Some estimates for the parabolic Anderson model

Transcription

1 Some esimaes for he parabolic Anderson model Samy Tindel Purdue Universiy Probabiliy Seminar - Urbana Champaign 2015 Collaboraors: Xia Chen, Yaozhong Hu, Jingyu Huang, Khoa Lê, David Nualar Samy T. (Purdue) Parabolic Anderson model UIUC / 32

2 Ouline 1 Inroducion Moivaions Aim of he alk 2 Main resuls 3 Elemens of proof Samy T. (Purdue) Parabolic Anderson model UIUC / 32

3 Ouline 1 Inroducion Moivaions Aim of he alk 2 Main resuls 3 Elemens of proof Samy T. (Purdue) Parabolic Anderson model UIUC / 32

4 Some (recen) hisory Philip Anderson: Born 1923 Wide range of achievemens In condensed maer physics Conribuion o Higgs mechanism Nobel prize in 1977 Sill Professor a Princeon One of Anderson s discoveries: For paricles moving in a disordered media Localized behavior insead of diffusion. Samy T. (Purdue) Parabolic Anderson model UIUC / 32

5 Equaion under consideraion Equaion: Sochasic hea equaion on R d : u,x = 1 2 u,x + u,x Ẇ,x, (1) wih 0, x R d. Ẇ general Gaussian noise, wih space-ime covariance srucure. u,x Ẇ,x differenial: Sraonovich or Skorohod sense. Samy T. (Purdue) Parabolic Anderson model UIUC / 32

6 Ouline 1 Inroducion Moivaions Aim of he alk 2 Main resuls 3 Elemens of proof Samy T. (Purdue) Parabolic Anderson model UIUC / 32

7 Moivaion 1: Resoluion of SPDEs More general equaion: u,x = Lu,x + G(u,x ) + F (u,x ) Ẇ,x, wih General ellipic operaor L Polynomial nonlineariy G Smooh nonlineariy F Samy T. (Purdue) Parabolic Anderson model UIUC / 32

8 Resoluion of SPDEs (2) Resoluion, Brownian W : Pesza-Zabczyk Dalang Resoluion, rough pahs case: Links: Caruana-Friz-Oberhauser Lejay, Gubinelli-T, Gubinelli-T, Deya-Gubinelli-T Hairer KPZ equaion (Gubinelli-Perkowski, Hairer, Zamboi) Filering, backward equaions, sochasic conrol (Friz e al.) Quesion: Can we say more abou u in he simple bilinear case u,x Ẇ,x? Samy T. (Purdue) Parabolic Anderson model UIUC / 32

9 Moivaion 2: Inermiency Equaion: u,x = 1 2 u,x + λ u,x Ẇ,x Phenomenon: The soluion u concenraes is energy in high peaks. Characerizaion: hrough momens Easy possible definiion of inermiency: for all k 1 > k 2 1 Resuls: [ E 1/k 1 u,x 1] k lim =. E 1/k 2 [ u,x k 2 ] Whie noise in ime: Khoshnevisan, Foondun, Conus, Joseph Fracional noise in ime: Balan-Conus Analysis hrough Feynman-Kac formula Samy T. (Purdue) Parabolic Anderson model UIUC / 32

10 u(,x) x u(,x) u(,x) x u(,x) x x Inemiency: illusraion (by Daniel Conus) Simulaions: for λ = 0.1, 0.5, 1 and 2. Samy T. (Purdue) Parabolic Anderson model UIUC / 32

11 Moivaion 3: Polymer measure Independen Wiener measure: d-dimensional Brownian moion B x, Wiener measure P B. Hamilonian for > 0: H (B x ) = 0 W (ds, Bx s ). Gibbs polymer measure: for β > 0, dg x (B) = e βh(bx ) u,x dp B. Sudies in he coninuous case: Rovira-T, Lacoin, Albers-Khanin-Quasel. Counerpar of inermiency: Localizaion. See Carmona-Hu, König-Lacoin-Mörers-Sidorova Samy T. (Purdue) Parabolic Anderson model UIUC / 32

12 Localizaion: illusraion 1 Figure: Simple random walk disribuion Samy T. (Purdue) Parabolic Anderson model UIUC / 32

13 Localizaion: illusraion 2 Figure: Disribuion of he direced polymer in srong disorder regime Samy T. (Purdue) Parabolic Anderson model UIUC / 32

