Large Deviation for (and by) amateur Raphal Chtrite CNRS, - - PowerPoint PPT Presentation

Large Deviation for (and by) amateur

Raphaël Chétrite

CNRS, Laboratoire Jean-Alexandre Dieudonné Nice

30/05/2011

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 1 / 16

Plan

1

Introduction

2

Large deviation for a Markovian process

3

Go Beyond the current

4

Generalization of Gallavotti-Cohen-Evans-Morriss symmetry

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 2 / 16

Caveat

”Lorsque l’on expose ...on peut supposer que chacun connait les variétés de Stein

• u les nombres de Betti d’un espace topologique ; mais si l’on a besoin d’une

intégrale stochastique, on doit définir à partir de zéro les filtrations, les processus prévisibles, les martingales, etc. Il y a la quelque chose d’anormal. Les raisons en sont bien sûr nombreuses, à commencer par le vocabulaire ésotérique des probabilistes”. Laurent Schwartz

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 3 / 16

Large Deviation theory

”Improbable events permit themselves the luxury of occurring.” C.Chan 1928

Heuristic of large deviation

Random variable AT which converges toward a Large deviation : How improbable for AT to converge towards a which is different from the typical value a (rare events) : LDP : 1AT =a ≍ exp(−TI(a)) I(a) is called the rate function (or fluctuation functional)(or action functional). Large deviation theory : Prove the LDP and calculate the rate function.

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 4 / 16

Large Deviation theory

”Improbable events permit themselves the luxury of occurring.” C.Chan 1928

Heuristic of large deviation

Random variable AT which converges toward a Large deviation : How improbable for AT to converge towards a which is different from the typical value a (rare events) : LDP : 1AT =a ≍ exp(−TI(a)) I(a) is called the rate function (or fluctuation functional)(or action functional). Large deviation theory : Prove the LDP and calculate the rate function.

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 4 / 16

Large Deviation theory

”Improbable events permit themselves the luxury of occurring.” C.Chan 1928

Heuristic of large deviation

Random variable AT which converges toward a Large deviation : How improbable for AT to converge towards a which is different from the typical value a (rare events) : LDP : 1AT =a ≍ exp(−TI(a)) I(a) is called the rate function (or fluctuation functional)(or action functional). Large deviation theory : Prove the LDP and calculate the rate function.

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 4 / 16

Scaled cumulant generating function (log-Laplace)

Λ(s) ≡ limT→∞

1 T ln exp(sTAT)

Varadhan Theorem : If AT satisfies the large deviation principle then Λ(s) ≡ limT→∞ 1

T ln

• da exp(sTa) exp (−TI(a))
• ≡

limT→∞ 1

T ln

• da exp(T (sa − I(a))
• = supa∈ℜ (sa − I(a)) ≡ I ⋆(s)

Gartner-Ellis Theorem : If Λ(s) exist and is differentiable, then the LDP exists, I(a) = sups∈ℜ (sa − Λ(s)) ≡ Λ⋆(a) and is strictly convex.

Why rare events can be important

Bibliography :

Den Hollander F, Large Deviations (Providence : Field Institute Monographs) 2000 Touchette H, The large deviation approach to statistical mechanics. Phys. Rep. 2009

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 5 / 16

Scaled cumulant generating function (log-Laplace)

Λ(s) ≡ limT→∞

1 T ln exp(sTAT)

Varadhan Theorem : If AT satisfies the large deviation principle then Λ(s) ≡ limT→∞ 1

T ln

• da exp(sTa) exp (−TI(a))
• ≡

limT→∞ 1

T ln

• da exp(T (sa − I(a))
• = supa∈ℜ (sa − I(a)) ≡ I ⋆(s)

Gartner-Ellis Theorem : If Λ(s) exist and is differentiable, then the LDP exists, I(a) = sups∈ℜ (sa − Λ(s)) ≡ Λ⋆(a) and is strictly convex.

Why rare events can be important

Bibliography :

Den Hollander F, Large Deviations (Providence : Field Institute Monographs) 2000 Touchette H, The large deviation approach to statistical mechanics. Phys. Rep. 2009

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 5 / 16

Scaled cumulant generating function (log-Laplace)

Λ(s) ≡ limT→∞

1 T ln exp(sTAT)

Varadhan Theorem : If AT satisfies the large deviation principle then Λ(s) ≡ limT→∞ 1

T ln

• da exp(sTa) exp (−TI(a))
• ≡

limT→∞ 1

T ln

• da exp(T (sa − I(a))
• = supa∈ℜ (sa − I(a)) ≡ I ⋆(s)

Gartner-Ellis Theorem : If Λ(s) exist and is differentiable, then the LDP exists, I(a) = sups∈ℜ (sa − Λ(s)) ≡ Λ⋆(a) and is strictly convex.

Why rare events can be important

Bibliography :

Den Hollander F, Large Deviations (Providence : Field Institute Monographs) 2000 Touchette H, The large deviation approach to statistical mechanics. Phys. Rep. 2009

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 5 / 16

Scaled cumulant generating function (log-Laplace)

Λ(s) ≡ limT→∞

1 T ln exp(sTAT)

Varadhan Theorem : If AT satisfies the large deviation principle then Λ(s) ≡ limT→∞ 1

T ln

• da exp(sTa) exp (−TI(a))
• ≡

limT→∞ 1

T ln

• da exp(T (sa − I(a))
• = supa∈ℜ (sa − I(a)) ≡ I ⋆(s)

Gartner-Ellis Theorem : If Λ(s) exist and is differentiable, then the LDP exists, I(a) = sups∈ℜ (sa − Λ(s)) ≡ Λ⋆(a) and is strictly convex.

