math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2)...

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1 / 28 Large deviation for the Ginzburg-Landau φ interface model with a conservation law Takao NISHIKAWA March 29, 2008

Transcript of math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2)...

Page 1: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

1 / 28

Large deviation for the Ginzburg-Landau ∇φ interface model with aconservation law

Takao NISHIKAWA

March 29, 2008

Page 2: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Model

Model

• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model

Main result

Rough sketch of theproof

2 / 28

Page 3: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Microscopic interface

Model

• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model

Main result

Rough sketch of theproof

3 / 28

Interface φ = {φ(x) ∈ R; x ∈ Zd}

Zd( )Rd

Á( )x

x

φ(x): the height at posision x

Page 4: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Energy of miscroscopic interface

Model

• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model

Main result

Rough sketch of theproof

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Energy of the microscopic interface φ = {φ(x) ∈ R; x ∈ Zd}

H(φ) =12

∑x,y∈Zd,|x−y|=1

V (φ(x) − φ(y))

(V : R → R is C2, symm., strictly convex)

Page 5: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Dynamics without conservation laws

Model

• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model

Main result

Rough sketch of theproof

5 / 28

Langevin eq.

dφ0t (x) = − ∂H

∂φ(x)(φ0

t )dt +√

2dwt(x), x ∈ ΓN (1)

• w = {wt(x); x ∈ ΓN}: independent 1D B.m.’s

• ∂H

∂φ(x)=

∑y:|x−y|=1

V ′(φ(x) − φ(y))

Page 6: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Dynamics without conservation laws

Model

• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model

Main result

Rough sketch of theproof

5 / 28

Langevin eq.

dφ0t (x) = − ∂H

∂φ(x)(φ0

t )dt +√

2dwt(x), x ∈ ΓN (1)

• w = {wt(x); x ∈ ΓN}: independent 1D B.m.’s

• ∂H

∂φ(x)=

∑y:|x−y|=1

V ′(φ(x) − φ(y))

Page 7: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Dynamics without conservation laws

Model

• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model

Main result

Rough sketch of theproof

5 / 28

Langevin eq.

dφ0t (x) = − ∂H

∂φ(x)(φ0

t )dt +√

2dwt(x), x ∈ ΓN (1)

• w = {wt(x); x ∈ ΓN}: independent 1D B.m.’s

• ∂H

∂φ(x)=

∑y:|x−y|=1

V ′(φ(x) − φ(y))

⇐⇒ Glauber dynamics

Page 8: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Dynamics with a conservation law (1)

Model

• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model

Main result

Rough sketch of theproof

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dφt(x) = Δ{

∂H

∂φ(·)(φt)}

(x)dt +√

2dwt(x), x ∈ ΓN (2)

• w = {wt(x); x ∈ ΓN}: Gaussian process with covariancestructure

E[ws(x)wt(y)] = Δ(x, y)s ∧ t

• Δ = ΔΓN: (discrete) Laplacian on ΓN

Δf(x) =∑

y∈ΓN ,|x−y|=1

(f(y) − f(x)), x ∈ ΓN

Page 9: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Dynamics with a conservation law (1)

Model

• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model

Main result

Rough sketch of theproof

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dφt(x) = Δ{

∂H

∂φ(·)(φt)}

(x)dt +√

2dwt(x), x ∈ ΓN (2)

• w = {wt(x); x ∈ ΓN}: Gaussian process with covariancestructure

E[ws(x)wt(y)] = Δ(x, y)s ∧ t

• Δ = ΔΓN: (discrete) Laplacian on ΓN

Δf(x) =∑

y∈ΓN ,|x−y|=1

(f(y) − f(x)), x ∈ ΓN

Page 10: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Dynamics with a conservation law (1)

Model

• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model

Main result

Rough sketch of theproof

6 / 28

dφt(x) = Δ{

∂H

∂φ(·)(φt)}

(x)dt +√

2dwt(x), x ∈ ΓN (2)

• w = {wt(x); x ∈ ΓN}: Gaussian process with covariancestructure

E[ws(x)wt(y)] = Δ(x, y)s ∧ t

• Δ = ΔΓN: (discrete) Laplacian on ΓN

Δf(x) =∑

y∈ΓN ,|x−y|=1

(f(y) − f(x)), x ∈ ΓN

Page 11: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Dynamics with a conservation law (2)

Model

• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model

Main result

Rough sketch of theproof

7 / 28

Remark 1. By Ito’s formula, it is easy to see∑x∈ΓN

φt(x) ≡∑

x∈ΓN

φ0(x) (= const.), t ≥ 0, (3)

that is, the total sum of the height variable (� number of particle) isconserved by this time evolution.

