Large and moderate deviations for random walks in random scenery

Transcription

Large and moderate deviations for
random walks in random scenery: a
review.
Fabienne CASTELL
Université Aix-Marseille I
Rencontres du projet ANR MEMEMO 2, Roscoff, June 2011 – p. 1/11
Outline of the talk
The model and the problem.
Outline of the talk
Some motivations.
Outline of the talk
Some motivations.
The Gaussian case: Link with the self intersection local
times.
Large deviations for self-intersection local times.
What does it say on large deviations for random walks
in random scenery?
Outline of the talk
Some motivations.
times.
in random scenery?
The general case.
The phase diagram.
Heuristics;
What has been proved?
Outline of proofs.
Outline of the talk
Some motivations.
times.
in random scenery?
The general case.
The phase diagram.
Heuristics;
What has been proved?
Outline of proofs.
Open Questions.
The random walk in random scenery.
Yt =
v(
)
(v(x), x ∈ Zd ): i.i.d. real random variables.
Law: P,
Expectation: E.
E [v(x)] = 0, P [v(x) > t] " exp(−ctq ) for t $ 1 and q > 1
The random walk in random scenery.
Yt =
!
t
v(Xs) ds
0
(v(x), x ∈ Zd ): i.i.d. real random variables.
Law: P,
Expectation: E.
E [v(x)] = 0, P [v(x) > t] " exp(−ctq ) for t $ 1 and q > 1
(Xs , s ≥ 0): continuous time Markov chain with value in
Zd , independent of v .
Law: Px ,
Expectation:
Ex .
"
Generator Lf (x) = y∈Zd µ(x − y)f (y),
µ symmetric in the basin of attraction of an α-stable law.
0 < α ≤ 2.
Large and moderate deviations. t $ 1
Annealed law: Pa = P ⊗ P0 .
Since v is centered, Ea (Yt ) = 0.
Problem: find rough asymptotics of type
Pa [Yt ≥ b(t)y] " exp (−a(t)J(y)) ,
#
for b(t) $ Ea (Yt2 ).
Classically,
#
Moderate Deviations (MD):
Ea (Yt2 ) ( b(t) ( t
Large Deviations (LD):
b(t) = t
Very Large Deviations (VLD): t ( b(t).
Large and moderate deviations. t $ 1
Order of Ea (Yt2 )
lt (x) =
Z
t
0
n
o
d
δx (Xs ) ds , Rt = x ∈ Z , lt (x) > 0 .
p
t
t
v(x)lt (x) "
.
Yt =
card(Rt ) = p
card(R
)
t
card(Rt )
x∈Rt
X
d<α
d=α
d>α
Trans/Rec
card(Rt )
Recurrent
tα
Recurrent
Transient
1
t/ log(t)
t
p
Ea (Yt2 )
1
t1− 2α
p
t log(t)
√
t
References
Kesten & Spitzer ’79, Borodin ’79
Bolthausen ’89
Deligiannidis & Utev ’10.
Kesten & Spitzer ’79, Borodin ’79
Motivation: Diffusion in layered flows
Advection-diffusion equation:
x
dZt = dBt + V (Zt )dt
v(x)
Shear Flow:
0
1
0
1
x
0
@
A
@
A.
z=
, V (z) =
y
v(x)
Polluant
y
0
B
Zt = B
@
(x)
Bt
(y)
Bt
+
Z
0
t
v(Bs(x) ) ds
1
C
C.
A
v(x) ∼ N (0, 1). LD for the SILT.
Link with the self-intersection local times (SILT).
Conditionnally on (Xt ; t ≥ 0), Yt ∼= N (0, It ) where
It =
Z
0
t
Z
t
E [v(Xs )v(Xu )] dsdu =
0
Z
t
0
Z
t
0
δ0 (Xs − Xu ) dsdu =
X
lt2 (x) .
x
Hence,
Pa [Yt ≥ b(t)y] = P
h√
i
It Z ≥ b(t)y , Z ∼ N (0, 1) independent of It .
The question is now to obtain LDP principles for It
Link with the self-intersection local times (SILT).
Conditionnally on (Xt ; t ≥ 0), Yt ∼= N (0, It ) where
It =
Z
0
t
Z
t
E [v(Xs )v(Xu )] dsdu =
0
Z
t
0
Z
t
0
δ0 (Xs − Xu ) dsdu =
X
lt2 (x) .
x
Hence,
Pa [Yt ≥ b(t)y] = P
h√
i
It Z ≥ b(t)y , Z ∼ N (0, 1) independent of It .