Why rare events can be important

Bibliography :

Den Hollander F, Large Deviations (Providence : Field Institute Monographs) 2000 Touchette H, The large deviation approach to statistical mechanics. Phys. Rep. 2009

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 5 / 16

Pedestrian approach of Level in Large Deviation

Level 1 : empirical mean

Large deviation for empirical mean of type Ae

T ≡ 1 T

R T

0 A(xt)dt

Level 2 : Measure empiric

Large deviation of local time spend in a state ρe

1,T(x) = 1 T

R T

0 δ (Xt − x) dt

Level 2 → Level 1 : contraction principle

The level 1 can be obtain by contraction of Level 2 because Ae

T =

R dxA(x)ρe

T(x).

Then δ (Ae

T − a) =

R

R dxA(x)ρ(x)=a d [ρ] δ (ρe T − ρ) ≍

R

R dxA(x)ρ(x)=a d [ρ] exp (−TI2 [ρ]) ≍ exp

` −T ` infρ/

R dxA(x)ρ(x)=a (I2 [ρ])

´´

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 6 / 16

Pedestrian approach of Level in Large Deviation

Level 1 : empirical mean

Large deviation for empirical mean of type Ae

T ≡ 1 T

R T

0 A(xt)dt

Level 2 : Measure empiric

Large deviation of local time spend in a state ρe

1,T(x) = 1 T

R T

0 δ (Xt − x) dt

Level 2 → Level 1 : contraction principle

The level 1 can be obtain by contraction of Level 2 because Ae

T =

R dxA(x)ρe

T(x).

Then δ (Ae

T − a) =

R

R dxA(x)ρ(x)=a d [ρ] δ (ρe T − ρ) ≍

R

R dxA(x)ρ(x)=a d [ρ] exp (−TI2 [ρ]) ≍ exp

` −T ` infρ/

R dxA(x)ρ(x)=a (I2 [ρ])

´´

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 6 / 16

Pedestrian approach of Level in Large Deviation

Level 1 : empirical mean

Large deviation for empirical mean of type Ae

T ≡ 1 T

R T

0 A(xt)dt

Level 2 : Measure empiric

Large deviation of local time spend in a state ρe

1,T(x) = 1 T

R T

0 δ (Xt − x) dt

Level 2 → Level 1 : contraction principle

The level 1 can be obtain by contraction of Level 2 because Ae

T =

R dxA(x)ρe

T(x).

Then δ (Ae

T − a) =

R

R dxA(x)ρ(x)=a d [ρ] δ (ρe T − ρ) ≍

R

R dxA(x)ρ(x)=a d [ρ] exp (−TI2 [ρ]) ≍ exp

` −T ` infρ/

R dxA(x)ρ(x)=a (I2 [ρ])

´´

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 6 / 16

Pedestrian approach of Level in Large Deviation

Level 1 : empirical mean

Large deviation for empirical mean of type Ae

T ≡ 1 T

R T

0 A(xt)dt

Level 2 : Measure empiric

Large deviation of local time spend in a state ρe

1,T(x) = 1 T

R T

0 δ (Xt − x) dt

Level 2 → Level 1 : contraction principle

The level 1 can be obtain by contraction of Level 2 because Ae

T =

R dxA(x)ρe

T(x).

Then δ (Ae

T − a) =

R

R dxA(x)ρ(x)=a d [ρ] δ (ρe T − ρ) ≍

R

R dxA(x)ρ(x)=a d [ρ] exp (−TI2 [ρ]) ≍ exp

` −T ` infρ/

R dxA(x)ρ(x)=a (I2 [ρ])

´´

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 6 / 16

Pedestrian approach of Level in Large Deviation

Level 1 : empirical mean

Large deviation for empirical mean of type Ae

T ≡ 1 T

R T

0 A(xt)dt

Level 2 : Measure empiric

Large deviation of local time spend in a state ρe

1,T(x) = 1 T

R T

0 δ (Xt − x) dt

Level 2 → Level 1 : contraction principle

The level 1 can be obtain by contraction of Level 2 because Ae

T =

R dxA(x)ρe

T(x).

Then δ (Ae

T − a) =

R

R dxA(x)ρ(x)=a d [ρ] δ (ρe T − ρ) ≍

R

R dxA(x)ρ(x)=a d [ρ] exp (−TI2 [ρ]) ≍ exp

` −T ` infρ/

R dxA(x)ρ(x)=a (I2 [ρ])

´´

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 6 / 16

Pedestrian approach of Level in Large Deviation

Level 1 : empirical mean

Large deviation for empirical mean of type Ae

T ≡ 1 T

R T

0 A(xt)dt

Level 2 : Measure empiric

Large deviation of local time spend in a state ρe

1,T(x) = 1 T

R T

0 δ (Xt − x) dt

Level 2 → Level 1 : contraction principle

The level 1 can be obtain by contraction of Level 2 because Ae

T =

R dxA(x)ρe

T(x).