Page 12: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Dynamics with a conservation law (2)

Model

• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model

Main result

Rough sketch of theproof

7 / 28

Remark 1. By Ito’s formula, it is easy to see∑x∈ΓN

φt(x) ≡∑

x∈ΓN

φ0(x) (= const.), t ≥ 0, (3)

that is, the total sum of the height variable (� number of particle) isconserved by this time evolution.

⇐⇒ Kawasaki dynamics

Page 13: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Macroscopic dynamics (scaling)

Model

• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model

Main result

Rough sketch of theproof

8 / 28

1. dynamics without conservation lawshN,0(t, θ) (t ∈ [0, t], θ ∈ [0, 1)d =: T

d)

hN,0(t, x/N) = N−1φ0N2t(x), x ∈ ΓN

(N−1 in space, N2 in time)

2. dynamics with a conservation lawhN (t, θ) (t ∈ [0, t], θ ∈ T

d)

hN (t, x/N) = N−1φN4t(x), x ∈ ΓN

(N−1 in space, N4 in time)

Page 14: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Macroscopic dynamics (scaling)

Model

• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model

Main result

Rough sketch of theproof

8 / 28

1. dynamics without conservation lawshN,0(t, θ) (t ∈ [0, t], θ ∈ [0, 1)d =: T

d)

hN,0(t, x/N) = N−1φ0N2t(x), x ∈ ΓN

(N−1 in space, N2 in time)

2. dynamics with a conservation lawhN (t, θ) (t ∈ [0, t], θ ∈ T

d)

hN (t, x/N) = N−1φN4t(x), x ∈ ΓN

(N−1 in space, N4 in time)

Page 15: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Hydrodynamic scaling limit (LLN)

Model

• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model

Main result

Rough sketch of theproof

9 / 28

1. dynamics without conservation laws ([FS97],[N03])

hN,0 −→ h0 :∂h0

∂t= div∇σ(∇h0),

where σ : Rd → R is surface tension introduced via

thermodynamic limit

2. dynamics with a conservation law ([N02])

hN −→ h :∂h

∂t= −Δ div∇σ(∇h)

Page 16: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Hydrodynamic scaling limit (LLN)

Model

• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model

Main result

Rough sketch of theproof

9 / 28

1. dynamics without conservation laws ([FS97],[N03])

hN,0 −→ h0 :∂h0

∂t= div∇σ(∇h0),

where σ : Rd → R is surface tension introduced via

thermodynamic limit

2. dynamics with a conservation law ([N02])

hN −→ h :∂h

∂t= −Δ div∇σ(∇h)

Page 17: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Hydrodynamic scaling limit (LLN)

Model

• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model

Main result

Rough sketch of theproof

9 / 28

1. dynamics without conservation laws ([FS97],[N03])

hN,0 −→ h0 :∂h0

∂t= div∇σ(∇h0),

where σ : Rd → R is surface tension introduced via

thermodynamic limit

2. dynamics with a conservation law ([N02])

hN −→ h :∂h

∂t= −Δ div∇σ(∇h)

Does Large deviation principle hold?

Page 18: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

LDP for the non-conservative model

Model

• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model

Main result

Rough sketch of theproof

10 / 28

Theorem 1 ([FN01]). hN,0 satisfies the large deviation principle inC([0, T ], L2

w(Td)) with the speed Nd. The rate functional has thefollowing expression if h is smooth enough:

I(h) =∫ T

0

∥∥∥∥ d

dth(t) − div∇σ(∇h(t))

∥∥∥∥2

L2

dt

Page 19: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

LDP for the non-conservative model

Model

• Microscopic interface• Energy ofmiscroscopic interface• Dynamics withoutconservation laws• Dynamics with aconservation law (1)• Dynamics with aconservation law (2)• Macroscopicdynamics (scaling)• Hydrodynamic scalinglimit (LLN)• LDP for thenon-conservative model

Main result

Rough sketch of theproof

10 / 28

Theorem 1 ([FN01]). hN,0 satisfies the large deviation principle inC([0, T ], L2

w(Td)) with the speed Nd. The rate functional has thefollowing expression if h is smooth enough:

I(h) =∫ T

0

∥∥∥∥ d

dth(t) − div∇σ(∇h(t))

∥∥∥∥2

L2

dt

LDP for the dynamics with conservation law?