14 Moivaion: summary Homogenizaion Pahwise PDEs Polymers u = 1 2 u + u Ẇ KPZ Inermiency Local imes Samy T. (Purdue) Parabolic Anderson model UIUC / 32

15 Ouline 1 Inroducion Moivaions Aim of he alk 2 Main resuls 3 Elemens of proof Samy T. (Purdue) Parabolic Anderson model UIUC / 32

16 Aim of he alk Equaion: Sochasic hea equaion on R d : u,x = 1 2 u,x + u,x Ẇ,x. Main issues: for a general Gaussian noise, Resoluion for Iô-Skorohod and Sraonovich equaions. Feynman-Kac represenaion. Links beween Feyman-Kac and pahwise (rough pahs) soluion. Inermiency esimaes. Samy T. (Purdue) Parabolic Anderson model UIUC / 32

17 Ouline 1 Inroducion Moivaions Aim of he alk 2 Main resuls 3 Elemens of proof Samy T. (Purdue) Parabolic Anderson model UIUC / 32

18 Descripion of he noise Encoding he noise as a random disribuion: W = {W (ϕ); ϕ H} cenered Gaussian family E [W (ϕ) W (ψ)] = ϕ, ψ H wih: ϕ, ψ H = ϕ(s, x)ψ(, y) γ(s ) Λ(x y) dx dy ds d R 2 + R2d = Fϕ(s, ξ) Fψ(, ξ) γ(s ) µ(dξ) ds d, R 2 + Rd γ, Λ posiive definie funcions. µ empered measure. Remark: This is sandard seing (Pesza-Zabczyk, Dalang). Samy T. (Purdue) Parabolic Anderson model UIUC / 32

19 Typical examples of noises Covariance Singulariy a 0 FT: sing. a Roughness γ() β No used B β/2 γ() δ() No used B 1/2 Λ(x) x η R d µ(dξ) 1+ ξ η < B η/2 Samy T. (Purdue) Parabolic Anderson model UIUC / 32

20 Possible soluions o he SHE Equaion: u,x = 1 2 u,x + u,x Ẇ,x, u 0,x = u 0 (x). Mild soluion: u,x = p u 0 (x) + 0 R d p s (x y)u s,y W (ds, dy), Feynman-Kac field: For a Brownian moion B independen of W, se V,x = 0 R d δ 0 (B x r y)w (dr, dy), [ ] u,x F = E B u0 (B x ) e V,x Samy T. (Purdue) Parabolic Anderson model UIUC / 32

21 Sraonovich seing Hypohesis on γ: The funcion γ saisfies 0 γ() C β β, wih β (0, 1). Hypohesis on µ: We assume he following inegrabiliy condiion, R d µ(dξ) 1 + ξ 2 2β <. Example: Riesz kernel in space, namely Λ(x) = x η. 0 < η 2 2β. 0 γ() C β β. Samy T. (Purdue) Parabolic Anderson model UIUC / 32

22 Iô s seing Dimension resricion: d = 1, ha is x R Hypohesis on γ: Ẇ whie noise in space, ha is γ() = δ() Hypohesis on µ: for 1/4 < H < 1/2 (very rough siuaion), µ(dξ) = ξ 1 2H dξ Samy T. (Purdue) Parabolic Anderson model UIUC / 32

23 Exisence-uniqueness resuls Theorem 1. Exisence-uniqueness, case 1: Under Sraonovich s seing, Exisence-uniqueness of a mild soluion In he rough pahs (Young) sense. Soluion in C β 2 ([0, T ]; B 1 β ) B 1 β weighed Besov space on R d. Exisence-uniqueness, case 2: Under Iô s seing, Exisence-uniqueness of a mild soluion In he Iô sense. Soluion in L 2 ([0, T ]; B 1/2 H ) Samy T. (Purdue) Parabolic Anderson model UIUC / 32

24 Feynman-Kac soluion Theorem 2. Case 1: Under Sraonovich s seing, Assume: u 0 C b (R d ). B Brownian moion, independen of W We se (Feynman-Kac formula): V,x = 0 R d δ 0 (B x r y)w (dr, dy), u F,x = E B [ u0 (B x ) e V,x ] Then u F well-defined and coincides wih soluion of SHE. Case 2: Under Iô s seing Feynman-Kac represenaion for momens Samy T. (Purdue) Parabolic Anderson model UIUC / 32