Then δ (Ae

T − a) =

R

R dxA(x)ρ(x)=a d [ρ] δ (ρe T − ρ) ≍

R

R dxA(x)ρ(x)=a d [ρ] exp (−TI2 [ρ]) ≍ exp

` −T ` infρ/

R dxA(x)ρ(x)=a (I2 [ρ])

´´

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 6 / 16

IID paradigm

Level 1 : Cramer theorem for independents bits

2 states 0 and 1, P(0) = p0 and P(1) = p1 = 1 − p0 Sequence of n bits : [x] = 011100100.... . Time average : X e

n = Pi=n

i=1 xi

n

. Typical behavior : X e

n → p1 = 1 − p0

Untypical behavior P (X e

n = µ) ≍?.

Level 2 : Sanov theorem for independents bits

Empirical density (”type” in statistics) : ρe

n(0) = ♯0 in [x] n

and ρe

n(1) = ♯1 in [x] n

Typical behavior : ρe

n(0) → p0 and ρe n(1) → p1.

Untypical behavior P (ρe

n(0) = µ0, ρe n(1) = µ1) = pnµ0

pnµ1

1 n! (nµ0)!(nµ1)!

Stirling : ln (P (ρe

n(0) = µ0, ρe n(1) = µ1)) = nµ0 ln p0 + nµ1 ln p1 + n ln n −

n − nµ0 ln nµ0 + nµ0 − nµ1 ln nµ1 + nµ1 = −n

• µ0 ln µ0

p0 + µ1 ln µ1 p1

• Case of general space states : I2(µ) ≡
• S(µ/ρ) if

µ ≪ ρ ∞ otherwise

• Raphaël Chétrite

(cnrs) Grandes déviation 30/05/2011 7 / 16

IID paradigm

Level 1 : Cramer theorem for independents bits

2 states 0 and 1, P(0) = p0 and P(1) = p1 = 1 − p0 Sequence of n bits : [x] = 011100100.... . Time average : X e

n = Pi=n

i=1 xi

n

. Typical behavior : X e

n → p1 = 1 − p0

Untypical behavior P (X e

n = µ) ≍?.

Level 2 : Sanov theorem for independents bits

Empirical density (”type” in statistics) : ρe

n(0) = ♯0 in [x] n

and ρe

n(1) = ♯1 in [x] n

Typical behavior : ρe

n(0) → p0 and ρe n(1) → p1.

Untypical behavior P (ρe

n(0) = µ0, ρe n(1) = µ1) = pnµ0

pnµ1

1 n! (nµ0)!(nµ1)!

Stirling : ln (P (ρe

n(0) = µ0, ρe n(1) = µ1)) = nµ0 ln p0 + nµ1 ln p1 + n ln n −

n − nµ0 ln nµ0 + nµ0 − nµ1 ln nµ1 + nµ1 = −n

• µ0 ln µ0

p0 + µ1 ln µ1 p1

• Case of general space states : I2(µ) ≡
• S(µ/ρ) if

µ ≪ ρ ∞ otherwise

• Raphaël Chétrite

(cnrs) Grandes déviation 30/05/2011 7 / 16

IID paradigm

Level 1 : Cramer theorem for independents bits

2 states 0 and 1, P(0) = p0 and P(1) = p1 = 1 − p0 Sequence of n bits : [x] = 011100100.... . Time average : X e

n = Pi=n

i=1 xi

n

. Typical behavior : X e

n → p1 = 1 − p0

Untypical behavior P (X e

n = µ) ≍?.

Level 2 : Sanov theorem for independents bits

Empirical density (”type” in statistics) : ρe

n(0) = ♯0 in [x] n

and ρe

n(1) = ♯1 in [x] n

Typical behavior : ρe

n(0) → p0 and ρe n(1) → p1.

Untypical behavior P (ρe

n(0) = µ0, ρe n(1) = µ1) = pnµ0

pnµ1

1 n! (nµ0)!(nµ1)!

Stirling : ln (P (ρe

n(0) = µ0, ρe n(1) = µ1)) = nµ0 ln p0 + nµ1 ln p1 + n ln n −

n − nµ0 ln nµ0 + nµ0 − nµ1 ln nµ1 + nµ1 = −n

• µ0 ln µ0

p0 + µ1 ln µ1 p1

• Case of general space states : I2(µ) ≡
• S(µ/ρ) if

µ ≪ ρ ∞ otherwise

• Raphaël Chétrite

(cnrs) Grandes déviation 30/05/2011 7 / 16

IID paradigm

Level 1 : Cramer theorem for independents bits

2 states 0 and 1, P(0) = p0 and P(1) = p1 = 1 − p0 Sequence of n bits : [x] = 011100100.... . Time average : X e

n = Pi=n

i=1 xi

n

. Typical behavior : X e

n → p1 = 1 − p0

Untypical behavior P (X e

n = µ) ≍?.

Level 2 : Sanov theorem for independents bits

Empirical density (”type” in statistics) : ρe

n(0) = ♯0 in [x] n

and ρe

n(1) = ♯1 in [x] n

Typical behavior : ρe

n(0) → p0 and ρe n(1) → p1.

Untypical behavior P (ρe

n(0) = µ0, ρe n(1) = µ1) = pnµ0

pnµ1

1 n! (nµ0)!(nµ1)!

Stirling : ln (P (ρe

n(0) = µ0, ρe n(1) = µ1)) = nµ0 ln p0 + nµ1 ln p1 + n ln n −

n − nµ0 ln nµ0 + nµ0 − nµ1 ln nµ1 + nµ1 = −n

• µ0 ln µ0

p0 + µ1 ln µ1 p1

• Case of general space states : I2(µ) ≡
• S(µ/ρ) if

µ ≪ ρ ∞ otherwise

• Raphaël Chétrite

(cnrs) Grandes déviation 30/05/2011 7 / 16

IID paradigm

Level 1 : Cramer theorem for independents bits

2 states 0 and 1, P(0) = p0 and P(1) = p1 = 1 − p0 Sequence of n bits : [x] = 011100100.... . Time average : X e

n = Pi=n

i=1 xi

n

. Typical behavior : X e

n → p1 = 1 − p0

Untypical behavior P (X e

n = µ) ≍?.