Page 20: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Main result

Model

Main result

• Assumptions• Large deviationprinciple• The rate functionalI(h)

Rough sketch of theproof

11 / 28

Page 21: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Assumptions

Model

Main result

• Assumptions• Large deviationprinciple• The rate functionalI(h)

Rough sketch of theproof

12 / 28

Assumption 1. We assume the following:

• d ≤ 3 (technical)

• A sequence of initial datum {hN (0)} is deterministic andsatisfying the following condition:

◦ 〈hN (0)〉 :=∫

Td

h(0, θ) dθ ≡ 0,

◦ ∃h0 ∈ H−1(Td) s.t. limN→∞

‖hN (0) − h0‖H−1 = 0,

◦ supN≥1

‖hN (0)‖H1 < ∞

Page 22: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Assumptions

Model

Main result

• Assumptions• Large deviationprinciple• The rate functionalI(h)

Rough sketch of theproof

12 / 28

Assumption 1. We assume the following:

• d ≤ 3 (technical)

• A sequence of initial datum {hN (0)} is deterministic andsatisfying the following condition:

◦ 〈hN (0)〉 :=∫

Td

h(0, θ) dθ ≡ 0,

◦ ∃h0 ∈ H−1(Td) s.t. limN→∞

‖hN (0) − h0‖H−1 = 0,

◦ supN≥1

‖hN (0)‖H1 < ∞

Page 23: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Assumptions

Model

Main result

• Assumptions• Large deviationprinciple• The rate functionalI(h)

Rough sketch of theproof

12 / 28

Assumption 1. We assume the following:

• d ≤ 3 (technical)

• A sequence of initial datum {hN (0)} is deterministic andsatisfying the following condition:

◦ 〈hN (0)〉 :=∫

Td

h(0, θ) dθ ≡ 0,

◦ ∃h0 ∈ H−1(Td) s.t. limN→∞

‖hN (0) − h0‖H−1 = 0,

◦ supN≥1

‖hN (0)‖H1 < ∞

Page 24: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Large deviation principle

Model

Main result

• Assumptions• Large deviationprinciple• The rate functionalI(h)

Rough sketch of theproof

13 / 28

Theorem 2. hN satisfying large deviation principle inC([0, T ], H−1

w (Td)) with the speed Nd and the rate functionalI(h). The rate functional I(h) has the following expression if h issmooth:

I(h) =∫ T

0

∥∥∥∥ d

dth(t) − Δ div∇σ(∇h(t))

∥∥∥∥2

H−1

dt

Page 25: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

The rate functional I(h)

Model

Main result

• Assumptions• Large deviationprinciple• The rate functionalI(h)

Rough sketch of theproof

14 / 28

Remark 2. Exactly saying, the rate functional is expressed asfollows:

I(h) = supJ∈C∞([0,T ]×Td),〈J(t)〉=0

I(h; J)

I(h; J) =∫

Td

J(T, θ)h(T, θ) dθ −∫

Td

J(0, θ)h0(θ) dθ

−∫ T

0dt

∫Td

∂J

∂t(t, θ)h(t, θ) dθ

+∫ T

0dt

∫Td

(∇ΔJ(t, θ),∇σ(∇h(t, θ)))Rd dθ

+∫ T

0dt

∫Td

J(t, θ)ΔJ(t, θ) dθ

Page 26: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

The rate functional I(h)

Model

Main result

• Assumptions• Large deviationprinciple• The rate functionalI(h)

Rough sketch of theproof

14 / 28

Remark 2. Exactly saying, the rate functional is expressed asfollows:

I(h) = supJ∈C∞([0,T ]×Td),〈J(t)〉=0

I(h; J)