The question is now to obtain LDP principles for It
Order of E(It )
It '
X „
x∈Rt
t
card(Rt )
d<α
E(It )
1
t2− α
«2
t2
'
.
card(Rt )
d=α
d>α
t log(t)
t
Very Large Deviations Results for the SILT
P [It ≥ c(t)y] " exp (−d(t)I(y)) , t2 ≥ c(t) ( E(It ) .
d(t)
d < 2α
t1−
2α
d
α
c(t) d
I(y)
References
α
K(α, d)y d
Chen, Rosen & Bass since ’04
Asselah PTRF ’08, Becker & König Preprint ’10
Laurent ’11
d > 2α
p
c(t)
√
K(L, d) y
Asselah & Castell PTRF ’07
Chen & Mörters J. Lond. Math. Soc. ’09
Asselah ALEA ’09, Laurent SPA ’10
d = 2α
p
c(t)
√
K(L, d) y
Castell AP ’10, Laurent SPA ’10
K(α, d) = K(L, d)
( R
)
α
2
1 Rd |ω| |ĝ(ω)| dω
K(α, d) = inf
; )g)2 = 1 ;
2
d
2
g:R #→R
)g)4
)
(
*g, Lg+
K(L, d) = inf
−
2 ; )g)2 = 1
g:Zd #→R
)g)4
Rough heuristics to explain the dichotomy d < or > 2α.
To produce many self-intersections, one strategy for the walk is to stay in a ball
Brt of radius rt within a time interval [0, τt ] ⊂ [0, t], and to run freely afterwards.
Gain: for z ∈ Brt , lτt (z) ' rτdt , and
t
p
τt2
d/2
It ' d = c(t) ⇔ τt = c(t)rt .
rt
t
p
τt2
d/2
It ' d = c(t) ⇔ τt = c(t)rt .
rt
Cost: this confinement happens with probability
p
d −α
τt
exp(− α ) " exp(− c(t)rt2 ) .
rt
t
p
τt2
d/2
It ' d = c(t) ⇔ τt = c(t)rt .
rt
Cost: this confinement happens with probability
p
d −α
τt
exp(− α ) " exp(− c(t)rt2 ) .
rt
Optimize the choice of rt under the constraint τt ≤ t.
d < 2α
d > 2α
τt = t
p
τt = c(t)
rt = t
2/d
c(t)
rt = 1
−1/d
d(t) = t
1− 2α
d
d(t) =
p
c(t)
α
d
c(t)
Almost a proof in the case d < 2α. Remind that τt = t.
rd
Scaled normalized local time: for x ∈ Rd , Lt (x) = tt lt ([xrt ]).
R
Then, Lt (x)dx = 1, and Lt satisfies a "weak" large deviations principle in
the space of probability measures, endowed with the weak convergence
topology.
„
«
t
P [Lt ' µ] " exp − α J (µ) ,
rt
8 R
< 1 )ω)α )ĝ(ω))2 dω if dµ = g 2 (x)dx
2
where J (µ) =
: +∞
otherwise.
R
It
Contraction principle: Note that Lt (x)2 dx = c(t)
. Hence
˜
ˆ
t
2
P [It ≥ c(t)y] = P )Lt )2 ≥ y " exp − α inf
rt
(
)!
‚ ‚2
‚ dµ ‚
‚
J (µ); ‚
.
‚ dx ‚ = y
2
Two problems with this "proof".
The LDP for Lt is a weak one, preventing the use of the contraction
principle. This problem is easy to fix, by comparison with the same quantity
for the random walk on the torus. Indeed, since
(x + y)2 ≥ x2 + y 2 for x, y positive real numbers ,
it is easy to see that
It ≤
(R)
It
, with
(R)
It
(R)
Now, Lt
‚
‚
‚ (R) ‚2
(R)
= ‚Lt ‚ and Lt periodization of Lt .
2
satisfies a full large deviations principle.
‚ dµ ‚
The function µ 0→ ‚ ‚ is not continuous in the topology of weak
dx
2
convergence. One needs to smooth Lt . Not so easy to do.
What does it say for the random walk in random scenery?
