Quantitative homogenization: Degenerate environments and stochastic - - PowerPoint PPT Presentation

Quantitative homogenization: Degenerate environments and stochastic interface model

Paul Dario Université Paris-Dauphine and École Normale Supérieure June 18th, 2019

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1

Introduction

2

Homogenization on percolation clusters

3

Homogenization applied to the ∇φ model

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Contents

1

Introduction

2

Homogenization on percolation clusters

3

Homogenization applied to the ∇φ model

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Introduction

Study of the elliptic equation ∇ ⋅ a(x)∇u = 0,

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Introduction

Study of the elliptic equation ∇ ⋅ a(x)∇u = 0, where a ∶= { Rd → S(Rd), x ↦ a(x) with S(Rd) is the set of symmetric matrices;

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Introduction

Study of the elliptic equation ∇ ⋅ a(x)∇u = 0, where a ∶= { Rd → S(Rd), x ↦ a(x) with S(Rd) is the set of symmetric matrices; A uniform ellipticity assumption: there exist 0 < λ ≤ Λ < ∞, for each x ∈ Rd, λId ≤ a(x) ≤ ΛId.

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Introduction

We assume that the environment a is random with two assumptions Stationarity: For each z ∈ Zd, the environments a(z + ⋅) and a have the same law.

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Introduction

We assume that the environment a is random with two assumptions Stationarity: For each z ∈ Zd, the environments a(z + ⋅) and a have the same law. Ergodicity: Ergodicity, mixing properties, concentration inequalities etc.

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Introduction

Goal: Prove that there exists a deterministic matrix a and a deterministic function u solution of ∇ ⋅ a∇u = 0, such that u is close to u.

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Introduction

Goal: Prove that there exists a deterministic matrix a and a deterministic function u solution of ∇ ⋅ a∇u = 0, such that u is close to u. Historical background: Qualitative theory: in the 80’s with Kozlov, Papanicolaou, Varadhan, Yurinski˘ ı etc.

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Introduction

Goal: Prove that there exists a deterministic matrix a and a deterministic function u solution of ∇ ⋅ a∇u = 0, such that u is close to u. Historical background: Qualitative theory: in the 80’s with Kozlov, Papanicolaou, Varadhan, Yurinski˘ ı etc. Quantitative theory: in the 10’s with Gloria, Otto, Neukamm, Armstrong, Mourrat, Kuusi etc.

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Three directions:

1 Homogenization on percolation clusters; 2 Homogenization of differentail forms; 3 Homogenization applied to ∇φ model. Paul Dario PhD Defense June 18th, 2019 7 / 33

Contents

1

Introduction

2

Homogenization on percolation clusters

3

Homogenization applied to the ∇φ model

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Homogenization on percolation clusters

Initial Goal: Extend the theory by relaxing λId ≤ a(x) ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ

this assumption

≤ ΛId.

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Homogenization on percolation clusters

Initial Goal: Extend the theory by relaxing λId ≤ a(x) ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ

this assumption

≤ ΛId. Some assumptions: Discrete setting: space Zd, discrete gradient, discrete laplacian etc.

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Homogenization on percolation clusters

Initial Goal: Extend the theory by relaxing λId ≤ a(x) ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ

this assumption

≤ ΛId. Some assumptions: Discrete setting: space Zd, discrete gradient, discrete laplacian etc. The environment a is defined on the edges ∇ ⋅ a∇u(x) = ∑

y∼x

a({x,y})(u(y) − u(x));

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Homogenization on percolation clusters

Initial Goal: Extend the theory by relaxing λId ≤ a(x) ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ

this assumption

≤ ΛId. Some assumptions: Discrete setting: space Zd, discrete gradient, discrete laplacian etc. The environment a is defined on the edges ∇ ⋅ a∇u(x) = ∑

y∼x

a({x,y})(u(y) − u(x)); The environment a takes only two values: 0 or 1, a ∶ Ed

• Set of edges

↦ {0,1};

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Homogenization on percolation clusters

Initial Goal: Extend the theory by relaxing λId ≤ a(x) ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ

this assumption

≤ ΛId. Some assumptions: Discrete setting: space Zd, discrete gradient, discrete laplacian etc. The environment a is defined on the edges ∇ ⋅ a∇u(x) = ∑

y∼x

a({x,y})(u(y) − u(x)); The environment a takes only two values: 0 or 1, a ∶ Ed

• Set of edges

↦ {0,1}; The random variables a(e) are i.i.d and characterized by the value p ∶= P(a(e) = 1).