I(h; J) =∫

Td

J(T, θ)h(T, θ) dθ −∫

Td

J(0, θ)h0(θ) dθ

−∫ T

0dt

∫Td

∂J

∂t(t, θ)h(t, θ) dθ

+∫ T

0dt

∫Td

(∇ΔJ(t, θ),∇σ(∇h(t, θ)))Rd dθ

+∫ T

0dt

∫Td

J(t, θ)ΔJ(t, θ) dθ

Page 27: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Rough sketch of the proof

Model

Main result

Rough sketch of theproof• Rough sketch of theproof

• Notations (1)

• Notations (2)

• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate

• Proof of Theorem 7

• Remaining Tasks

15 / 28

Page 28: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Rough sketch of the proof

Model

Main result

Rough sketch of theproof• Rough sketch of theproof

• Notations (1)

• Notations (2)

• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate

• Proof of Theorem 7

• Remaining Tasks

16 / 28

The proof is by H−1-method in

• Landim-Yau, Commun. Pure Appl. Math. (’95)

• Funaki-N., Probab. Theory Relat. Fields (’01)

Page 29: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Rough sketch of the proof

Model

Main result

Rough sketch of theproof• Rough sketch of theproof

• Notations (1)

• Notations (2)

• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate

• Proof of Theorem 7

• Remaining Tasks

16 / 28

The proof is by H−1-method in

• Landim-Yau, Commun. Pure Appl. Math. (’95)

• Funaki-N., Probab. Theory Relat. Fields (’01)

Page 30: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Rough sketch of the proof

Model

Main result

Rough sketch of theproof• Rough sketch of theproof

• Notations (1)

• Notations (2)

• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate

• Proof of Theorem 7

• Remaining Tasks

16 / 28

The proof is by H−1-method in

• Landim-Yau, Commun. Pure Appl. Math. (’95)

• Funaki-N., Probab. Theory Relat. Fields (’01)

Page 31: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Notations (1)

Model

Main result

Rough sketch of theproof• Rough sketch of theproof

• Notations (1)

• Notations (2)

• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate

• Proof of Theorem 7

• Remaining Tasks

17 / 28

• (Zd)∗: all oriented bonds in Zd, i.e.

(Zd)∗ = {(x, y) ∈ Zd × Z

d; |x − y| = 1}

• Γ∗N : all oriented bonds in ΓN

• X : all η ∈ R(Z

d)∗ satisfying the following conditions:

1. η(b) = −η(−b),where −b = (y, x) for b = (x, y).

2. For every closed loops C∑b∈C

η(b) = 0

holds.

Page 32: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Notations (1)

Model

Main result

Rough sketch of theproof• Rough sketch of theproof

• Notations (1)

• Notations (2)

• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate

• Proof of Theorem 7

• Remaining Tasks

17 / 28

• (Zd)∗: all oriented bonds in Zd, i.e.

(Zd)∗ = {(x, y) ∈ Zd × Z

d; |x − y| = 1}

• Γ∗N : all oriented bonds in ΓN

• X : all η ∈ R(Z

d)∗ satisfying the following conditions:

1. η(b) = −η(−b),where −b = (y, x) for b = (x, y).

2. For every closed loops C∑b∈C

η(b) = 0

holds.

Page 33: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Notations (1)

Model

Main result

Rough sketch of theproof• Rough sketch of theproof

• Notations (1)

• Notations (2)

• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate

• Proof of Theorem 7

• Remaining Tasks

17 / 28

• (Zd)∗: all oriented bonds in Zd, i.e.

(Zd)∗ = {(x, y) ∈ Zd × Z

d; |x − y| = 1}

• Γ∗N : all oriented bonds in ΓN

• X : all η ∈ R(Z

d)∗ satisfying the following conditions:

1. η(b) = −η(−b),where −b = (y, x) for b = (x, y).

2. For every closed loops C∑b∈C

η(b) = 0

holds.

