ic ellip ici y hypo hesis, Mo ien has checked ha
uε
,x possesses a C∞densi y y→pε
,x(y) wi h espec o he Lebesgue
measu e. Conside ing he s ochas ic Bu ge s equa ion, i.e. g( )= 2/2,
and assuming a nondegene acy condi ion on he diffusion coefficien ,
Zaidi and Nuala [26] ha e p o ed ha he law o he solu ion is ab-
solu ely con inuous. Applying echniques o Mallia in Calculus oge he
wi h he Cole-Hop ans o ma ion and assuming ha he dispe sion σ
does no depend on uε
,x and 1/K ≤σ≤K o some cons an K>0,
J. L´eon e al. [18] ha e shown ha uε
,x has a smoo h densi y a all poin
( , x)∈(0,T]×(0,1).
Le Hdeno e he Came on-Ma in space associa ed wi h he B ownian
shee {W ,x,( , x)∈[0,T]×[0,1]}, and se hH=T
01
0|˙
hs,y|2ds dy1/2,
wi h ˙
hs,y =∂2hs,y/∂s∂y. Fo any h∈H, le {Sh
,x,( , x)∈[0,T]×[0,1]}
be he solu ion o he de e minis ic e olu ion equa ion
(1.3) Sh
,x =G (x, ξ)+
01
0
G −s(x, y)σ(Sh
s,y)˙
hs,y ds dy
−
01
0
∂G −s
∂y (x, y)g(Sh
s,y)ds dy +
01
0
G −s(x, y) (Sh
s,y)ds dy.
Densi y Es ima es on a Pa abolic SPDE 79
We se , o y∈R,
d2(y) = in 1
2h2
H,h∈H,S
h
,x =y.(1.4)
Ou fi s aim is o p o e he called Da ies es ima e o he densi y pε
,x,
which is he uppe bound e sion o A onson’s es ima es. The hea ke nel
case was s udied by Da ies [10]. Kusuoka and S oock [14] ha e deal
wi h he diffusion p ocesses, hey ha e ob ained a comple e app oach in
a small ime using he scaling p ope y o he B ownian mo ion. Fo
he semig oup p (x, ·) associa ed wi h he gene a o o a diffusion, hey
ob ained he uppe and he lowe bounds o he o m 1/√ imes an
exponen ial e m ela ed o he dis ance associa ed wi h he gene a o .
In ou pape we do no ha e he scaling p ope y, we will wo k in
e ms o pa ame e εwhich p oduces small pe u ba ions o he solu ion
o (1.1). Combining exponen ial es ima es o he ail p obabili ies and
Mallia in Calculus, we will p o e ha he uppe bound o pε
,x(y)iso
he o m 1/ε imes an exponen ial e m o he ype
−C|y−S0
,x|2
ε2
o e e y ε∈(0,1), y∈R, whe e S0
,x is he solu ion o (1.3) as h=0.
A simila esul o one-dimensional wa e equa ion pe u bed by a whi e
noise has been analysed by L´eand e and Russo [17].
Secondly we analyse he loga i hmic es ima es o he densi y pε
,x,
hese es ima es a e known as Va adhan-L´eand e es ima es. Assuming
some condi ions on he coefficien s as in [20], we p o e ha , o fixed
( , x), he densi y pε
,x dec eases exponen ially as εcon e ges o 0 as
ollows
exp −d2(y)
ε2.
In he diffusion case, due o scaling p ope y, his p oblem is ela ed o
he s udy o he densi y in small ime. We e e o [15], [16] o such
kind o es ima es. The eac ion-diffusion p oblem, i.e. g= 0 in (1.1),
has been ea ed by Mille and Sanz-Sol´e[19].
The pape is o ganized as ollows. In he nex sec ion we o mula e
he s a emen s o he main esul s as Theo ems 2.1 and 2.2. Sec ion 3
is de o ed o he p oo o Theo em 2.1. In Sec ion 4, we p o e Theo-
em 2.2 and analyse he fini eness o d2(y) defined in (1.4). In Sec ion 5
we apply he esul o Sec ion 3 o he eac ion-diffusion equa ion. The
a gumen s o Sec ions 3 and 4 depend on accu a e es ima es o he G een
unc ion G (x, y), which a e gi en as an Appendix. Fo all no ions and
80 D. M´
a quez-Ca e as, M. Mellouk
no a ions conce ning he Mallia in Calculus, using along he pape , we
e e o [21], [22]. As usual, all cons an s a e deno ed by C, indepen-
den ly o hei alues.
2. S a emen o he main esul s
This sec ion is de o ed o enuncia e he main esul s o he a icle.
We in oduce he ollowing hypo hesis on he coefficien s and he ini-
ial condi ion:
(H1) ,g,σ:R→Rand C∞- unc ions wi h bounded de i a i es o any
o de g ea e han one, σis uni o mly bounded and ξ∈C([0,1]).
(H2) The e exis s C>0 such ha in {|σ(x)|;x∈R}≥C.
Along he pape we fix ∈(0,T] and x∈(0,1).
Theo em 2.1 (Da ies es ima e).Assume (H1) and (H2). Then, he e
exis some cons an s C1,C
2>0such ha
pε
,x(y)≤C1
εexp −|y−S0
,x|2
C2ε2,
o any y∈Rand ε∈(0,1).
Rema k. Al hough we assume (H1) in o de o ob ain Theo em 2.1, he
p oo s ill goes h ough unde weake condi ions.
Theo em 2.2 (Va adhan-L´eand e es ima e).Unde (H1) and (H2),
lim
ε↓0ε2log pε
,x(y)=−d2(y),(2.1)
wi h d2(y)defined in (1.4).
Rema k. The boundedness o σis needed o ensu e exis ence and
smoo hness o pε
,x [20].
3. Da ies es ima e
In his sec ion ou main pu pose is he p oo o Theo em 2.1. In
o de o p o e i we need some echnical lemmas. The fi s one is an
exponen ial es ima e o he ail p obabili ies.
Lemma 3.1. Assume ,gLipschi z and σLipschi z and bounded. Fo
any p∈[1,∞), he e exis s ρ>0la ge enough such ha
sup
0<ε≤1
Eexp p|uε
,x −S0
,x|2
ρε2<∞.
Densi y Es ima es on a Pa abolic SPDE 81
P oo : Fo ( , x)∈[0,T]×[0,1], acco ding o (1.2) and (1.3), clea ly
|uε
,x −S0
,x|≤
01
0
∂G −s
∂y (x, y)g(uε
s,y)−g(S0
s,y)ds dy
+
01
0
G −s(x, y) (uε
s,y)− (S0
s,y)ds dy
+ε
01
0
G −s(x, y)σ(uε
s,y)W(ds, dy).
Hence, Lipschi z’s condi ions on and g, Schwa z’s inequali y, (6.1) and
(6.2) yield he exis ence o a cons an C>0 such ha
|uε
,x −S0
,x|2≤ε2
01
0
G −s(x, y)σ(uε
s,y)W(ds, dy)
2
+C
0
1
√ −ssup
0≤y≤1|uε
s,y −S0
s,y|2ds.
Using G onwall’s Lemma, we ob ain
sup
0≤ ≤T
sup
0≤x≤1|uε
,x −S0
,x|2≤Cε2


