Couplage du principe des grandes déviations et de l homogénisation dans le cas des EDP paraboliques: (le cas constant)

Transcription

1 Couplage du principe des grandes déviaions e de l homogénisaion dans le cas des EDP paraboliques: (le cas consan) Alioune COULIBALY U.F.R Sciences e Technologie Universié Assane SECK de Ziguinchor Probabilié e Analyse - CIMPA 2014

2 Plan 1 Hisory 2 Inro 3 LDP 4 Convergence

3 Hisory Hisory P. Baldi (1991) Freidlin and Sowers (1999) Diédhiou and Manga (2007) Diédhiou (Cimpa 2014)

4 Inroducion absrac We consider he coupling of homogenizaion and large deviaion principle in parial differenial equaion (PDE). We compare hem wih he help of he raio δ/ɛ beween he small viscosiy parameer (ɛ) and he homogenizaion parameer (δ); his comparison is required as ɛ and δ end o zero. We use some large deviaion esimaes o sudy he behavior of he PDE soluion.

5 PDE PDE u ɛ,δ (, x) = L ɛ,δu ɛ,δ (, x) + 1 ɛ f ( x δ, uɛ,δ (, x) ) (1) u ɛ,δ (0, x) = g(x), x R d

6 assumpion f is a nonlinear funcion 1-periodic and verify: and for all x, f (x, 1) = 0 here is c a bounded funcion : f (x, y) = c(x, y).y c(x, y) > 0 for all x and y in (0,1) c(x, y) 0 if no max c(x, y) = c(x) and g C ( R d, R +) a bounded funcion : sup x R d g(x) = ḡ <. Take G 0 = { x R d : g(x) > 0 }, g is coninuous one noes G 0 = G 0

7 definiion Le (Ω, F, P) a probabiliy riple on which a d-dimensional Brownian moion ( W 1,..., W d) is defined. Le E he corresponding expecaion operaor. We have already defined.,. as he sandard euclidian inner produc on R d ; le. be he associaed norm. Also le T d be he d-dimensional orus of size 1 and C ( T d ; R d) be he space of coninuous mapping from T d o R d ; le. C(T d ;R d ) be he associaed supremum norm. Also we define P ( T d) as he collecion of all probabiliy measure on T d.

8 Approach Consider he Markov diffusion process X x,ɛ,δ R d governed by he operaor : L ɛ,δ = ɛ 2 d ( x ) (σσ 2 ) ij + δ x i x j i,j=1 d i=1 ( B ɛ,δ x ) δ x i The rajecories of his process can be consruced wih he help of he SDE: = ( ) ( ) X ɛσ x,ɛ,δ δ dw + B ɛ,δ X x,ɛ,δ δ d dx x,ɛ,δ X x,ɛ,δ 0 = x

9 assumpion σ : R d R d d and B ɛ,δ : R d R d are regular applicaions. The vecor-valued funcion B ɛ,δ is given by : B ɛ,δ = ɛ δ B 0 + B 1 + B ɛ,δ 2 B 0, B 1 and B ɛ,δ 2 are C ( R d, R d) for all ɛ, δ > 0 and lim ɛ,δ 0 Bɛ,δ 2 Cp(R d,r d ) = 0

10 hypohesis Since he wo parameers δ (homogenizaion) and ɛ (large deviaion) end o zero, we consider a new defined parameer δ ɛ = δ δ Suppose ha lim ɛ ɛ 0 ɛ = γ, where γ > 0 a consan. There resuls ha he homogenizaion parameer and he large deviaion parameer go a he same rae

11 some rescalings covering map Define Then X x,ɛ,δɛ by : X x,ɛ,δɛ { d x,ɛ,δ X = σ X x,ɛ,δ 0 = x δ ɛ = 1 δ ɛ X( x,ɛ,δɛ ) δɛ 2 ɛ ( ) X x,ɛ,δ d W ( ) + δɛ ɛ X Bɛ,δ x,ɛ,δ d where ɛ,δɛ W = ɛ δ ɛ W( ɛ,δɛ δɛ ɛ ) 2 is a Brownian moion.

12 new consideraion he orus T d { } Thereafer, consider he process X ɛ,δ ɛ : 0 T d -valued which generaor is defined by : L ɛ,δɛ = 1 2 d (σσ 2 ) ij (x) + δ ɛ x i x j ɛ i,j=1 d i=1 B ɛ,δɛ (x) x i Consider a = σσ, he above generaor converges o he operaor L γ = 1 2 d 2 (a) ij (x) + x i x j i,j=1 d i=1 B 0 i (x) x i + γ d i=1 B 1 i (x) x i

13 reference ools The basic se of calculaions of his subjec involves deriving he Varadhan formula and in idenifying he consan C 2. The main echnique for showing ha is he following resul : (Baxendale and Sroock (1988), corollary 1.12, p[183] o p[185])