Page 34: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Notations (2)

Model

Main result

Rough sketch of theproof• Rough sketch of theproof

• Notations (1)

• Notations (2)

• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate

• Proof of Theorem 7

• Remaining Tasks

18 / 28

• ∇: discrete gradient

∇φ(b) = φ(x) − φ(y), b = (x, y)

• μu: shift-invariant ergodic (canonical) Gibbs meas. on gradientfield X

• F (u): expected value of F : RX → R with respect to μu

F (u) = Eμu [F (η)]

• Λl = {x ∈ Zd; max |xi| ≤ l}

• τy : spatical shift by y ∈ Zd

• AvΛ F (η): sample mean of F on Λ � ΓN , that is,

Page 35: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Notations (2)

Model

Main result

Rough sketch of theproof• Rough sketch of theproof

• Notations (1)

• Notations (2)

• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate

• Proof of Theorem 7

• Remaining Tasks

18 / 28

• ∇: discrete gradient

∇φ(b) = φ(x) − φ(y), b = (x, y)

• μu: shift-invariant ergodic (canonical) Gibbs meas. on gradientfield X

• F (u): expected value of F : RX → R with respect to μu

F (u) = Eμu [F (η)]

• Λl = {x ∈ Zd; max |xi| ≤ l}

• τy : spatical shift by y ∈ Zd

• AvΛ F (η): sample mean of F on Λ � ΓN , that is,

Page 36: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Notations (2)

Model

Main result

Rough sketch of theproof• Rough sketch of theproof

• Notations (1)

• Notations (2)

• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate

• Proof of Theorem 7

• Remaining Tasks

18 / 28

• ∇: discrete gradient

∇φ(b) = φ(x) − φ(y), b = (x, y)

• μu: shift-invariant ergodic (canonical) Gibbs meas. on gradientfield X

• F (u): expected value of F : RX → R with respect to μu

F (u) = Eμu [F (η)]

• Λl = {x ∈ Zd; max |xi| ≤ l}

• τy : spatical shift by y ∈ Zd

• AvΛ F (η): sample mean of F on Λ � ΓN , that is,

Page 37: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Notations (2)

Model

Main result

Rough sketch of theproof• Rough sketch of theproof

• Notations (1)

• Notations (2)

• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate

• Proof of Theorem 7

• Remaining Tasks

18 / 28

• ∇: discrete gradient

∇φ(b) = φ(x) − φ(y), b = (x, y)

• μu: shift-invariant ergodic (canonical) Gibbs meas. on gradientfield X

• F (u): expected value of F : RX → R with respect to μu

F (u) = Eμu [F (η)]

• Λl = {x ∈ Zd; max |xi| ≤ l}

• τy : spatical shift by y ∈ Zd

• AvΛ F (η): sample mean of F on Λ � ΓN , that is,

Page 38: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Notations (2)

Model

Main result

Rough sketch of theproof• Rough sketch of theproof

• Notations (1)

• Notations (2)

• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate

• Proof of Theorem 7

• Remaining Tasks

18 / 28

• ∇: discrete gradient

∇φ(b) = φ(x) − φ(y), b = (x, y)

• μu: shift-invariant ergodic (canonical) Gibbs meas. on gradientfield X

• F (u): expected value of F : RX → R with respect to μu

F (u) = Eμu [F (η)]

• Λl = {x ∈ Zd; max |xi| ≤ l}

• τy : spatical shift by y ∈ Zd

• AvΛ F (η): sample mean of F on Λ � ΓN , that is,

Page 39: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Notations (3)

Model

Main result

Rough sketch of theproof• Rough sketch of theproof

• Notations (1)

• Notations (2)

• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate

• Proof of Theorem 7

• Remaining Tasks

19 / 28

• ηlx: sample mean of η ∈ X on Λl + x, i.e.,

ηlx = (2l + 1)−d

∑y∈Λl+x

d∑i=1

η(ei + y)ei ∈ Rd, (4)

where ei is i-th unit vector in Rd

• WFl : the difference between the sample mean and the

ensemble mean

WFl (η) = AvΛl

F (η) − F (ηl0), η ∈ X (5)

Page 40: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Notations (3)

Model

Main result

Rough sketch of theproof• Rough sketch of theproof

• Notations (1)

• Notations (2)

• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate

• Proof of Theorem 7

• Remaining Tasks

19 / 28

• ηlx: sample mean of η ∈ X on Λl + x, i.e.,

ηlx = (2l + 1)−d

∑y∈Λl+x

d∑i=1

η(ei + y)ei ∈ Rd, (4)

where ei is i-th unit vector in Rd

• WFl : the difference between the sample mean and the

ensemble mean

WFl (η) = AvΛl

F (η) − F (ηl0), η ∈ X (5)

Page 41: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Exponential a-priori estimate

Model

Main result

Rough sketch of theproof• Rough sketch of theproof

• Notations (1)

• Notations (2)

• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate

• Proof of Theorem 7

• Remaining Tasks

20 / 28

Proposition 3 (A priori estimate). There exists a constant κ > 0such that for all T ≥ 0, 0 < β ≤ 1, N ≥ 1 and all initial datum

E

[exp

{βe−κT

∫ T

0

(∑b∈Γ∗

N

V ′(∇φNs (b))∇φN

s (b)

+ Nd‖hN (s)‖2H−1

)ds

}]

≤ exp{βNd‖hN‖2H−1 + 2βNd/κ}.