·
01
0
G·−s(∗,y)σ(uε
s,y)W(ds, dy)



2
∞
.
The e o e, since σis uni o mly bounded, an exponen ial inequali y o
s ochas ic in eg als in ol ing he G een ke nel G (x, y) (see Lemma 3.2
in [23] o also [24]) implies ha he e exis some posi i e cons an s 0
and C0, such ha
P|uε
,x −S0
,x|2
ε2>

≤P


·
01
0
G·−s(∗,y)σ(uε
s,y)W(ds, dy)



2
∞
>
C≤exp −
C0,
o any ≥ 0.

82 D. M´
a quez-Ca e as, M. Mellouk
Now, le 0,C
0>0 be as be o e and choose ρ>0 la ge enough such
ha C0p<ρ. Then, Fubini’s s ochas ic heo em and he sui able choice
o ρgi e
Eexp p|uε
,x −S0
,x|2
ρε2≤ep 0/ρ +E1
ε2|uε
,x−S0
,x|2
0
p
ρep
ρydy
≤ep 0/ρ +∞
0
p
ρe(p
ρ−1
C0)ydy
<+∞.
This concludes he p oo o he lemma.
Fo any ε∈(0,1), we conside he andom a iable defined by
ˆuε
,x =uε
,x −S0
,x
ε.
Assume (H1). S anda d a gumen s based on Bu kholde ’s, H¨olde ’s and
G onwall’s inequali ies (see [20, P oposi ion 5.1]) yield o any k∈N,
p≥1,
sup
0<ε<1
sup
,x uε
,xk,p ≤C,(3.1)
sup
0<ε<1
sup
,x ˆuε
,xk,p ≤C,(3.2)
whe e ·
k,p deno es he no m o he Sobole space Dk,p, ha is, o
k∈N,p≥1,
Fp
k,p =E(|F|p)+
k

