Some estimates for the parabolic Anderson model Samy Tindel Purdue - - PowerPoint PPT Presentation

Some estimates for the parabolic Anderson model

Samy Tindel

Purdue University

Probability Seminar - Urbana Champaign 2015 Collaborators: Xia Chen, Yaozhong Hu, Jingyu Huang, Khoa Lê, David Nualart

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Outline

1

Introduction Motivations Aim of the talk

2

Main results

3

Elements of proof

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Outline

1

Introduction Motivations Aim of the talk

2

Main results

3

Elements of proof

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Some (recent) history

Philip Anderson: Born 1923 Wide range of achievements ֒ → In condensed matter physics Contribution to Higgs mechanism Nobel prize in 1977 Still Professor at Princeton One of Anderson’s discoveries: For particles moving in a disordered media ֒ → Localized behavior instead of diffusion.

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Equation under consideration

Equation: Stochastic heat equation on Rd: ∂tut,x = 1 2∆ut,x + ut,x ˙ Wt,x, (1) with t ≥ 0, x ∈ Rd. ˙ W general Gaussian noise, with space-time covariance structure. ut,x ˙ Wt,x differential: Stratonovich or Skorohod sense.

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Outline

1

Introduction Motivations Aim of the talk

2

Main results

3

Elements of proof

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Motivation 1: Resolution of SPDEs

More general equation: ∂tut,x = Lut,x + G(ut,x) + F(ut,x) ˙ Wt,x, with General elliptic operator L Polynomial nonlinearity G Smooth nonlinearity F

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Resolution of SPDEs (2)

Resolution, Brownian W : Peszat-Zabczyk Dalang Resolution, rough paths case: Caruana-Friz-Oberhauser Lejay, Gubinelli-T, Gubinelli-T, Deya-Gubinelli-T Hairer Links: KPZ equation (Gubinelli-Perkowski, Hairer, Zambotti) Filtering, backward equations, stochastic control (Friz et al.) Question: Can we say more about u in the simple bilinear case ut,x ˙ Wt,x?

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Motivation 2: Intermittency

Equation: ∂tut,x = 1

2∆ut,x + λ ut,x ˙

Wt,x Phenomenon: The solution u concentrates its energy in high peaks. Characterization: through moments ֒ → Easy possible definition of intermittency: for all k1 > k2 ≥ 1 lim

t→∞

E1/k1

• |ut,x|k1
• E1/k2 [|ut,x|k2] = ∞ .

Results: White noise in time: Khoshnevisan, Foondun, Conus, Joseph Fractional noise in time: Balan-Conus Analysis through Feynman-Kac formula

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Intemittency: illustration (by Daniel Conus)

Simulations: for λ = 0.1, 0.5, 1 and 2.

t x u(t,x) t x u(t,x) t x u(t,x) t x u(t,x)

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Motivation 3: Polymer measure

Independent Wiener measure: d-dimensional Brownian motion Bx, Wiener measure PB. Hamiltonian for t > 0: −Ht(Bx) =

t

0 W (ds, Bx s ).

Gibbs polymer measure: for β > 0, dGx

t (B) = e−βHt(Bx)

ut,x dPB. Studies in the continuous case: Rovira-T, Lacoin, Alberts-Khanin-Quastel. Counterpart of intermittency: Localization. ֒ → See Carmona-Hu, König-Lacoin-Mörters-Sidorova

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Localization: illustration 1

Figure: Simple random walk distribution

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Localization: illustration 2

Figure: Distribution of the directed polymer in strong disorder regime

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Motivation: summary

∂tu = 1

2∆u + u ˙

W KPZ Polymers

Local times

Homogenization Pathwise PDEs Intermittency

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Outline

1

Introduction Motivations Aim of the talk

2

Main results

3

Elements of proof

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Aim of the talk

Equation: Stochastic heat equation on Rd: ∂tut,x = 1 2∆ut,x + ut,x ˙ Wt,x. Main issues: for a general Gaussian noise, Resolution for Itô-Skorohod and Stratonovich equations. Feynman-Kac representation. Links between Feyman-Kac and pathwise (rough paths) solution. Intermittency estimates.