Level 2 : Sanov theorem for independents bits

Empirical density (”type” in statistics) : ρe

n(0) = ♯0 in [x] n

and ρe

n(1) = ♯1 in [x] n

Typical behavior : ρe

n(0) → p0 and ρe n(1) → p1.

Untypical behavior P (ρe

n(0) = µ0, ρe n(1) = µ1) = pnµ0

pnµ1

1 n! (nµ0)!(nµ1)!

Stirling : ln (P (ρe

n(0) = µ0, ρe n(1) = µ1)) = nµ0 ln p0 + nµ1 ln p1 + n ln n −

n − nµ0 ln nµ0 + nµ0 − nµ1 ln nµ1 + nµ1 = −n

• µ0 ln µ0

p0 + µ1 ln µ1 p1

• Case of general space states : I2(µ) ≡
• S(µ/ρ) if

µ ≪ ρ ∞ otherwise

• Raphaël Chétrite

(cnrs) Grandes déviation 30/05/2011 7 / 16

IID paradigm

Level 1 : Cramer theorem for independents bits

2 states 0 and 1, P(0) = p0 and P(1) = p1 = 1 − p0 Sequence of n bits : [x] = 011100100.... . Time average : X e

n = Pi=n

i=1 xi

n

. Typical behavior : X e

n → p1 = 1 − p0

Untypical behavior P (X e

n = µ) ≍?.

Level 2 : Sanov theorem for independents bits

Empirical density (”type” in statistics) : ρe

n(0) = ♯0 in [x] n

and ρe

n(1) = ♯1 in [x] n

Typical behavior : ρe

n(0) → p0 and ρe n(1) → p1.

Untypical behavior P (ρe

n(0) = µ0, ρe n(1) = µ1) = pnµ0

pnµ1

1 n! (nµ0)!(nµ1)!

Stirling : ln (P (ρe

n(0) = µ0, ρe n(1) = µ1)) = nµ0 ln p0 + nµ1 ln p1 + n ln n −

n − nµ0 ln nµ0 + nµ0 − nµ1 ln nµ1 + nµ1 = −n

• µ0 ln µ0

p0 + µ1 ln µ1 p1

• Case of general space states : I2(µ) ≡
• S(µ/ρ) if

µ ≪ ρ ∞ otherwise

• Raphaël Chétrite

(cnrs) Grandes déviation 30/05/2011 7 / 16

IID paradigm

Level 1 : Cramer theorem for independents bits

2 states 0 and 1, P(0) = p0 and P(1) = p1 = 1 − p0 Sequence of n bits : [x] = 011100100.... . Time average : X e

n = Pi=n

i=1 xi

n

. Typical behavior : X e

n → p1 = 1 − p0

Untypical behavior P (X e

n = µ) ≍?.

Level 2 : Sanov theorem for independents bits

Empirical density (”type” in statistics) : ρe

n(0) = ♯0 in [x] n

and ρe

n(1) = ♯1 in [x] n

Typical behavior : ρe

n(0) → p0 and ρe n(1) → p1.

Untypical behavior P (ρe

n(0) = µ0, ρe n(1) = µ1) = pnµ0

pnµ1

1 n! (nµ0)!(nµ1)!

Stirling : ln (P (ρe

n(0) = µ0, ρe n(1) = µ1)) = nµ0 ln p0 + nµ1 ln p1 + n ln n −

n − nµ0 ln nµ0 + nµ0 − nµ1 ln nµ1 + nµ1 = −n

• µ0 ln µ0

p0 + µ1 ln µ1 p1

• Case of general space states : I2(µ) ≡
• S(µ/ρ) if

µ ≪ ρ ∞ otherwise

• Raphaël Chétrite

(cnrs) Grandes déviation 30/05/2011 7 / 16

IID paradigm

Level 1 : Cramer theorem for independents bits

2 states 0 and 1, P(0) = p0 and P(1) = p1 = 1 − p0 Sequence of n bits : [x] = 011100100.... . Time average : X e

n = Pi=n

i=1 xi

n

. Typical behavior : X e

n → p1 = 1 − p0

Untypical behavior P (X e

n = µ) ≍?.

Level 2 : Sanov theorem for independents bits

Empirical density (”type” in statistics) : ρe

n(0) = ♯0 in [x] n

and ρe

n(1) = ♯1 in [x] n

Typical behavior : ρe

n(0) → p0 and ρe n(1) → p1.

Untypical behavior P (ρe

n(0) = µ0, ρe n(1) = µ1) = pnµ0

pnµ1

1 n! (nµ0)!(nµ1)!

Stirling : ln (P (ρe

n(0) = µ0, ρe n(1) = µ1)) = nµ0 ln p0 + nµ1 ln p1 + n ln n −

n − nµ0 ln nµ0 + nµ0 − nµ1 ln nµ1 + nµ1 = −n

• µ0 ln µ0

p0 + µ1 ln µ1 p1

• Case of general space states : I2(µ) ≡
• S(µ/ρ) if

µ ≪ ρ ∞ otherwise

• Raphaël Chétrite

(cnrs) Grandes déviation 30/05/2011 7 / 16

IID paradigm

Level 1 : Cramer theorem for independents bits

2 states 0 and 1, P(0) = p0 and P(1) = p1 = 1 − p0 Sequence of n bits : [x] = 011100100.... . Time average : X e

n = Pi=n

i=1 xi

n

. Typical behavior : X e

n → p1 = 1 − p0

Untypical behavior P (X e

n = µ) ≍?.