α 1
γ = max( , ) .
d 2
bt # t3/2
γ
t3/2 # bt # t1− 2
3/2
t
√
# bt # t log(t)
1
t3/2 # bt # t1− 2α
γ
p
t1− 2 # bt # var(Yt )
p
√
t log(t) # bt # var(Yt )
doesn’t exist
Strategy
Pa [Yt ≥ bt y]
«
„
2
b2
y
exp − t2t 2
bt
It maximal
It % lt (0)2 % t2
VLD for It
d>α
d=α
d<α
0
2γ
γ+1
(b y)
exp @−c t2γ−1
t γ+1
d<α
A
It %
2
γ+1
bt
(2γ−1)+
t γ+1
no constraint on It
d>α
d=α
1
„
b2
exp − tt
y2
2c
«
It % E(It ) % tc
A proof of the deviations of the RWRS (for d ≥ 2α, i.e γ = 12 ).
Case bt # t3/2 : Remind that It ≤ t2 .
ˆ√
˜
Upper bound: Pa [Yt ≥ bt y] = P
It Z ≥ bt y ≤ P [tZ ≥ bt y].
˜
ˆ
˜
ˆ√
It Z ≥ b t y
≥ P It ≥ t2 (1 − ")2 P [t(1 − ")Z ≥ bt y]
Lower bound: P
Case t3/2
≥ P (T ≥ t(1 − "))P [t(1 − ")Z ≥ bt y]
„
«
b2
y2
t
' exp(−t(1 − ")) exp − 2t2 (1−#)2
#
"
r
ˆ√
˜
It
Z
# bt # t3/4 : P
It Z ≥ b t y = P
4/3 1/3 ≥ y .
bt
−4/3
VLD for bt
−1/3
It and LD for bt
bt
Z at the same speed + contraction principle:
hp
i
P
It Z ≥ bt y ' exp
2/3
−bt
inf
(
2
√
z √
K u+
; uz = y
2
)!
.
√
Case t3/4 # bt # t: Remind that E(It ) % tc.
ˆ√
˜
Lower bound: P
It Z ≥ bt y ≥ P [It ≥ tc(1 − ")] P [tc(1 − ")Z ≥ bt y] ' P [tc(1 − ")Z ≥ bt y]
hp
i
˜
ˆ√
It Z ≥ b t y
≤ P [It ≥ tc(1 + ")] + P
tc(1 + ")Z ≥ bt y
Upper Bound P
„
«
2
` √ "´
b2
y
t
' exp − tc + exp − 2tc(1+#)
Need to have large deviations estimates for It , up to its mean (only known for α = 2, d = 2, 3, d ≥ 5).
General case: phase diagram
Phase diagram for d > α.
1< d/α < 2
2
d/α >= 2
t2
t
b =t1+1/q
t
bt
bt
V
II
t
IV
t
III
(q+1)(2α−d)/(2α−q(d−α))
b =t
t
(d+α)/(d+2α)
t
1−α/2d
t
t2/3
2/3
t
I
b =t1−1/(q+2)
t
1/2
t
1
2(d/α−1)
d/α
2
q
3
t1/2
1
2
q
3
d/α
General case: phase diagram
Phase diagram for d > α.
More precisely,
d/α = 1.05
T2
d/α = 1.2
T2
d/α = 1.5
T2
1+1/q
T
b
bT
bT
bT=T
(q+1)(2α−d)/(2α−q(d−α))
b =T
T
T
T
1/2
T
T
1/2
1 d/α
3
T
1/2
1
q
d/α = 1.7
T2
d/α
IV
3
T
d/α = 1.95
T2
1
q
T2
d/α
3
q
d/α >= 2
V
T
b
bT
bT
T
T
1−1/(q+2)
b =T
II
IV
T
T
III
T1/2
1
T(d+α)/(d+2α)
T1−α/2d
T1−α/2d
d/α
3
q
T1/2
I
1
d/α
q
3
T1/2
1
q
d/α
3
General case: heuristics
Rough heuristics to explain the different regions.
Yt =
Z
t
v(Xs )ds =
0
X
v(x)lt (x) .
x∈Zd
To make Yt large, a possible strategy is:
- for the walk, to stay confined in a ball Brt of radius rt within a time interval [0, τt ] ⊂ [0, t], and to run
freely afterwards.
P
τt
Gain: Yt has increased of an amount %
x∈Brt v(x) r d .