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Outline

1

Introduction Motivations Aim of the talk

2

Main results

3

Elements of proof

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Description of the noise

Encoding the noise as a random distribution: W = {W (ϕ); ϕ ∈ H} centered Gaussian family E [W (ϕ) W (ψ)] = ϕ, ψH with: ϕ, ψH =

• R2

+×R2d ϕ(s, x)ψ(t, y) γ(s − t) Λ(x − y) dx dy ds dt

=

• R2

+×Rd Fϕ(s, ξ) Fψ(t, ξ) γ(s − t) µ(dξ) ds dt,

γ, Λ positive definite functions. µ tempered measure. Remark: This is standard setting (Peszat-Zabczyk, Dalang).

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Typical examples of noises

Covariance Singularity at 0 FT: sing. at ∞ Roughness γ(t) |t|−β Not used B−β/2 γ(t) δ(t) Not used B−1/2 Λ(x) |x|−η

• Rd

µ(dξ) 1+|ξ|η < ∞

B−η/2

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Possible solutions to the SHE

Equation: ∂tut,x = 1 2∆ut,x + ut,x ˙ Wt,x , u0,x = u0(x). Mild solution: ut,x = ptu0(x) +

t

• Rd pt−s(x − y)us,y W (ds, dy),

Feynman-Kac field: For a Brownian motion B independent of W , set Vt,x =

t

• Rd δ0(Bx

t−r − y)W (dr, dy),

uF

t,x = EB

• u0(Bx

t ) eVt,x

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Stratonovich setting

Hypothesis on γ: The function γ satisfies 0 ≤ γ(t) ≤ Cβ|t|−β, with β ∈ (0, 1). Hypothesis on µ: We assume the following integrability condition,

• Rd

µ(dξ) 1 + |ξ|2−2β < ∞ . Example: Riesz kernel in space, namely Λ(x) = |x|−η . 0 < η ≤ 2 − 2β. 0 ≤ γ(t) ≤ Cβ|t|−β.

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Itô’s setting

Dimension restriction: d = 1, that is x ∈ R Hypothesis on γ: ˙ W white noise in space, that is γ(t) = δ(t) Hypothesis on µ: for 1/4 < H < 1/2 (very rough situation), µ(dξ) = |ξ|1−2H dξ

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Existence-uniqueness results

Existence-uniqueness, case 1: Under Stratonovich’s setting, Existence-uniqueness of a mild solution ֒ → In the rough paths (Young) sense. Solution in C

β 2 ([0, T]; B1−β)

֒ → B1−β weighted Besov space on Rd. Existence-uniqueness, case 2: Under Itô’s setting, Existence-uniqueness of a mild solution ֒ → In the Itô sense. Solution in L2([0, T]; B1/2−H) Theorem 1.

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Feynman-Kac solution

Case 1: Under Stratonovich’s setting, Assume: u0 ∈ Cb(Rd). B Brownian motion, independent of W We set (Feynman-Kac formula): Vt,x =

t

• Rd δ0(Bx

t−r − y)W (dr, dy),

uF

t,x = EB

• u0(Bx

t ) eVt,x

Then uF well-defined and coincides with solution of SHE. Case 2: Under Itô’s setting ֒ → Feynman-Kac representation for moments Theorem 2.

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Moments estimates

Suppose: c0|t|−β ≤ γ(t) ≤ C0|t|−β. c1|x|−η ≤ Λ(x) ≤ C1|x|−η. Then, whenever it is defined, uF satisfies: exp

• c2 t

4−2β−η 2−η k 4−η 2−η

• ≤ E
• uk

t,x

• ≤ exp
• C2 t

4−2β−η 2−η k 4−η 2−η

• .

Theorem 3.