Level 2 : Sanov theorem for independents bits

Empirical density (”type” in statistics) : ρe

n(0) = ♯0 in [x] n

and ρe

n(1) = ♯1 in [x] n

Typical behavior : ρe

n(0) → p0 and ρe n(1) → p1.

Untypical behavior P (ρe

n(0) = µ0, ρe n(1) = µ1) = pnµ0

pnµ1

1 n! (nµ0)!(nµ1)!

Stirling : ln (P (ρe

n(0) = µ0, ρe n(1) = µ1)) = nµ0 ln p0 + nµ1 ln p1 + n ln n −

n − nµ0 ln nµ0 + nµ0 − nµ1 ln nµ1 + nµ1 = −n

• µ0 ln µ0

p0 + µ1 ln µ1 p1

• Case of general space states : I2(µ) ≡
• S(µ/ρ) if

µ ≪ ρ ∞ otherwise

• Raphaël Chétrite

(cnrs) Grandes déviation 30/05/2011 7 / 16

IID paradigm

Level 1 : Cramer theorem for independents bits

2 states 0 and 1, P(0) = p0 and P(1) = p1 = 1 − p0 Sequence of n bits : [x] = 011100100.... . Time average : X e

n = Pi=n

i=1 xi

n

. Typical behavior : X e

n → p1 = 1 − p0

Untypical behavior P (X e

n = µ) ≍?.

Level 2 : Sanov theorem for independents bits

Empirical density (”type” in statistics) : ρe

n(0) = ♯0 in [x] n

and ρe

n(1) = ♯1 in [x] n

Typical behavior : ρe

n(0) → p0 and ρe n(1) → p1.

Untypical behavior P (ρe

n(0) = µ0, ρe n(1) = µ1) = pnµ0

pnµ1

1 n! (nµ0)!(nµ1)!

Stirling : ln (P (ρe

n(0) = µ0, ρe n(1) = µ1)) = nµ0 ln p0 + nµ1 ln p1 + n ln n −

n − nµ0 ln nµ0 + nµ0 − nµ1 ln nµ1 + nµ1 = −n

• µ0 ln µ0

p0 + µ1 ln µ1 p1

• Case of general space states : I2(µ) ≡
• S(µ/ρ) if

µ ≪ ρ ∞ otherwise

• Raphaël Chétrite

(cnrs) Grandes déviation 30/05/2011 7 / 16

Ergodic Markov processes : Donsker-Varadhan theory for Level 2 ( Sanov theorem )

Donsker, M.D. , Varadhan, S.R. : Asymptotic evaluation of certain Markov process expectations for large times, I. Comm. Pure Appl. Math. 28 (1975) Level 2 : empirical density ρe

1,T(x) = 1 T

R T

0 δ (Xt − x) dt

Typical behavior : ρe

1,T(x) → ρinv(x)

La théorie des probabilités n’est au fond que le bon sens réduit au calcul : elle fait apprécier avec exactitude, ce que les esprits justes sentent par une sorte d’instinct. Marquis Simon de Laplace, Essai philosophique sur les probabilités. ˙ δ ` ρe

1,T − ρ

´¸ ≍ exp (−TI2 [ρ]) with I2 [ρ] = inf[f ] `R dxρ(x) (exp (−f ) L exp f ) (x) ´ Case with discrete time (Markov chain) I2 [ρ] = inf[f ] `R dxρ(x) ln [(exp (−f ) M exp f ) (x)] ´ Impossible to have a explicit form in the general case, except for the reversible continuous time case. Maybe it exist a level more detailed such that the level 2 can be see as a contraction ? We can’t obtain the entropy functional as a contraction of the level 2....we must to go forward in a more detailed level, but what is entropy functional ?

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 8 / 16

Ergodic Markov processes : Donsker-Varadhan theory for Level 2 ( Sanov theorem )

Donsker, M.D. , Varadhan, S.R. : Asymptotic evaluation of certain Markov process expectations for large times, I. Comm. Pure Appl. Math. 28 (1975) Level 2 : empirical density ρe

1,T(x) = 1 T

R T

0 δ (Xt − x) dt

Typical behavior : ρe

1,T(x) → ρinv(x)

La théorie des probabilités n’est au fond que le bon sens réduit au calcul : elle fait apprécier avec exactitude, ce que les esprits justes sentent par une sorte d’instinct. Marquis Simon de Laplace, Essai philosophique sur les probabilités. ˙ δ ` ρe

1,T − ρ

´¸ ≍ exp (−TI2 [ρ]) with I2 [ρ] = inf[f ] `R dxρ(x) (exp (−f ) L exp f ) (x) ´ Case with discrete time (Markov chain) I2 [ρ] = inf[f ] `R dxρ(x) ln [(exp (−f ) M exp f ) (x)] ´ Impossible to have a explicit form in the general case, except for the reversible continuous time case. Maybe it exist a level more detailed such that the level 2 can be see as a contraction ? We can’t obtain the entropy functional as a contraction of the level 2....we must to go forward in a more detailed level, but what is entropy functional ?