j=1
E(DjFp
Hj)
(see [22] o basic defini ions).
I only emains o s udy he Mallia in ma ix γε
,x o uε
,x.
Lemma 3.2. Assume (H1) and (H2). Fo any p≥1,ε∈(0,1),
(γε
,x)−1p≤Cε−2,(3.3)
whe e γε
,x =
01
0|D ,zuε
,x|2d dz and ·pis he Lp(Ω)-no m.
Densi y Es ima es on a Pa abolic SPDE 83
P oo : Le Mε
,x( , z) be he solu ion o
Mε
,x( , z)=G − (x, z)+ε
1
0
G −s(x, y)σ(uε
s,y)Mε
s,y( , z)W(ds, dy)
−
1
0
∂G −s
∂y (x, y)g(uε
s,y)Mε
s,y( , z)ds dy
+
1
0
G −s(x, y) (uε
s,y)Mε
s,y( , z)ds dy.
Clea ly, he Mallia in de i a i e o uε
,x is gi en by he ollowing equa ion
D ,zuε
,x =1
{ < }εσ(uε
,z)Mε
,x( , z).
Hence,
γε
,x =ε2
01
0
σ(uε
,z)2Mε
,x( , z)2d dz.
Compu a ions simila o hose used o p o e P oposi ion 5.2 in [20] show
ha he e exis s a cons an C>0 such ha
sup
0<ε<1
E
01
0
σ2(uε
,z)Mε
,x( , z)2d dz−p
<C,
o any p≥1. Consequen ly, (3.3) is sa isfied.
We a e now eady o gi e he p oo o Theo em 2.1.
P oo o Theo em 2.1: Le y∈Rand ρ>0 la ge enough. By a change
o a iable and he s ochas ic in eg a ion by pa s o mula o Mallia in
Calculus (see, o ins ance, P oposi ion 3.2.1 in [22]), i δ{y}deno es he
84 D. M´
a quez-Ca e as, M. Mellouk
Di ac δ- unc ion a y, hen
pε
,x(y)=Eδ{y}(uε
,x)
= exp −|y−S0
,x|2
ρε2Eδ{y}(uε
,x) exp |uε
,x −S0
,x|2
ρε2
=1
εexp −|y−S0
,x|2
ρε2Eδ{0}(ˆuε
,x) exp |uε
,x −S0
,x|2
ρε2
=1
εexp −|y−S0
,x|2
ρε2E1{ˆuε
,x>0}D∗Dˆuε
,x(ˆγε
,x)−1
×exp |uε
,x −S0
,x|2
ρε2,
(3.4)
whe e ˆγε
,x =γε
,x/ε2and D∗deno es he adjoin ope a o o D, also
called he Sko ohod in eg al (see [22]).
Fi s , no ice ha (3.4) is well-defined. Indeed, by Lemma 3.1, he
exponen ial e m o (3.4) belongs o D1,p
loc uni o mly in ε∈(0,1) o
any p≥1 and ρ>0 la ge enough. Finally, simila a gumen s as in
P oposi ion 6 in [13] (see also [2, Lemma 3.36]), oge he (3.2) and
Lemma 3.2, yield
E
1{ˆuε
,x>0}D∗Dˆuε
,x(ˆγε
,x)−1exp |uε
,x −S0
,x|2
ρε2
<∞,
and his comple es he p oo o he heo em.
Rema k. As g= 0, we deal wi h he well-known s ochas ic hea equa ion
and S0
,x in Theo em 2.1 is he solu ion o he ollowing de e minis ic
e olu ion equa ion
,x =G (x, ξ)+1
01
0
G −s(x, y) ( s,y)ds dy.
4. Va adhan-L´eand e es ima e
In o de o p o e Theo em 2.2, we need wo lemmas p o ed by Nu-
ala [22]. These lemmas a e p esen ed o gene al Wiene unc ion-
als ollowing he o mula ion in he case o diffusions p ocesses o Ben
A ous’s and L´eand e’s me hod (see [3]).
Densi y Es ima es on a Pa abolic SPDE 85
Le {W(h),h∈H}be an a bi a y Gaussian amily. We ecall ha a
