PAM at IPAM Samy Tindel Universit de Lorraine Rough paths: theory - - PowerPoint PPT Presentation

PAM at IPAM

Samy Tindel

Université de Lorraine

Rough paths: theory and applications - Los Angeles 2014 Joint work with Yaozhong Hu, Jingyu Huang and David Nualart

Samy T. (Nancy) PAM at IPAM LA 2014 1 / 30

Outline

1

Introduction Motivations Aim of the talk

2

Main results Results for the Stratonovich equation Results for the Skorohod equation

3

Elements of proof

Samy T. (Nancy) PAM at IPAM LA 2014 2 / 30

Outline

1

Introduction Motivations Aim of the talk

2

Main results Results for the Stratonovich equation Results for the Skorohod equation

3

Elements of proof

Samy T. (Nancy) PAM at IPAM LA 2014 3 / 30

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.

Samy T. (Nancy) PAM at IPAM LA 2014 4 / 30

Outline

1

Introduction Motivations Aim of the talk

2

Main results Results for the Stratonovich equation Results for the Skorohod equation

3

Elements of proof

Samy T. (Nancy) PAM at IPAM LA 2014 5 / 30

Resolution of SPDEs

More general equation: ∂tut,x = Lut,x + G(ut,x) + F(ut,x) ˙ Wt,x, with (many contributors at the conference) General elliptic operator L Polynomial nonlinearity G Smooth nonlinearity F Links: KPZ equation Filtering, backward equations, stochastic control Question: Can we say more about u in the simple bilinear case ut,x ˙ Wt,x?

Samy T. (Nancy) PAM at IPAM LA 2014 6 / 30

Homogenization

Problem: Asymptotic regime for ∂tuε

t,x = 1

2∆uε

t,x + ε−βut,x V t

εα , x ε,

where V stationary random field. Link with SPDEs: Under certain regimes for α, β, V we have uε → u Analysis through Feynman-Kac formula See Iftimie-Pardoux-Piatnitski, Bal-Gu

Samy T. (Nancy) PAM at IPAM LA 2014 7 / 30

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→∞

E

• |ut,x|k1
• E [|ut,x|k2] = ∞ .

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

Samy T. (Nancy) PAM at IPAM LA 2014 8 / 30

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)

Samy T. (Nancy) PAM at IPAM LA 2014 9 / 30

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

Samy T. (Nancy) PAM at IPAM LA 2014 10 / 30

Localization: illustration 1 (by Frédéric Cérou)

Figure : Simple random walk distribution

Samy T. (Nancy) PAM at IPAM LA 2014 11 / 30

Localization: illustration 2 (by Frédéric Cérou)

Figure : Distribution of the directed polymer in strong disorder regime

Samy T. (Nancy) PAM at IPAM LA 2014 12 / 30

Outline

1

Introduction Motivations Aim of the talk

2

Main results Results for the Stratonovich equation Results for the Skorohod equation

3

Elements of proof

Samy T. (Nancy) PAM at IPAM LA 2014 13 / 30

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 Skorohod and Stratonovich equations. Feynman-Kac representation. Links between Feyman-Kac and pathwise (rough paths) solution. Intermittency estimates.

Samy T. (Nancy) PAM at IPAM LA 2014 14 / 30

Outline

1

Introduction Motivations Aim of the talk

2

Main results Results for the Stratonovich equation Results for the Skorohod equation

3

Elements of proof

Samy T. (Nancy) PAM at IPAM LA 2014 15 / 30

Description of the noise

Encoding of 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).

Samy T. (Nancy) PAM at IPAM LA 2014 16 / 30

Outline

1

Introduction Motivations Aim of the talk

2

Main results Results for the Stratonovich equation Results for the Skorohod equation

3

Elements of proof

Samy T. (Nancy) PAM at IPAM LA 2014 17 / 30

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.

