Couplage du principe des grandes
déviations et de l’homogénisation dans le
cas des EDP paraboliques:
(le cas constant)
Alioune COULIBALY
U.F.R Sciences et Technologie
Université Assane SECK de Ziguinchor
Probabilité et Analyse - CIMPA 2014
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History
Intro
LDP
Convergence
Plan
1
History
2
Intro
3
LDP
4
Convergence
beamer-tu-log
History
Intro
LDP
Convergence
History
History
P. Baldi (1991)
Freidlin and Sowers (1999)
Diédhiou and Manga (2007)
Diédhiou (Cimpa 2014)
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History
Intro
LDP
Convergence
Introduction
abstract
We consider the coupling of homogenization and large
deviation principle in partial differential equation (PDE). We
compare them with the help of the ratio δ/ between the small
viscosity parameter () and the homogenization parameter (δ);
this comparison is required as and δ tend to zero. We use
some large deviation estimates to study the behavior of the
PDE solution.
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PDE
PDE

u ,δ



(t, x) = L,δ u ,δ (t, x) + 1 f
∂t


 u ,δ (0, x) = g(x),
x ∈ Rd
x
,δ
δ , u (t, x)
(1)
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Convergence
assumption
f is a nonlinear function 1-periodic and verify:
for all x, f (x, 1) = 0
there is c a bounded function : f (x, y) = c(x, y).y
and
c(x, y) > 0 for all x and y in (0,1)
c(x, y) ≤ 0 if not
max c(x, y) = c(x)
and
g ∈ C Rd , R+ a bounded function
: supx∈Rd g(x) = ḡ < ∞.
d
Take G0 = x ∈ R : g(x) > 0 , g is continuous one notes
G̊0 = G0
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definition
Let (Ω, F, P) a probability tripleon which a d-dimensional
Brownian motion W 1 , . . . , W d is defined. Let E the
corresponding expectation operator. We have already defined
h., .i as the standard euclidian inner product on Rd ; let k . k be
the associated norm. Also let Td be the d-dimensional torus of
size 1 and C Td ; Rd be the space of continuous mapping from
Td to Rd ; let k . kC (Td ;Rd ) be the associated supremum norm.
Also we define P Td as the collection of all probability
measure on Td .
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History
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LDP
Convergence
Approach
Consider the Markov diffusion process Xtx,,δ ∈ Rd governed by
the operator :
L,δ =
d
d
x ∂2
x ∂
X
X
(σσ ∗ )ij
+
B ,δ
2
δ ∂xi ∂xj
δ ∂xi
i,j=1
i=1
The trajectories of this process can be constructed with the
help of the SDE:

 dX x,,δ = √σ Xtx,,δ dW + B ,δ Xtx,,δ dt
t
t
δ
δ
 x,,δ
X0
=x
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Convergence
assumption
σ : Rd −→ Rd×d and B ,δ : Rd −→ Rd are regular applications.
The vector-valued function B ,δ is given by :
B0 , B1 and B2,δ
B ,δ = B0 + B1 + B2,δ
δ
are C ∞ Rd , Rd for all , δ > 0 and
lim k B2,δ kCp (Rd ,Rd ) = 0
,δ→0
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hypothesis
Since the two parameters δ (homogenization) and (large
deviation) tend to zero, we consider a new defined parameter
δ = δ
Suppose that lim↓0 δ = γ, where γ > 0 a constant . There
results that the homogenization parameter and the large
deviation parameter go at the same rate
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History
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some rescalings
covering map
Define X̃tx,,δ by : X̃tx,,δ =
1 x,,δ
δ X δ 2
√
t
Then
(
d X̃tx,,δ = σ X̃tx,,δ d W̃t +
X̃0x,,δ =
where W̃t,δ =
δ ,δ
B
X̃tx,,δ dt
x
δ
√
,δ
δ W δ 2
√
is a Brownian motion.
t
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History
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Convergence
new consideration
the torus Td
n
o
Thereafter, consider the process X̃t,δ : t ≥ 0 Td -valued
which generator is defined by :
L̃,δ =
d
d
1X
∂2
δ X ,δ
∂
(σσ ∗ )ij (x)
+
B (x)
2
∂xi ∂xj
∂xi
i,j=1
i=1
Consider a = σσ ∗ , the above generator converges to the
operator
Lγ =
d
d
d
X
X
∂2
∂
∂
1X
(a)ij (x)
+
Bi0 (x)
+γ
Bi1 (x)
2
∂xi ∂xj
∂xi
∂xi
i,j=1
i=1
i=1
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reference
tools
The basic set of calculations of this subject involves deriving the
Varadhan formula and in identifying the constant C 2 . The main
technique for showing that is the following result : (Baxendale
and Stroock (1988), corollary 1.12, p[183] to p[185])
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History
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Convergence
LDP
rate function
Let J 2 (θ) defined by Freidlin and Sowers (1999), the limit of
1
hθ, X x,,δ i
log E exp
J 2 (θ) = n
Z
inf
φ∈C ∞ Td
o n
sup
o
µ∈P Td
+
Td
1
γ

d
1 X

2 k =1
2
h(I − ∇φ) σk (z), θi + h(I − ∇φ) B1 (z), θi
h(B0 − L0 φ) (z), θi µ(dz)
Define the Legendre-fenchel of J 2 (θ)
D 0E
0
J 2 (θ) = sup
θ, θ − J 2 (θ )
0
{θ ∈Rd }
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History
Intro
LDP
Convergence
large deviation principle
LDP
Define in addition :
2
S0,T
(φ)
=
 Z