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Growth rate

Under Itô’s setting, with 1/4 < H < 1/2 we have: lim

R→∞

1 [ln(R)]

1 1+H ln

• max

|x|≤R u(t, x)

• = cH,

where cH is solution to a variational problem. Theorem 4.

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Comments

Remarks: (i) Moment estimates imply intermittency. (ii) Important step: ֒ → exponential integrability of Feynman-Kac functional. (iii) Proof for moment estimates: ֒ → Feynman-Kac representation, small ball estimates. (iv) Exponent

1 1+H in growth rate:

֒ → extension of KPZ exponent 2

3 for space-time white noise

Extensions: (i) Extension 1: Non linear cases with σ(u) Lipschitz (ii) Extension 2: Skorohod (vs. Stratonovich) setting

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Outline

1

Introduction Motivations Aim of the talk

2

Main results

3

Elements of proof

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Feynman-Kac functional

Suppose γ and µ satisfy (with β ∈ (0, 1)): 0 ≤ γ(t) ≤ Cβ|t|−β, and

• Rd

µ(dξ) 1 + |ξ|2−2β < ∞ . Set: Vt,x =

t

• Rd δ0(Bx

t−r − y)W (dr, dy),

Then for any λ ∈ R and T > 0: sup

t∈[0,T], x∈Rd E [exp (λ Vt,x)] < ∞.

Proposition 5.

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Proof 1: Gaussian computations

Easy Gaussian step: conditionally to B, Vt,x is Gaussian. Thus E [exp (λ Vt,x)] = EB

• exp

λ2

2 Y

• ,

where Y = 2

• 0≤r≤s≤t γ(r − s)Λ(Br − Bs)drds .

Aim: Control singularities in r − s in moments of Y . Method: Inspired by Le Gall’s renormalization of self intersection local times. ֒ → Partition of simplex 0 ≤ r ≤ s ≤ t.

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Proof 2: Le Gall’s partition

t

t 8 t 4 3t 8 t 2 5t 8 3t 4 7t 8

t

t/8 t/4 3t/8 t/2 5t/8 3t/4 7t/8

Output: Partition {An,k; n ≥ 1, k = 1, . . . , 2n−1} of the simplex

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Proof 2: Le Gall’s partition

t

t 8 t 4 3t 8 t 2 5t 8 3t 4 7t 8

t

t/8 t/4 3t/8 t/2 5t/8 3t/4 7t/8

A1,1 Output: Partition {An,k; n ≥ 1, k = 1, . . . , 2n−1} of the simplex

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Proof 2: Le Gall’s partition

t

t 8 t 4 3t 8 t 2 5t 8 3t 4 7t 8

t

t/8 t/4 3t/8 t/2 5t/8 3t/4 7t/8

A1,1 A2,1 A2,2 Output: Partition {An,k; n ≥ 1, k = 1, . . . , 2n−1} of the simplex

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Proof 2: Le Gall’s partition

t

t 8 t 4 3t 8 t 2 5t 8 3t 4 7t 8

t

t/8 t/4 3t/8 t/2 5t/8 3t/4 7t/8

A1,1 A2,1 A2,2

A3,1 A3,2 A3,3 A3,4

Output: Partition {An,k; n ≥ 1, k = 1, . . . , 2n−1} of the simplex

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Proof 3: removing singularities

Familly of random variables: we set an,k =

• An,k

γ(r − s)Λ(Br − Bs)dr ds. Relation with Y : We have Y = ∞

n=1

2n−1

k=1 an,k

For fixed n ֒ → Random variables {an,k; k = 1, . . . , 2n−1} are independent. Identity in law: for 2 independent Brownian motions B, ˜ B, an,k

(d)

=

• t

2n

• t

2n

γ(r + s) Λ(Br + Bs) ds dr Thus nasty singularity (r − s)−1 → nicer singularity (r + s)−1. Remainder of the proof: integral computations with p, γ, Λ.

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