“t
”
τt
Cost: This confinement happens with probability ' exp − rα
t
Yt =
Z
t
v(Xs )ds =
0
X
v(x)lt (x) .
x∈Zd
To make Yt large, a possible strategy is:
- for the walk, to stay confined in a ball Brt of radius rt within a time interval [0, τt ] ⊂ [0, t], and to run
freely afterwards.
P
τt
Gain: Yt has increased of an amount %
x∈Brt v(x) r d .
“t
”
τt
Cost: This confinement happens with probability ' exp − rα
t
P
d −1
- for the scenery, to make x∈Br v(x) % bt rt τt
t
d/2
Cost If rt
)
d
b t rt
τt
) rtd , we are in a regime of moderate deviations for
probability of this event is about
exp
if
d
b t rt
τt
b2t rt2d
− 2 d
τ t rt
!
' exp
b2t rtd
− 2
τt
# rtd , we are in a regime of very large deviations for
of this event is about
„
„ «q «
bt
d
exp −rt
.
τt
!
P
P
x∈Brt
v(x). The
;
x∈Brt
v(x). The probability
Optimizing in (rt , τt ) leads to the phase diagram.
Case d ≥ 2α
Random Walk
Region I
Scenery
moderate deviations for
P
x∈R v(x)
no constraint
τt =
Region II
v(0) %
τt = bt
rt %
Region IV
rt %
Region V
rt %
1
d+α
bt
“
t
bt
d+α
d
d+α
bt
t
1
d+α d+α
t
τt = t
” q
q
q+1
bt
1
q+1
bt
large deviations for
P
x∈Br v(x)
τt = t
“ ” q
t
bt
b2t t−1
t
q
q+1
bt
rt % 1
Region III
Speed of the deviations
1
t d+α
very large deviations for
P
x∈Br v(x)
t
v(0) %
bt
t
“
t
bt
”
αq
d+α
“
bt
t
d
t d+α
”q
Optimizing in (rt , τt ) leads to the phase diagram.
Case d < 2α
Random Walk
Region I
Scenery
P
MD for x∈R v(x)
no constraint
τt =
Region II
q
q+1
bt
v(0) %
rt % 1
Region III
rt %
Region IV
rt %
Region V
rt %
τt = t
„ «
t3
b2
t
1
d+α
τt = t
“ ” q
t
bt
“
1
d+α d+α
t
τt = t
” q
t
bt
d+α
1
t d+α
MD for
P
VLD for
P
Speed of the deviations
b2t t−1
t
1
q+1
bt
x∈Brt
x∈Brt
v(0) %
bt
t
v(x)
v(x)
q
q+1
bt
d
t d+α
“
t
bt
”
“
bt
t
αq
d+α
“
bt
t
”
2α
d+α
d
t d+α
”q
General case: results
What has been done?
Except in the Gaussian case, only the case α = 2 has been considered.
Castell & Pradeille SPA ’01;
Gaussian case + correlations in space. Brownian motion instead of random walk;
1< d/α < 2
t2
d/α >= 2
t2
bt
bt
V
II
t
IV
t
III
t(d+α)/(d+2α)
1−α/2d
2/3
t
t2/3
t
I
1/2
t
1/2
1
2(d/α−1)
2
d/α q
3
t
1
2
q
3
d/α
What has been done?
Castell & Pradeille SPA ’01; Asselah & Castell PTRF ’03;
Bounded scenery. Brownian motion instead of random walk;
1< d/α < 2
t2
d/α >= 2
t2
bt
bt
V
II
t
IV
t
III
t(d+α)/(d+2α)
1−α/2d
2/3
t
t2/3
t
I
1/2
t
1/2
1
2(d/α−1)
2
d/α q
3
t
1
2
q
3
d/α
What has been done?
Castell & Pradeille SPA ’01; Asselah & Castell PTRF ’03; Castell Ann. IHP ’04;
Gaussian case + correlations in space. Brownian motion instead of random walk;
1< d/α < 2
t2
d/α >= 2
t2
bt
bt
V
II
t
IV
t
III
t(d+α)/(d+2α)
1−α/2d
2/3
t
t2/3
t
I
1/2
t
1/2
1
2(d/α−1)
2
d/α q
3
t
1
2
q
3
d/α
What has been done?
Gantert, König & Shi Ann IHP ’07;
1< d/α < 2
t2
d/α >= 2
t2
bt
bt
V
II
t
IV
t
III
t(d+α)/(d+2α)
t1−α/2d
t2/3
t2/3
I
1/2
1/2
t
1
2(d/α−1)
2
d/α q
3
t
1
2
q
3
d/α
What has been done?