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Homogenization on percolation clusters

Figure 1: supercritical with p = 0.7

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Homogenization on percolation clusters

We want to study harmonic function on C∞. ☞ Related to the behavior of the random walk on C∞:

1 Invariance principles Sidoravicius, Snitzman 04, Berger, Biskup 07,

Mathieu, Piatnitski 07 etc.

2 Gaussian bounds on the heat kernel Mathieu, Remy 04, Barlow 04. 3 Local limit theorem Barlow, Hambly 09. Paul Dario PhD Defense June 18th, 2019 11 / 33

Homogenization on percolation clusters

Theorem: Quantitative Homogenization on C∞

There exist two exponents α > 0 and s > 0 and a nonnegative random variable X satisfying P(X > r) ≤ exp(−crs), such that ➢ for every R > 0 such that R ≥ X,

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Homogenization on percolation clusters

Theorem: Quantitative Homogenization on C∞

There exist two exponents α > 0 and s > 0 and a nonnegative random variable X satisfying P(X > r) ≤ exp(−crs), such that ➢ for every R > 0 such that R ≥ X, ➢ for every C∞-harmonic function u ∶ C∞ ∩ BR → R,

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Homogenization on percolation clusters

Theorem: Quantitative Homogenization on C∞

There exist two exponents α > 0 and s > 0 and a nonnegative random variable X satisfying P(X > r) ≤ exp(−crs), such that ➢ for every R > 0 such that R ≥ X, ➢ for every C∞-harmonic function u ∶ C∞ ∩ BR → R, ➢ there exists an harmonic function u ∶ BR → R such that ∥u − u∥L2(C∞∩BR/2) ≤ CR−α ∥u∥L2(C∞∩BR) .

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Homogenization on percolation clusters

We let Ak ∶= {u ∶ C∞ → R ∶ u is C∞-harmonic and lim

r→∞r−k−1 ∥u∥L2(Br) = 0}

and also Ak ∶= {Harmonic polynomials of degree less than k}. The space Ak is finite dimensional and dimAk = (d + k − 1 k ) + (d + k − 2 k − 1 ).

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Theorem (Regularity theory on C∞)

There exist two exponents s,α > 0 and a nonnegative random variable X satisfying P(X > r) ≤ exp(−crs), such that the following hold:

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Theorem (Regularity theory on C∞)

There exist two exponents s,α > 0 and a nonnegative random variable X satisfying P(X > r) ≤ exp(−crs), such that the following hold: (i) For every u ∈ Ak, there exists p ∈ Ak such that for every r ≥ X, ∥u − p∥L2(C∞∩Br) ≤ Cr−α ∥p∥L2(Br) ;

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Theorem (Regularity theory on C∞)

There exist two exponents s,α > 0 and a nonnegative random variable X satisfying P(X > r) ≤ exp(−crs), such that the following hold: (i) For every u ∈ Ak, there exists p ∈ Ak such that for every r ≥ X, ∥u − p∥L2(C∞∩Br) ≤ Cr−α ∥p∥L2(Br) ; (ii) For every p ∈ Ak, there exists u ∈ Ak such that for every r ≥ X the previous estimate holds;

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Theorem (Regularity theory on C∞)

There exist two exponents s,α > 0 and a nonnegative random variable X satisfying P(X > r) ≤ exp(−crs), such that the following hold: (iii) For every C∞-harmonic function u, there exists φ ∈ Ak such that, for every R ≥ r ≥ X, one has ∥u − φ∥L2(C∞∩Br) ≤ C ( r R )

k+1

∥u∥L2(C∞∩BR) .

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Consequences and the corrector

Application: Almost-surely, dimAk = dimAk = (d + k − 1 k ) + (d + k − 2 k − 1 ).

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Consequences and the corrector

Application: Almost-surely, dimAk = dimAk = (d + k − 1 k ) + (d + k − 2 k − 1 ). A specific space of interest is the space A1: from the previous theorem, each u ∈ A1 can be written, for some p ∈ Rd, u = lp + c + χp

• the corrector

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Consequences and the corrector

Application: Almost-surely, dimAk = dimAk = (d + k − 1 k ) + (d + k − 2 k − 1 ). A specific space of interest is the space A1: from the previous theorem, each u ∈ A1 can be written, for some p ∈ Rd, u = lp + c + χp

• the corrector

and

• sc

C∞∩BR

χp ≤ R1−α.