(6)

holds.

Page 42: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Superexponential local equilibrium

Model

Main result

Rough sketch of theproof• Rough sketch of theproof

• Notations (1)

• Notations (2)

• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate

• Proof of Theorem 7

• Remaining Tasks

21 / 28

Theorem 4. For all α > 0 we have

limε↓0

limN→∞

N−d log E

⎡⎣exp

⎛⎝α

∫ T

0

∑x∈ΓN

∣∣τxWFNε(∇φN

t )∣∣ dt

⎞⎠⎤⎦ ≤ 0.

Page 43: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Superexponential local equilibrium

Model

Main result

Rough sketch of theproof• Rough sketch of theproof

• Notations (1)

• Notations (2)

• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate

• Proof of Theorem 7

• Remaining Tasks

21 / 28

Theorem 4. For all α > 0 we have

limε↓0

limN→∞

N−d log E

⎡⎣exp

⎛⎝α

∫ T

0

∑x∈ΓN

∣∣τxWFNε(∇φN

t )∣∣ dt

⎞⎠⎤⎦ ≤ 0.

This theorem is the core of the proof of LDP

Page 44: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Superexponential 1-block estimate

Model

Main result

Rough sketch of theproof• Rough sketch of theproof

• Notations (1)

• Notations (2)

• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate

• Proof of Theorem 7

• Remaining Tasks

22 / 28

Theorem 5. For all α > 0 we have

liml→∞

limN→∞

N−d log E

⎡⎣exp

⎛⎝α

∫ T

0

∑x∈ΓN

∣∣τxWFl (∇φN

t )∣∣ dt

⎞⎠⎤⎦ ≤ 0.

Page 45: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Proof of Theorem 5 (1)

Model

Main result

Rough sketch of theproof• Rough sketch of theproof

• Notations (1)

• Notations (2)

• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate

• Proof of Theorem 7

• Remaining Tasks

23 / 28

Since we have

EφN

[exp

∫ T

0

∑x∈ΓN

∣∣τxWFl (∇φN

t )∣∣ dt

)]

≤(

supφN

EφN

[exp

∫ N−dT

0

∑x∈ΓN

∣∣τxWFl (∇φN

t )∣∣ dt

)])Nd

by Markov property, it is sufficient to show that

liml→∞

limN→∞

log supφN

EφN

[exp

∫ N−dT

0

∑x∈ΓN

∣∣τxWFl (∇φN

t )∣∣ dt

)]≤ 0.

Page 46: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Proof of Theorem 5 (2)

Model

Main result

Rough sketch of theproof• Rough sketch of theproof

• Notations (1)

• Notations (2)

• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate

• Proof of Theorem 7

• Remaining Tasks

24 / 28

Here, since Portenko’s argument(cf. Simon’s book) is applicapable,and the superexponetial estimate can be reduced to the ordinary (noexponential!).Therefore, once we have

liml→∞

limN→∞

supφN

EφN

⎡⎣∫ N−dT

0

∑x∈ΓN

∣∣τxWFl (∇φN

t )∣∣ dt

⎤⎦ ≤ 0,

we obtain the conclusion.

Page 47: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Proof of Theorem 5 (2)

Model

Main result

Rough sketch of theproof• Rough sketch of theproof

• Notations (1)

• Notations (2)

• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate

• Proof of Theorem 7

• Remaining Tasks

24 / 28

Here, since Portenko’s argument(cf. Simon’s book) is applicapable,and the superexponetial estimate can be reduced to the ordinary (noexponential!).Therefore, once we have

liml→∞

limN→∞

supφN

EφN

⎡⎣∫ N−dT

0

∑x∈ΓN

∣∣τxWFl (∇φN

t )∣∣ dt

⎤⎦ ≤ 0,

we obtain the conclusion.