T
.
J 2 φ (s) ds
if φ absolutely continuous and φ(0) = x
0

∞
if not
Theorem (Freidlin and Sowers 1999)
n
o
Fix T > 0 and x ∈ Rd . The family Xtx,,δ : 0 ≤ t ≤ T
of
>0
d
C [0, T ] , R -valued random variables has a large deviation
2 (φ) for all φ ∈ C [0, T ] , Rd .
principle with rate function S0,T
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History
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Convergence
barrier
constant
Now define
Z
C2 =
sup
c(z)µ(dz)
µ∈P (Td ) Td
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History
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LDP
Convergence
fundemental theorem
Varadhan formula
Theorem
Let c be an element of C ∞ Td and D a Borel subset of
C [0, t] , Rd . Then




t
Z
1

x,,δ
c
Xs
0


x,,δ
lim inf log E 
)e
1D (Xt
!
ds
δ
2
 ≥tC 2 − inf S0,t
(φ)

◦


↓0


x,,δ
lim sup log E 
)e
1D (Xt
↓0






φ∈D
t
Z
1


x,,δ
c
0
Xs
δ
!
ds



2
 ≤tC 2 − inf S0,t
(φ)

φ∈D
{
}


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History
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LDP
Convergence
behavior of u ,δ
Feynman Kac

u ,δ (t, x) = E g Xtx,,δ e
1
Rt
0
c
x,,δ
Xs
δ
,Ysx,,δ

ds

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History
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LDP
Convergence
BSDE
Pardoux Peng 1992

R  Ysx,,δ = g (X x,,δ ) + 1 t f Xrx,,δ ,Yrx,,δ
t
δ
s
o
nR
t
 E
|Zrx,,δ |2 dr < ∞
s
dr
−
1
√
Rt
s
x,,δ
Zr
dWr
,0 ≤ s ≤ t
and
Y0x,,δ = u ,δ (t, x)
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viscosity solution
Pradeilles 1998
Since u ,δ (t, x) > 0, denote v ,δ (t, x) = log u ,δ (t, x) and
observe, that v ,δ (t, x) is a viscosity solution of :






2
1
x
∂v ,δ
(t, x) = L,δ v ,δ (t, x) + ∇v ,δ (t, x)σ (x) + c
, u ,δ (t, x)
∂t
2
δ
v ,δ (0, x) = log (g(x)) ,
x ∈ G0




d
 lim v ,δ (t, x) = −∞,
x ∈ R \G0
t→0
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some notation
definition
Let us define a distance in R+ × Rd , by for
(t, x), (s, y ) ∈ R+ × Rd :
d {(t, x), (s, y)} = max {|t − s|, kx − y k}
Let us now introduce some notation :
Z
A=
inf
sup
{φ∈C ∞ (Td )} µ∈P (Td )
(I − ∇φ) a (I − ∇φ) (z)µ(dz)
Td
Z
B=
inf
sup
{φ∈C ∞ (Td )} {µ∈P (Td )}
ρ2 (t, x, G0 ) =
(I − ∇φ) B1 +
Td
1
γ
(B0 − L0 φ) (z)µ(dz)
inf
S20,τ (φ)
{φ∈C ([0,t],Rd );φ0 =x;φt ∈G0 }
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asymptotic behavior
Define
v (t, x) = lim sup v ,δ (s, y) : (s, y) ∈ B {(t, x), η}
η→0
v (t, x) = lim inf v ,δ (s, y) : (s, y) ∈ B {(t, x), η}
η→0
Theorem
v and v are sub and super viscosity solutions of :

1
∂w
d


max
(t, x) − hA∇w(t, x), ∇w(t, x)i − hB, ∇w(t, x)i − C 2 = 0, x ∈ R , t > 0


w

∂t
2

w(0, x) = 0,
x ∈ G0




d
 lim w(t, x) = −∞,

x ∈ R \G0
t→0
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asymptotic behavior
definition
Let O is open subset in R+ × Rd , define the function τ on
[0 , ∞ [ × C [0 , ∞ [ × Rd
τ = τ (t, φ) = inf {s : (t − s, φ(s)) ∈ O}
Take Θt the set of Markov functions τ .
Use the process
(
V (t, x) = inf
τ ∈Θt
C 2τ −
inf
{φ∈C ([0,t],Rd );φ0 =x;φt ∈G0 }
)
S20,τ (φ)
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History
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LDP
Convergence
asymptotic behavior
using the fact (Pradeilles 1998):
−ρ2 (t, x, G0 ) ≤ v (t, x) ≤ v (t, x) ≤ min Kt − ρ2 (t, x, G0 ) , 0
Theorem
For (t, x) ∈ R∗+ × Rd ,
lim log u ,δ (t, x) = V (t, x)
↓0
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asymptotic behavior
corollary
Let M and E be a partition of R+ × Rd , such that:
n
o
M = (t, x) ∈ R+ × Rd : V (t, x) = 0
n
o
E = (t, x) ∈ R+ × Rd : V (t, x) < 0
Theorem
We have
(
lim u ,δ (t,x) =
↓0
0 uniformly from any compact K of E
0
◦
1 uniformly from any compact K of M
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