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 8 / 16

Ergodic Markov processes : Donsker-Varadhan theory for Level 2 ( Sanov theorem )

Donsker, M.D. , Varadhan, S.R. : Asymptotic evaluation of certain Markov process expectations for large times, I. Comm. Pure Appl. Math. 28 (1975) Level 2 : empirical density ρe

1,T(x) = 1 T

R T

0 δ (Xt − x) dt

Typical behavior : ρe

1,T(x) → ρinv(x)

La théorie des probabilités n’est au fond que le bon sens réduit au calcul : elle fait apprécier avec exactitude, ce que les esprits justes sentent par une sorte d’instinct. Marquis Simon de Laplace, Essai philosophique sur les probabilités. ˙ δ ` ρe

1,T − ρ

´¸ ≍ exp (−TI2 [ρ]) with I2 [ρ] = inf[f ] `R dxρ(x) (exp (−f ) L exp f ) (x) ´ Case with discrete time (Markov chain) I2 [ρ] = inf[f ] `R dxρ(x) ln [(exp (−f ) M exp f ) (x)] ´ Impossible to have a explicit form in the general case, except for the reversible continuous time case. Maybe it exist a level more detailed such that the level 2 can be see as a contraction ? We can’t obtain the entropy functional as a contraction of the level 2....we must to go forward in a more detailed level, but what is entropy functional ?

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 8 / 16

Ergodic Markov processes : Donsker-Varadhan theory for Level 2 ( Sanov theorem )

Donsker, M.D. , Varadhan, S.R. : Asymptotic evaluation of certain Markov process expectations for large times, I. Comm. Pure Appl. Math. 28 (1975) Level 2 : empirical density ρe

1,T(x) = 1 T

R T

0 δ (Xt − x) dt

Typical behavior : ρe

1,T(x) → ρinv(x)

La théorie des probabilités n’est au fond que le bon sens réduit au calcul : elle fait apprécier avec exactitude, ce que les esprits justes sentent par une sorte d’instinct. Marquis Simon de Laplace, Essai philosophique sur les probabilités. ˙ δ ` ρe

1,T − ρ

´¸ ≍ exp (−TI2 [ρ]) with I2 [ρ] = inf[f ] `R dxρ(x) (exp (−f ) L exp f ) (x) ´ Case with discrete time (Markov chain) I2 [ρ] = inf[f ] `R dxρ(x) ln [(exp (−f ) M exp f ) (x)] ´ Impossible to have a explicit form in the general case, except for the reversible continuous time case. Maybe it exist a level more detailed such that the level 2 can be see as a contraction ? We can’t obtain the entropy functional as a contraction of the level 2....we must to go forward in a more detailed level, but what is entropy functional ?

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 8 / 16

Ergodic Markov processes : Donsker-Varadhan theory for Level 2 ( Sanov theorem )

Donsker, M.D. , Varadhan, S.R. : Asymptotic evaluation of certain Markov process expectations for large times, I. Comm. Pure Appl. Math. 28 (1975) Level 2 : empirical density ρe

1,T(x) = 1 T

R T

0 δ (Xt − x) dt

Typical behavior : ρe

1,T(x) → ρinv(x)

La théorie des probabilités n’est au fond que le bon sens réduit au calcul : elle fait apprécier avec exactitude, ce que les esprits justes sentent par une sorte d’instinct. Marquis Simon de Laplace, Essai philosophique sur les probabilités. ˙ δ ` ρe

1,T − ρ

´¸ ≍ exp (−TI2 [ρ]) with I2 [ρ] = inf[f ] `R dxρ(x) (exp (−f ) L exp f ) (x) ´ Case with discrete time (Markov chain) I2 [ρ] = inf[f ] `R dxρ(x) ln [(exp (−f ) M exp f ) (x)] ´ Impossible to have a explicit form in the general case, except for the reversible continuous time case. Maybe it exist a level more detailed such that the level 2 can be see as a contraction ? We can’t obtain the entropy functional as a contraction of the level 2....we must to go forward in a more detailed level, but what is entropy functional ?

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 8 / 16

Ergodic Markov processes : Donsker-Varadhan theory for Level 2 ( Sanov theorem )

Donsker, M.D. , Varadhan, S.R. : Asymptotic evaluation of certain Markov process expectations for large times, I. Comm. Pure Appl. Math. 28 (1975) Level 2 : empirical density ρe

1,T(x) = 1 T

R T

0 δ (Xt − x) dt

Typical behavior : ρe

1,T(x) → ρinv(x)

La théorie des probabilités n’est au fond que le bon sens réduit au calcul : elle fait apprécier avec exactitude, ce que les esprits justes sentent par une sorte d’instinct. Marquis Simon de Laplace, Essai philosophique sur les probabilités. ˙ δ ` ρe

1,T − ρ

´¸ ≍ exp (−TI2 [ρ]) with I2 [ρ] = inf[f ] `R dxρ(x) (exp (−f ) L exp f ) (x) ´ Case with discrete time (Markov chain) I2 [ρ] = inf[f ] `R dxρ(x) ln [(exp (−f ) M exp f ) (x)] ´ Impossible to have a explicit form in the general case, except for the reversible continuous time case. Maybe it exist a level more detailed such that the level 2 can be see as a contraction ? We can’t obtain the entropy functional as a contraction of the level 2....we must to go forward in a more detailed level, but what is entropy functional ?