Samy T. (Nancy) PAM at IPAM LA 2014 18 / 30

Feynman-Kac solution

Assume: Previous assumptions on γ and µ, and u0 ∈ Cb(Rd). 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

Then uF well-defined and solves the equation: ut,x = ptu0(x) +

t

• Rd pt−s(x − y)us,y W (ds, dy),

interpreted in the Malliavin - Stratonovich sense. Theorem 1. Proof: ֒ → Exponential integrability of Vt,x.

Samy T. (Nancy) PAM at IPAM LA 2014 19 / 30

Pathwise solution

Assume the previous hypothesis, plus:

• Rd

µ(dξ) 1 + |ξ|2−2β−ε < ∞ . Consider the equation: ut,x = ptu0(x) +

t

• Rd pt−s(x − y)us,y W (ds, dy),

(2) interpreted in the Young sense. Then:

• Eq. (2) admits a unique solution in C

β 2 ([0, T]; B1−β).

B1−β is a weighted Besov space on Rd. The unique solution to (2) is uF. Theorem 2.

Samy T. (Nancy) PAM at IPAM LA 2014 20 / 30

Moments estimates

Suppose: c0|t|−β ≤ γ(t) ≤ C0|t|−β. c1|x|−η ≤ Λ(x) ≤ C1|x|−η. Then, whenever they are defined, both u⋄ and uF satisfy: 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. Remarks: (i) This result implies intermittency. (ii) Extensions: other kind of Λ including δ0, time independent case. (iii) Proof: Feynman-Kac representation, small ball estimates.

Samy T. (Nancy) PAM at IPAM LA 2014 21 / 30

Outline

1

Introduction Motivations Aim of the talk

2

Main results Results for the Stratonovich equation Results for the Skorohod equation

3

Elements of proof

Samy T. (Nancy) PAM at IPAM LA 2014 22 / 30

Skorohod setting

Hypothesis on γ: The function γ lies in L1

loc.

Hypothesis on µ: We assume the following integrability condition,

• Rd

µ(dξ) 1 + |ξ|2 < ∞. Example 1: Riesz kernel in space, namely Λ(x) = |x|−η . µ(dξ) = cη,d|ξ|−(d−η) dξ. 0 < η < 2. Example 2: White noise in dimension 1.

Samy T. (Nancy) PAM at IPAM LA 2014 23 / 30

Skorohod: existence and uniqueness

Assume: Previous assumptions on γ and µ. u0 ∈ Cb(Rd). Then Skorohod equation: ut,x = ptu0(x) +

t

• Rd pt−s(x − y)us,y δWs,y

admits a unique solution. Theorem 4. Proof: ֒ → Wiener chaos expansions and estimates.

Samy T. (Nancy) PAM at IPAM LA 2014 24 / 30

Skorohod: representation of moments

Assume: Previous assumptions on γ and µ, and u0 ≡ 1. u0 ∈ Cb(Rd). Then the solution u⋄ of the Skorohod equation satisfies: E

• (u⋄

t,x)k

= EB

 exp  

• 1≤i<j≤k

t t

0 γ(s − r)Λ(Bi s − Bj r)dsdr

   

for a family of i.i.d Brownian motions in Rd. Theorem 5. Proof: ֒ → Feynman-Kac formula for approximations.

Samy T. (Nancy) PAM at IPAM LA 2014 25 / 30

Outline

1

Introduction Motivations Aim of the talk

2

Main results Results for the Stratonovich equation Results for the Skorohod equation

3

Elements of proof

Samy T. (Nancy) PAM at IPAM LA 2014 26 / 30

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 6.

Samy T. (Nancy) PAM at IPAM LA 2014 27 / 30

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.

Samy T. (Nancy) PAM at IPAM LA 2014 28 / 30

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

Samy T. (Nancy) PAM at IPAM LA 2014 29 / 30

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

Samy T. (Nancy) PAM at IPAM LA 2014 29 / 30

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

Samy T. (Nancy) PAM at IPAM LA 2014 29 / 30

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

Samy T. (Nancy) PAM at IPAM LA 2014 29 / 30

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.

Samy T. (Nancy) PAM at IPAM LA 2014 30 / 30