Page 48: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Proof of Theorem 5 (2)

Model

Main result

Rough sketch of theproof• Rough sketch of theproof

• Notations (1)

• Notations (2)

• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate

• Proof of Theorem 7

• Remaining Tasks

24 / 28

Here, since Portenko’s argument(cf. Simon’s book) is applicapable,and the superexponetial estimate can be reduced to the ordinary (noexponential!).Therefore, once we have

liml→∞

limN→∞

supφN

EφN

⎡⎣∫ N−dT

0

∑x∈ΓN

∣∣τxWFl (∇φN

t )∣∣ dt

⎤⎦ ≤ 0,

we obtain the conclusion.

This estimate is the ordinary 1-block estimateif d ≤ 3, and therefore we can prove it by astandard way!

Page 49: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Superexponential 2-blocks estimate

Model

Main result

Rough sketch of theproof• Rough sketch of theproof

• Notations (1)

• Notations (2)

• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate

• Proof of Theorem 7

• Remaining Tasks

25 / 28

Theorem 6. For every α > 0 we have

liml→∞

limε↓0

limN→∞

N−d log EφN[exp

∫ T

0

∑x∈ΓN

∣∣∣W f,Nl,ε (∇φN

t )∣∣∣ dt

)]≤ 0.

Once we have following theorem, we can obtain the above.

Theorem 7. For every α > 0 we have

liml→∞

limε↓0

limN→∞

N−d log EφN⎡⎣exp

⎛⎝α |ΛNε|−1

∫ T

0

∑x∈ΓN

∑y∈ΛNε�Λ2l

∣∣ηlx − ηl

x+y

∣∣2 dt

⎞⎠⎤⎦ ≤ 0.

Page 50: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Superexponential 2-blocks estimate

Model

Main result

Rough sketch of theproof• Rough sketch of theproof

• Notations (1)

• Notations (2)

• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate

• Proof of Theorem 7

• Remaining Tasks

25 / 28

Theorem 6. For every α > 0 we have

liml→∞

limε↓0

limN→∞

N−d log EφN[exp

∫ T

0

∑x∈ΓN

∣∣∣W f,Nl,ε (∇φN

t )∣∣∣ dt

)]≤ 0.

Once we have following theorem, we can obtain the above.

Theorem 7. For every α > 0 we have

liml→∞

limε↓0

limN→∞

N−d log EφN⎡⎣exp

⎛⎝α |ΛNε|−1

∫ T

0

∑x∈ΓN

∑y∈ΛNε�Λ2l

∣∣ηlx − ηl

x+y

∣∣2 dt

⎞⎠⎤⎦ ≤ 0.

Page 51: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Proof of Theorem 7

Model

Main result

Rough sketch of theproof• Rough sketch of theproof

• Notations (1)

• Notations (2)

• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate

• Proof of Theorem 7

• Remaining Tasks

26 / 28

By a similar manner to [FN01], it is sufficient to show the following:

liml→∞

limε↓0

limN→∞

N−d log EφN

[exp

⎛⎝α |ΛNε|−1

∫ T

0

∑y∈ΛNε

⎛⎝∑

b∈Γ∗N

Ψy(b;∇φNt ) − 4Nd

⎞⎠ dt

⎞⎠] ≤ 0

(7)

• Ψy(b; η) ={V ′(η(b + y)) − V ′(η(b))

} {η(b + y) − η(b)}• Thermodynamic Identity:

d∑i=1

E[V ′(η(ei))η(ei)] = ∇σ(u) · ui + 1

Page 52: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Proof of Theorem 7

Model

Main result

Rough sketch of theproof• Rough sketch of theproof

• Notations (1)

• Notations (2)

• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate

• Proof of Theorem 7

• Remaining Tasks

26 / 28

By a similar manner to [FN01], it is sufficient to show the following:

liml→∞

limε↓0

limN→∞

N−d log EφN

[exp

⎛⎝α |ΛNε|−1

∫ T

0

∑y∈ΛNε

⎛⎝∑

b∈Γ∗N

Ψy(b;∇φNt ) − 4Nd

⎞⎠ dt

⎞⎠] ≤ 0

(7)