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Theorem: Optimal scaling estimates for the corrector

For each dimension d ≥ 3, there exist an exponent s > 0 and a constant C < ∞ such that, for each x,y ∈ Zd, and each p ∈ B1, ∣χp(x) − χp(y)∣1{x,y∈C∞} ≤ Os (C). In dimension 2, the growth of the corrector behaves like the square root of the logarithm, ∣χp(x) − χp(y)∣1{x,y∈C∞} ≤ Os (C log

1 2 ∣x − y∣), Paul Dario PhD Defense June 18th, 2019 16 / 33

Theorem: Optimal scaling estimates for the corrector

For each dimension d ≥ 3, there exist an exponent s > 0 and a constant C < ∞ such that, for each x,y ∈ Zd, and each p ∈ B1, ∣χp(x) − χp(y)∣1{x,y∈C∞} ≤ Os (C). In dimension 2, the growth of the corrector behaves like the square root of the logarithm, ∣χp(x) − χp(y)∣1{x,y∈C∞} ≤ Os (C log

1 2 ∣x − y∣),

where the notation Os(K) is used to measure stochastic integrability X ≤ Os(K) ⇐ ⇒ E[exp((X K )

s

)] ≤ 2.

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Idea of the proof: A renormalization structure for C∞

☞ Idea: we wish to apply the theory developed in the uniformly elliptic setting ⇒ We need to understand the geometry of C∞.

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Idea of the proof: A renormalization structure for C∞

☞ Idea: we wish to apply the theory developed in the uniformly elliptic setting ⇒ We need to understand the geometry of C∞. ☞ We appeal to renormalization. General Ansatz: The infinite cluster fills the entire space and it coexists with small isolated clusters.

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Idea of the proof: A renormalization structure for C∞

☞ Idea: we wish to apply the theory developed in the uniformly elliptic setting ⇒ We need to understand the geometry of C∞. ☞ We appeal to renormalization. General Ansatz: The infinite cluster fills the entire space and it coexists with small isolated clusters.

Definition: Good cube

Given a discrete cube Q of size R and a percolation configuration, we say that a cube is good if

1 There exists one large cluster which connects all the faces of the cube; 2 The diameter of all the remaining clusters is smaller than N

10.

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Good cubes

Figure 2: A good cube of size 50

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Good cubes

Figure 2: A good cube of size 50

Theorem, Penrose and Pisztora 1996

Let Q be a cube of size R, then P(Q is good) ≥ 1 − C exp(−cR).

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A partition of good cubes

A partition of good cubes

There exists a partition P of Zd into triadic cubes of varying sizes such that

1 Every cube ◻ ∈ P is a good cube, 2 Two neighboring cubes ◻,◻′ ∈ P have comparable sizes;

1 3 ≤ size(◻) size(◻′) ≤ 3.

3 For x ∈ Zd, if we denote by ◻P(x) the unique cube of the partition P

containing x, then the size of ◻P(x) has an exponential tail P(size(◻P(x)) > R) ≤ C exp(−cR).

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A partition of good cubes

Figure 3: A realization of the partition P

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Application of the partition P

What can this partition be used to ☞ Prove functional inequalities on the percolation cluster: Poincaré (see Barlow 04, Mathieu, Remy 04, Benjamini, Mossel 03), Sobolev, Meyers etc.

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Contents

1

Introduction

2

Homogenization on percolation clusters

3

Homogenization applied to the ∇φ model

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The ∇φ model

Model of stochastic interface: Consider a scalar field φ ∶ Zd → R ⇒ we model the interface by {(x,φ(x))} ⊆ Zd × R. the value φ(x) is the height of the interface at x.

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The ∇φ model

Model of stochastic interface: Consider a scalar field φ ∶ Zd → R ⇒ we model the interface by {(x,φ(x))} ⊆ Zd × R. the value φ(x) is the height of the interface at x. We associate an energy to the configuration φ thanks to the Hamiltonian H(φ) ∶= ∑

x∼y

V (φ(x) − φ(y)) where V ∶ R → R is an elastic potential, uniformly convex.