andom a iable F:Ω→Ris said o be nondegene a e i F∈D∞(R)=
k≥1p≥1Dk,p(R) and he Mallia in ma ix γF=DF,DFHsa isfies
γ−1
F∈∩
p≥1Lp(Ω).
Lemma 4.1 ([22, P oposi ion 4.4.1]).Conside a amily {Fε,0<ε<
1}o nondegene a e andom a iables, and a unc ion Φ∈C1
p(H,R)such
ha
lim
ε↓0
1
εFεw+h
ε−Φ(h)=Z(h),
in he opology o D∞, o each h∈H, whe e Z(h)is a andom a iable
in he fi s Wiene chaos wi h a iance γΦ(h). Define
d2
R(y) = in 1
2h2
H,Φ(h)=y, γΦ(h)>0,y∈R.
Then, i pεdeno es he densi y o Fε,
lim in
ε↓0ε2log pε(y)≥−d2
R(y).
Lemma 4.2 ([22, P oposi ion 4.4.2]).Le {Fε,ε∈(0,1)}be a amily
o nondegene a e andom a iables sa is ying
i) sup
0<ε<1Fεk,p <∞, o each k≥1,p∈[1,∞).
ii) Fo any p≥1, he e exis s N(p)∈[1,∞)such ha γ−1
Fεp≤
ε−N(p) o e e y ε∈(0,1].
iii) The amily {Fε,ε∈(0,1)}sa isfies a la ge de ia ion p inciple on
Rwi h a e unc ion I(y),y∈R.
Then, i pεdeno es he densi y o Fε,
lim sup
ε↓0
ε2log pε(y)≤−I(y).
We nex check ha uε
,x sa isfies he equie emen s o Lemma 4.1 and
Lemma 4.2. These wo lemmas gi e, epec i ely, a lowe and an uppe
bound o
lim
ε↓0ε2log pε
,x(y).
Assump ion (H1) implies ha o fixed ( , x)∈[0,T]×[0,1], he map-
ping h∈H→Sh
,x, defined in (1.3), is infini ely F ´eche diffe en iable.
Fu he mo e, he F ´eche de i a i e o Sh
,x is gi en by
DSh
,x(k)=T
01
0
D ,zSh
,x ˙
k ,z d dz, k ∈H,
92D.M
´
a quez-Ca e as, M. Mellouk
Rema k. In his pa icula case ( he s ochas ic hea equa ion), o any
y0∈R, we a e able o find a pa icula elemen o Hwi h a special
s uc u e such ha applied o he skele on is equal o y0.
P oo o Co olla y 5.1: Le ( , x)∈[0,T]×[0,1] be fixed. We will p o e
ha , o any y0∈R, he e exis s h(0) ∈Hsa is ying ψh(0)
,x =y0and
h(0)H≤C|y0−ψ0
,x|, o some posi i e cons an Cdepending on σ,
and he G een ke nel. Then, ¯
d2(y0)≤1
2h(0)2
H, and his ac implies
(5.3).
Fo any (u, z)∈[0,T]×[0,1], define
(5.4) 7(u, z)=Gu(z,ξ)+u
01
0
Gu−s(z,y) (ψ0
s,y)ds dy
+u
01
0
Gu−s(z,y)˙
k ,x(s, y)ds dy,
whe e
˙
k ,x(s, y)=G −s(x, y)(y0−ψ0
,x)
01
0
G2
− (x, )d d −1
.
Then, k ,x(·,·)∈Hand i sa isfies 7( , x)=y0.
Fo any (s, y)∈[0,T]×[0,1] se
˙
h(0)
s,y =− (7(s, y)) − (ψ0
s,y)−˙
k ,x(s, y)
σ(7(s, y)) .(5.5)
Then, (5.4) and (5.5) imply
7(u, z)=Gu(z,ξ)+u
01
0
Gu−s(z,y)σ(7(s, y))˙
h(0)
s,y ds dy
+u
01
0
Gu−s(z,y) (7(s, y)) ds dy,
and, by uniqueness o solu ion, 7(u, z)=ψh(0)
u,z o any (u, z)∈[0,T]×
[0,1]. In pa icula ψh(0)
,x =y0. Mo eo e , om (5.5), we ha e
h(0)2
H≤C(A1+A2),