• Ψy(b; η) ={V ′(η(b + y)) − V ′(η(b))

} {η(b + y) − η(b)}• Thermodynamic Identity:

d∑i=1

E[V ′(η(ei))η(ei)] = ∇σ(u) · ui + 1

Page 53: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Proof of Theorem 7

Model

Main result

Rough sketch of theproof• Rough sketch of theproof

• Notations (1)

• Notations (2)

• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate

• Proof of Theorem 7

• Remaining Tasks

26 / 28

By a similar manner to [FN01], it is sufficient to show the following:

liml→∞

limε↓0

limN→∞

N−d log EφN

[exp

⎛⎝α |ΛNε|−1

∫ T

0

∑y∈ΛNε

⎛⎝∑

b∈Γ∗N

Ψy(b;∇φNt ) − 4Nd

⎞⎠ dt

⎞⎠] ≤ 0

(7)

• Ψy(b; η) ={V ′(η(b + y)) − V ′(η(b))

} {η(b + y) − η(b)}• Thermodynamic Identity:

d∑i=1

E[V ′(η(ei))η(ei)] = ∇σ(u) · ui + 1

Page 54: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Remaining Tasks

Model

Main result

Rough sketch of theproof• Rough sketch of theproof

• Notations (1)

• Notations (2)

• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate

• Proof of Theorem 7

• Remaining Tasks

27 / 28

• LDP in the case of the dimension d ≥ 4

To obtain 1-block estimate, the stationary measuresneed to be characterized.(Gibbs measures and reversible measures arecharacterized completely, but stationary measuresare not.)

• LDP in the case of Dirichlet-type boundary condition

Even for the non-conservative dynamics, LDP is stillopen.Problem: how to get the enough regularity toestablish 2-blocks estimate

Page 55: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Remaining Tasks

Model

Main result

Rough sketch of theproof• Rough sketch of theproof

• Notations (1)

• Notations (2)

• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate

• Proof of Theorem 7

• Remaining Tasks

27 / 28

• LDP in the case of the dimension d ≥ 4

To obtain 1-block estimate, the stationary measuresneed to be characterized.(Gibbs measures and reversible measures arecharacterized completely, but stationary measuresare not.)

• LDP in the case of Dirichlet-type boundary condition

Even for the non-conservative dynamics, LDP is stillopen.Problem: how to get the enough regularity toestablish 2-blocks estimate

Page 56: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Remaining Tasks

Model

Main result

Rough sketch of theproof• Rough sketch of theproof

• Notations (1)

• Notations (2)

• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate

• Proof of Theorem 7

• Remaining Tasks

27 / 28

• LDP in the case of the dimension d ≥ 4

To obtain 1-block estimate, the stationary measuresneed to be characterized.(Gibbs measures and reversible measures arecharacterized completely, but stationary measuresare not.)

• LDP in the case of Dirichlet-type boundary condition

Even for the non-conservative dynamics, LDP is stillopen.Problem: how to get the enough regularity toestablish 2-blocks estimate

Page 57: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Remaining Tasks

Model

Main result

Rough sketch of theproof• Rough sketch of theproof

• Notations (1)

• Notations (2)

• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate

• Proof of Theorem 7

• Remaining Tasks

27 / 28

• LDP in the case of the dimension d ≥ 4

To obtain 1-block estimate, the stationary measuresneed to be characterized.(Gibbs measures and reversible measures arecharacterized completely, but stationary measuresare not.)

• LDP in the case of Dirichlet-type boundary condition

Even for the non-conservative dynamics, LDP is stillopen.Problem: how to get the enough regularity toestablish 2-blocks estimate

Page 58: math.bme.humath.bme.hu/~balazs/fritzfest/ffnishikawa.pdf · Dynamics with a conservation law (2) Model • Microscopic interface • Energy of miscroscopic interface • Dynamics

Model

Main result

Rough sketch of theproof• Rough sketch of theproof

• Notations (1)

• Notations (2)

• Notations (3)• Exponential a-prioriestimate• Superexponentiallocal equilibrium• Superexponential1-block estimate• Proof of Theorem 5(1)• Proof of Theorem 5(2)• Superexponential2-blocks estimate

• Proof of Theorem 7

• Remaining Tasks

28 / 28

Thank you for your attention!