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 8 / 16

Ergodic Markov processes : Donsker-Varadhan theory for Level 2 ( Sanov theorem )

Donsker, M.D. , Varadhan, S.R. : Asymptotic evaluation of certain Markov process expectations for large times, I. Comm. Pure Appl. Math. 28 (1975) Level 2 : empirical density ρe

1,T(x) = 1 T

R T

0 δ (Xt − x) dt

Typical behavior : ρe

1,T(x) → ρinv(x)

La théorie des probabilités n’est au fond que le bon sens réduit au calcul : elle fait apprécier avec exactitude, ce que les esprits justes sentent par une sorte d’instinct. Marquis Simon de Laplace, Essai philosophique sur les probabilités. ˙ δ ` ρe

1,T − ρ

´¸ ≍ exp (−TI2 [ρ]) with I2 [ρ] = inf[f ] `R dxρ(x) (exp (−f ) L exp f ) (x) ´ Case with discrete time (Markov chain) I2 [ρ] = inf[f ] `R dxρ(x) ln [(exp (−f ) M exp f ) (x)] ´ Impossible to have a explicit form in the general case, except for the reversible continuous time case. Maybe it exist a level more detailed such that the level 2 can be see as a contraction ? We can’t obtain the entropy functional as a contraction of the level 2....we must to go forward in a more detailed level, but what is entropy functional ?

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 8 / 16

Functional entropy creation

Reversed trajectory at time T : [x] ⇒ R[x] Action functional : WT

0 such that

exp ` −WT

0 [x]

´ ≡

R∗Pρr

0,[0,T][x]

Pρ0,[0,T][x]

Functional fluctuating entropy creation σT if ρr

0(x) = ρT(x) ≡

R dyρ0(y)PT

0 (y, x)

Fluctuating entropy production in the environment JT if ρr

0(x) = ρ0(x) = 1

P0,T(dx; dy; w)exp(−w) = P0,T(dy; dx; −w)

Link Fluctuation Relation-Martingale theory-FDT :

• R. Chetrite, S. Gupta : Two Refreshing Views of Fluctuation Theorems Through Kinematics

Elements and Exponential Martingale. J Stat Phys (2011)

Pure Jump process with transition rates W (x, y)

JT [x] = PNt

i=1 ln

“ W (xi−1,xi )

W (xi ,xi−1)

” = P

0≤s≤T,∆xs=0 ln

„

W (X −

s ,Xs)

W (Xs,X −

s )

«

Diffusion process with generator L = u.∇ + ∇.d

2.∇ JT [x] = R T ` 2b u(xt).d−1(xt) ◦ dxt ´

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 9 / 16

Functional entropy creation

Reversed trajectory at time T : [x] ⇒ R[x] Action functional : WT

0 such that

exp ` −WT

0 [x]

´ ≡

R∗Pρr

0,[0,T][x]

Pρ0,[0,T][x]

Functional fluctuating entropy creation σT if ρr

0(x) = ρT(x) ≡

R dyρ0(y)PT

0 (y, x)

Fluctuating entropy production in the environment JT if ρr

0(x) = ρ0(x) = 1

P0,T(dx; dy; w)exp(−w) = P0,T(dy; dx; −w)

Link Fluctuation Relation-Martingale theory-FDT :

• R. Chetrite, S. Gupta : Two Refreshing Views of Fluctuation Theorems Through Kinematics

Elements and Exponential Martingale. J Stat Phys (2011)

Pure Jump process with transition rates W (x, y)

JT [x] = PNt

i=1 ln

“ W (xi−1,xi )

W (xi ,xi−1)

” = P

0≤s≤T,∆xs=0 ln

„

W (X −

s ,Xs)

W (Xs,X −

s )

«

Diffusion process with generator L = u.∇ + ∇.d

2.∇ JT [x] = R T ` 2b u(xt).d−1(xt) ◦ dxt ´

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 9 / 16

Functional entropy creation

Reversed trajectory at time T : [x] ⇒ R[x] Action functional : WT

0 such that

exp ` −WT

0 [x]

´ ≡

R∗Pρr

0,[0,T][x]

Pρ0,[0,T][x]

Functional fluctuating entropy creation σT if ρr

0(x) = ρT(x) ≡

R dyρ0(y)PT

0 (y, x)

Fluctuating entropy production in the environment JT if ρr

0(x) = ρ0(x) = 1

P0,T(dx; dy; w)exp(−w) = P0,T(dy; dx; −w)

Link Fluctuation Relation-Martingale theory-FDT :

• R. Chetrite, S. Gupta : Two Refreshing Views of Fluctuation Theorems Through Kinematics

Elements and Exponential Martingale. J Stat Phys (2011)

Pure Jump process with transition rates W (x, y)

JT [x] = PNt

i=1 ln

“ W (xi−1,xi )

W (xi ,xi−1)

” = P

0≤s≤T,∆xs=0 ln

„

W (X −

s ,Xs)

W (Xs,X −

s )

«

Diffusion process with generator L = u.∇ + ∇.d

2.∇ JT [x] = R T ` 2b u(xt).d−1(xt) ◦ dxt ´

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 9 / 16

Functional entropy creation

Reversed trajectory at time T : [x] ⇒ R[x] Action functional : WT

0 such that

exp ` −WT

0 [x]

´ ≡

R∗Pρr

0,[0,T][x]

Pρ0,[0,T][x]

Functional fluctuating entropy creation σT if ρr

0(x) = ρT(x) ≡

R dyρ0(y)PT

0 (y, x)