Densi y Es ima es on a Pa abolic SPDE 93
wi h
A1=T
01
0 (7(s, y)) − (ψ0
s,y)
σ(7(s, y)) 2
ds dy,
A2=T
01
0˙
k ,x(s, y)
σ(7(s, y))2
ds dy.
Then, i is easy o check ha h(0)2
H≤C|y0−ψ0
,x|2.
6. Appendix
Le G (x, y) deno e he undamen al solu ion o he hea equa ion
wi h Di ichle ’s bounda y condi ions. Tha means
G (x, y)= 1
√4π
+∞

n=−∞exp −(y−x−2n)2
4 −exp (y+x−2n)2
4 .
We ecall he ollowing p ope ies, o e e y x, y ∈[0,1], ∈[0,T], β>0,
G (x, y)≤C
√ exp −(y−x)2
4 ,
sup
0≤x≤11
0|G (x, y)|βdy ≤C −β
2+1
2,(6.1)
sup
0≤x≤11
0
∂G
∂y (x, y)
β
dy ≤C −β+1
2,(6.2)
 +h
1
0
∂G +h−s
∂y (x, y)
β
dy ds ≤Cβh3
2−β,(6.3)
o h>0,0<β<3
2.
We e e o [7] and [20] o he p oo o esul s on his G een ke nel.
Lemma 6.1 (Lemma 3.1 in [19]).The e exis s a≥1such ha o any
( , x)∈(0,T]×(0,1),0<µ<in  , x2
a2,(1−x)2
a2,

−µx+√µ
x−√µ
G2
−s(x, y)ds dy ≥C√µ,
whe e C=1
4$2
π1−1
√2π.
94D.M
´
a quez-Ca e as, M. Mellouk
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96D.M
´
a quez-Ca e as, M. Mellouk
Facul a de Ma em`a iques
Uni e si a de Ba celona
G an Via 585
08007 Ba celona
Spain
E-mail add ess:[email p o ec ed]
E-mail add ess:[email p o ec ed]
P ime a e si´o ebuda el 14 de eb e de 2001,
da e a e si´o ebuda el 18 d’oc ub e de 2001.