14 LDP rae funcion Le J 2 (θ) defined by Freidlin and Sowers (1999), he limi of [ ( )] 1 ɛ log E exp ɛ θ, X x,ɛ,δɛ J 2 (θ) = inf {φ C ( T d )} { sup µ P (T d )} 1 d (I φ) σ k (z), θ 2 + (I φ) B 1 (z), θ T d 2 k=1 + 1 ) γ (B 0 L 0 φ) (z), θ µ(dz) Define he Legendre-fenchel of J 2 (θ) J 2 (θ) = sup θ, θ J 2 (θ ) {θ R d }

15 large deviaion principle LDP Define in addiion : T ( S0,T 2 J 2. ) φ (s) ds if φ absoluely coninuous and φ(0) = x (φ) = 0 if no Theorem (Freidlin and Sowers 1999) { Fix T > 0 and x R d. The family X x,ɛ,δɛ } : 0 T of ɛ>0 C ( [0, T ], R d) -valued random variables has a large deviaion principle wih rae funcion S 2 0,T (φ) for all φ C ( [0, T ], R d).

16 barrier consan Now define C 2 = sup c(z)µ(dz) µ P(T d ) T d

17 fundemenal heorem Varadhan formula Theorem Le c be an elemen of C ( T d) and D a Borel subse of C ( [0, ], R d). Then lim inf ɛ 0 lim sup ɛ 0 ɛ log E 1 D (X x,ɛ,δɛ ) e ɛ log E 1 D (X x,ɛ,δɛ ) e ( 1 X x,ɛ,δɛ c s )ds ɛ δɛ 0 ( 1 X x,ɛ,δɛ c s )ds ɛ δɛ 0 C 2 inf } S0,(φ) 2 { φ D C 2 inf {φ D} S2 0,(φ)

18 behavior of u ɛ,δ ɛ Feynman Kac ( u ɛ,δɛ (, x) = E g X x,ɛ,δɛ ) 1 ɛ e ( 0 c Xs x,ɛ,δɛ ) δɛ,ys x,ɛ,δɛ ds

19 BSDE Pardoux Peng 1992 Ys x,ɛ,δɛ E and { s = g (X x,ɛ,δɛ ) + 1 ɛ } x,ɛ,δɛ Zr 2 dr < s f ( ) Xr x,ɛ,δɛ δɛ,yr x,ɛ,δɛ dr 1 ɛ Y x,ɛ,δɛ 0 = u ɛ,δɛ (, x) s Z r x,ɛ,δɛ dw r, 0 s

20 viscosiy soluion Pradeilles 1998 Since u ɛ,δɛ (, x) > 0, denoe v ɛ,δɛ (, x) = ɛ log u ɛ,δɛ (, x) and observe, ha v ɛ,δɛ (, x) is a viscosiy soluion of : v ɛ,δɛ (, x) = L ɛ,δɛ v ɛ,δɛ (, x) + 1 ( ) v ɛ,δɛ (, x)σ (x) 2 x + c, u ɛ,δɛ (, x) 2 δ ɛ v ɛ,δɛ (0, x) = ɛ log (g(x)), x G 0 lim v ɛ,δɛ (, x) =, x R d \G 0 0

21 some noaion definiion Le us define a disance in R + R d, by for (, x), (s, y) R + R d : d {(, x), (s, y)} = max { s, x y } Le us now inroduce some noaion : A = inf sup (I φ) a (I φ) (z)µ(dz) {φ C (T d )} µ P(T d ) T d [ ] B = inf sup (I φ) B (B 0 L 0 φ) (z)µ(dz) {φ C (T d )} {µ P(T d )} T d γ ρ 2 (, x, G 0 ) = inf {φ C([0,],R d );φ 0 =x;φ G 0} S0,τ 2 (φ)

22 asympoic behavior Define { v(, x) = lim sup v ɛ,δ ɛ (s, y) : (s, y) B {(, x), η} } η 0 v(, x) = lim inf η 0 { v ɛ,δ ɛ (s, y) : (s, y) B {(, x), η} } Theorem v and v are sub and super viscosiy soluions of : ( w max w (, x) 1 ) A w(, x), w(, x) B, w(, x) C 2 = 0, x R d, > 0 2 w(0, x) = 0, x G 0 lim 0 w(, x) =, x Rd \G 0

23 asympoic behavior definiion Le O is open subse in R + R d, define he funcion τ on [0, [ C ( [0, [ R d) τ = τ(, φ) = inf {s : ( s, φ(s)) O} Take Θ he se of Markov funcions τ. Use he process { V (, x) = inf τ Θ C 2 τ inf {φ C([0,],R d );φ 0 =x;φ G 0} S0,τ 2 (φ) }

24 asympoic behavior using he fac (Pradeilles 1998): ρ 2 (, x, G 0 ) v(, x) v(, x) min ( K ρ 2 (, x, G 0 ), 0 ) Theorem For (, x) R + R d, lim ɛ log u ɛ,δɛ (, x) = V (, x) ɛ 0

25 asympoic behavior corollary Le M and E be a pariion of R + R d, such ha: { } M = (, x) R + R d : V (, x) = 0 { } E = (, x) R + R d : V (, x) < 0 Theorem We have lim u ɛ,δɛ(,x) = ɛ 0 { 0 uniformly from any compac K of E 1 uniformly from any compac K of M