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The ∇φ model

☞ Associated to the Langevin dynamics dφt(x) = − ∑

y∼x

V ′ (φt(x) − φt(y))dt + √ 2dBt (x). ☞ Random interface distributed according to the Gibbs measure 1 Z exp(−H(φ))∏

x

dφ(x).

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The ∇φ model

Figure 4: A realization of the random surface (by C. Gu)

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The surface tension

Goal: Prove that the properly rescaled version of the interface converges to a deterministic interface u which is a critical point of W (u) ∶= ∫ ν (∇u(x)) dx, where ν is the surface tension associated to the ∇φ model. Two questions: ☞ What is the surface tension ν? ☞ Can we obtain a rate of convergence?

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The surface tension

☞ Given a cube Q ⊆ Zd and a slope p ∈ Rd, ZQ,p ∶= ∫ e−H(φ) ∏

x∈Q

dφ(x) ∏

x∈∂Q

δφ(x)=p⋅x. It is the partition function associated to the model with in the cube with affine boundary condition.

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The surface tension

☞ Given a cube Q ⊆ Zd and a slope p ∈ Rd, ZQ,p ∶= ∫ e−H(φ) ∏

x∈Q

dφ(x) ∏

x∈∂Q

δφ(x)=p⋅x. It is the partition function associated to the model with in the cube with affine boundary condition. ☞ The finite volume surface tension ν(Q,p) ∶= − 1 ∣Q∣ lnZQ,p.

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The surface tension

Theorem, Funaki-Spohn

For each p ∈ Rd, ν (Q,p) →

∣Q∣→∞ ν(p).

☞ Subadditivity argument.

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The surface tension

Theorem, Funaki-Spohn

For each p ∈ Rd, ν (Q,p) →

∣Q∣→∞ ν(p).

☞ Subadditivity argument.

Theorem

There exists α > 0 such that, for each p ∈ B1, each cube Q of size R, ∣ν (Q,p) − ν(p)∣ ≤ CR−α.

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Ideas of the proof

Use ideas from stochastic homogenization: The finite volume surface tension plays a similar role as the energy associated to the corrector νhom(Q,p) ∶= inf

u∈lp+H1

0(Q)⨏Q ∇u ⋅ a∇u.

A few important properties of this quantity: Variational formulation; Uniformly convex.

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A variational formulation for ν

The finite volume surface tension has the variational formulation: ν (Q,p) = inf

P∈P(lp+h1

0(Q))

F (P)

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A variational formulation for ν

The finite volume surface tension has the variational formulation: ν (Q,p) = inf

P∈P(lp+h1

0(Q))

F (P) with F (P) ∶=

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A variational formulation for ν

The finite volume surface tension has the variational formulation: ν (Q,p) = inf

P∈P(lp+h1

0(Q))

F (P) with F (P) ∶= 1 ∣Q∣ EP [H(φ)] ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ Energy +

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A variational formulation for ν

The finite volume surface tension has the variational formulation: ν (Q,p) = inf

P∈P(lp+h1

0(Q))

F (P) with F (P) ∶= 1 ∣Q∣ EP [H(φ)] ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ Energy + 1 ∣Q∣ H (P) ÜÜÜÜÜ Entropy , where H(P) is the differential entropy of P: H (P) ∶= ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ∫ dP dLeb(x)ln( dP dLeb(x)) dx if meaningful, +∞ otherwise.

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Convexity of F

Convexity using optimal transport: ☞ Let P0,P1 be two probability measures on lp + h1

0(Q);

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Convexity of F

Convexity using optimal transport: ☞ Let P0,P1 be two probability measures on lp + h1

0(Q);

☞ Consider the optimal coupling µ between P0 and P1;

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Convexity of F

Convexity using optimal transport: ☞ Let P0,P1 be two probability measures on lp + h1

0(Q);

☞ Consider the optimal coupling µ between P0 and P1; ☞ For t ∈ [0,1], consider Pt ∶= ((1 − t)φ0 + tφ1)∗ µ(dφ0,dφ1) .

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1 The mapping t ↦ EPt [H(φ)] is uniformly convex. Paul Dario PhD Defense June 18th, 2019 32 / 33

1 The mapping t ↦ EPt [H(φ)] is uniformly convex. 2 The mapping t ↦ H (Pt) is convex McCann 95. Paul Dario PhD Defense June 18th, 2019 32 / 33

Thank you for your attention!

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