Gantert, König & Shi Ann IHP ’07; Asselah & Castell PTRF ’07;
Only the speeds of deviations, and d > 4
1< d/α < 2
t2
d/α >= 2
t2
bt
bt
V
II
t
IV
t
III
t(d+α)/(d+2α)
1−α/2d
2/3
t
t2/3
t
I
t1/2
1
2(d/α−1)
2
d/α q
3
t1/2
1
2
q
3
d/α
What has been done?
Gantert, König & Shi Ann IHP ’07; Asselah & Castell PTRF ’07; Fleischman,
Mörters & Wachtel SPA ’08;
1< d/α < 2
t2
d/α >= 2
t2
bt
bt
V
II
t
IV
t
III
t(d+α)/(d+2α)
t1−α/2d
t2/3
t2/3
I
1/2
1/2
t
1
2(d/α−1)
2
d/α q
3
t
1
2
q
3
d/α
What has been done?
Mörters & Wachtel SPA ’08; Asselah ALEA ’09
Only d > 4.
1< d/α < 2
t2
d/α >= 2
t2
bt
bt
V
II
t
IV
t
III
t(d+α)/(d+2α)
1−α/2d
2/3
t
t2/3
t
I
t1/2
1
2(d/α−1)
2
d/α q
3
t1/2
1
2
q
3
d/α
What has been done?
Mörters & Wachtel SPA ’08; Asselah ALEA ’09
Corollary of the deviations for the SILT.
1< d/α < 2
t2
d/α >= 2
t2
bt
bt
V
II
t
IV
t
III
t(d+α)/(d+2α)
1−α/2d
2/3
t
t2/3
t
I
t1/2
1
2(d/α−1)
2
d/α q
3
t1/2
1
2
q
3
d/α
Open Questions.
Moderate deviations for the SILT in the upper-critical
case, i.e. find the rough asymptotics for#
P [It − E(It ) ≥ c(t)y] for E(It ) ≥ c(t) $ var(It ).
Open Questions.
Describe the typical behavior of the random walk when it
makes a lot of self intersections.
Open Questions.
For the random walk in random scenery, fill in the holes
in the phase diagram.
Open Questions.
For the random walk in random scenery, fill in the holes
in the phase diagram.
Functional large deviations for the random walk in
random scenery.
References.
Central limit theorems for the RWRS.
Borodin, A. N. A limit theorem for sums of independent random variables defined on a recurrent
random walk. (Russian) Dokl. Akad. Nauk SSSR 246 (1979), no. 4, 786–787.
Borodin, A. N. Limit theorems for sums of independent random variables defined on a transient
random walk. Investigations in the theory of probability distributions, IV. Zap. Nauchn. Sem.
Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 85 (1979), 17–29.
Kesten, H.; Spitzer, F. A limit theorem related to a new class of self-similar processes. Z.
Wahrsch. Verw. Gebiete 50 (1979), 89–121.
Deligiannidis, G.; Utev, S. An asymptotic variance of the self-intersections of random walks. Arxiv
2010.
Large deviations for intersection local times
About 20 papers by Xia Chen, R. Bass & J. Rosen since 2004. All this work is the material of a
very exhaustive book by X. Chen:
Chen, X. Random Walk Intersections: Large Deviations and Related Topics. Mathematical
Surveys and Monographs, AMS. (2009) Vol. 157, Providence, RI.
A review paper by W. König
Upper tails of self-intersection local times of random walks: survey of proof techniques, available
on his homepage.
References.
Large deviations for intersection local times
Asselah, A., Castell, F. : Self intersection times for random walks, and Random walk in random
scenery in dimensions d ≥ 5. Probab. Theory Related Fields 138 (2007), no. 1-2, 1–32.
Asselah, A. Large deviations estimates for self-intersection local times for simple random walk in
Z3 . Probab. Theory Related Fields 141 (2008), no. 1-2, 19–45.
Chen, Xia; Mörters, P.: Upper tails for intersection local times of random walks in supercritical
dimensions. J. Lond. Math. Soc. (2) 79 (2009), no. 1, 186–210.
Asselah, A. Large deviation principle for self-intersection local times for random walk in Zd with
d ≥ 5. ALEA Lat. Am. J. Probab. Math. Stat. 6 (2009), 281Ð322.
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Large and moderate deviations for random walks in random scenery

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