25 Momens esimaes Theorem 3. Suppose: c 0 β γ() C 0 β. c 1 x η Λ(x) C 1 x η. Then, whenever i is defined, u F saisfies: ( exp c 2 4 2β η 2 η ) k 4 η 2 η E [ u,x k ] exp (C 2 4 2β η 2 η ) k 4 η 2 η. Samy T. (Purdue) Parabolic Anderson model UIUC / 32

26 Growh rae Theorem 4. Under Iô s seing, wih 1/4 < H < 1/2 we have: lim R 1 [ln(r)] 1 1+H ln ( max u(, x) x R where c H is soluion o a variaional problem. ) = c H, Samy T. (Purdue) Parabolic Anderson model UIUC / 32

27 Commens Remarks: (i) Momen esimaes imply inermiency. (ii) Imporan sep: exponenial inegrabiliy of Feynman-Kac funcional. (iii) Proof for momen esimaes: Feynman-Kac represenaion, small ball esimaes. 1 (iv) Exponen in growh rae: 1+H exension of KPZ exponen 2 for space-ime whie noise 3 Exensions: (i) Exension 1: Non linear cases wih σ(u) Lipschiz (ii) Exension 2: Skorohod (vs. Sraonovich) seing Samy T. (Purdue) Parabolic Anderson model UIUC / 32

28 Ouline 1 Inroducion Moivaions Aim of he alk 2 Main resuls 3 Elemens of proof Samy T. (Purdue) Parabolic Anderson model UIUC / 32

29 Feynman-Kac funcional Proposiion 5. Suppose γ and µ saisfy (wih β (0, 1)): 0 γ() C β β, and R d µ(dξ) 1 + ξ 2 2β <. Se: V,x = δ 0 (B x 0 R d r y)w (dr, dy), Then for any λ R and T > 0: sup E [exp (λ V,x )] <. [0,T ], x R d Samy T. (Purdue) Parabolic Anderson model UIUC / 32

30 Proof 1: Gaussian compuaions Easy Gaussian sep: condiionally o B, V,x is Gaussian. Thus ( )] λ 2 E [exp (λ V,x )] = E B [exp 2 Y, where Y = 2 γ(r s)λ(b r B s )drds. 0 r s Aim: Conrol singulariies in r s in momens of Y. Mehod: Inspired by Le Gall s renormalizaion of self inersecion local imes. Pariion of simplex 0 r s. Samy T. (Purdue) Parabolic Anderson model UIUC / 32

31 Proof 2: Le Gall s pariion 7/8 3/4 5/8 /2 3/8 /4 / Oupu: Pariion {A n,k ; n 1, k = 1,..., 2 n 1 } of he simplex Samy T. (Purdue) Parabolic Anderson model UIUC / 32

32 Proof 2: Le Gall s pariion 7/8 3/4 5/8 /2 3/8 /4 /8 A 1, Oupu: Pariion {A n,k ; n 1, k = 1,..., 2 n 1 } of he simplex Samy T. (Purdue) Parabolic Anderson model UIUC / 32

33 Proof 2: Le Gall s pariion 7/8 3/4 5/8 /2 3/8 /4 /8 A 2,1 A 1,1 A 2, Oupu: Pariion {A n,k ; n 1, k = 1,..., 2 n 1 } of he simplex Samy T. (Purdue) Parabolic Anderson model UIUC / 32

34 Proof 2: Le Gall s pariion 7/8 A 2,2 A 3,4 3/4 5/8 A 1,1 A 3,3 /2 3/8 A 2,1 A 3,2 /4 /8 A 3, Oupu: Pariion {A n,k ; n 1, k = 1,..., 2 n 1 } of he simplex Samy T. (Purdue) Parabolic Anderson model UIUC / 32

35 Proof 3: removing singulariies Familly of random variables: we se a n,k = γ(r s)λ(b r B s )dr ds. A n,k Relaion wih Y : We have Y = n=1 2 n 1 k=1 a n,k For fixed n Random variables {a n,k ; k = 1,..., 2 n 1 } are independen. Ideniy in law: for 2 independen Brownian moions B, B, a n,k (d) = 0 2 n 0 2 n γ(r + s) Λ(B r + B s ) ds dr Thus nasy singulariy (r s) 1 nicer singulariy (r + s) 1. Remainder of he proof: inegral compuaions wih p, γ, Λ. Samy T. (Purdue) Parabolic Anderson model UIUC / 32