Fluctuating entropy production in the environment JT if ρr

0(x) = ρ0(x) = 1

P0,T(dx; dy; w)exp(−w) = P0,T(dy; dx; −w)

Link Fluctuation Relation-Martingale theory-FDT :

• R. Chetrite, S. Gupta : Two Refreshing Views of Fluctuation Theorems Through Kinematics

Elements and Exponential Martingale. J Stat Phys (2011)

Pure Jump process with transition rates W (x, y)

JT [x] = PNt

i=1 ln

“ W (xi−1,xi )

W (xi ,xi−1)

” = P

0≤s≤T,∆xs=0 ln

„

W (X −

s ,Xs)

W (Xs,X −

s )

«

Diffusion process with generator L = u.∇ + ∇.d

2.∇ JT [x] = R T ` 2b u(xt).d−1(xt) ◦ dxt ´

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 9 / 16

Functional entropy creation

Reversed trajectory at time T : [x] ⇒ R[x] Action functional : WT

0 such that

exp ` −WT

0 [x]

´ ≡

R∗Pρr

0,[0,T][x]

Pρ0,[0,T][x]

Functional fluctuating entropy creation σT if ρr

0(x) = ρT(x) ≡

R dyρ0(y)PT

0 (y, x)

Fluctuating entropy production in the environment JT if ρr

0(x) = ρ0(x) = 1

P0,T(dx; dy; w)exp(−w) = P0,T(dy; dx; −w)

Link Fluctuation Relation-Martingale theory-FDT :

• R. Chetrite, S. Gupta : Two Refreshing Views of Fluctuation Theorems Through Kinematics

Elements and Exponential Martingale. J Stat Phys (2011)

Pure Jump process with transition rates W (x, y)

JT [x] = PNt

i=1 ln

“ W (xi−1,xi )

W (xi ,xi−1)

” = P

0≤s≤T,∆xs=0 ln

„

W (X −

s ,Xs)

W (Xs,X −

s )

«

Diffusion process with generator L = u.∇ + ∇.d

2.∇ JT [x] = R T ` 2b u(xt).d−1(xt) ◦ dxt ´

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 9 / 16

Functional entropy creation

Reversed trajectory at time T : [x] ⇒ R[x] Action functional : WT

0 such that

exp ` −WT

0 [x]

´ ≡

R∗Pρr

0,[0,T][x]

Pρ0,[0,T][x]

Functional fluctuating entropy creation σT if ρr

0(x) = ρT(x) ≡

R dyρ0(y)PT

0 (y, x)

Fluctuating entropy production in the environment JT if ρr

0(x) = ρ0(x) = 1

P0,T(dx; dy; w)exp(−w) = P0,T(dy; dx; −w)

Link Fluctuation Relation-Martingale theory-FDT :

• R. Chetrite, S. Gupta : Two Refreshing Views of Fluctuation Theorems Through Kinematics

Elements and Exponential Martingale. J Stat Phys (2011)

Pure Jump process with transition rates W (x, y)

JT [x] = PNt

i=1 ln

“ W (xi−1,xi )

W (xi ,xi−1)

” = P

0≤s≤T,∆xs=0 ln

„

W (X −

s ,Xs)

W (Xs,X −

s )

«

Diffusion process with generator L = u.∇ + ∇.d

2.∇ JT [x] = R T ` 2b u(xt).d−1(xt) ◦ dxt ´

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 9 / 16

Functional entropy creation

Reversed trajectory at time T : [x] ⇒ R[x] Action functional : WT

0 such that

exp ` −WT

0 [x]

´ ≡

R∗Pρr

0,[0,T][x]

Pρ0,[0,T][x]

Functional fluctuating entropy creation σT if ρr

0(x) = ρT(x) ≡

R dyρ0(y)PT

0 (y, x)

Fluctuating entropy production in the environment JT if ρr

0(x) = ρ0(x) = 1

P0,T(dx; dy; w)exp(−w) = P0,T(dy; dx; −w)

Link Fluctuation Relation-Martingale theory-FDT :

• R. Chetrite, S. Gupta : Two Refreshing Views of Fluctuation Theorems Through Kinematics

Elements and Exponential Martingale. J Stat Phys (2011)

Pure Jump process with transition rates W (x, y)

JT [x] = PNt

i=1 ln

“ W (xi−1,xi )

W (xi ,xi−1)

” = P

0≤s≤T,∆xs=0 ln

„

W (X −

s ,Xs)

W (Xs,X −

s )

«

Diffusion process with generator L = u.∇ + ∇.d

2.∇ JT [x] = R T ` 2b u(xt).d−1(xt) ◦ dxt ´

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 9 / 16

Ergodic Markov : the Level 2,5

Ergodic Markov Chains : The Large deviation becomes explicit for the pair empirical measure : ρe

2,N(x, y) = 1 N

N

i=1 δ (Xi − x) δ (Xi+1 − y)

Ergodic pure jump process : LD explicit for joint empirical density and ”empirical jump distribution” : ρe

2,T(x, y) ≡ 1 T

• 0≤s≤T,∆xs=0 δ (X −

s − x) δ (X + s − y) then

• δ
• ρe

1,T − ρ

• δ
• ρe

2,T − C

• ≍ exp (−TI2,5 [ρ, C]) with

I2,5 [ρ, C] = ( R dxdy “ −C(x, y) + ρ(x)W (x, y) + C(x, y) ln

C(x,y) ρ(x)W (x,y)

” si R dxC(x, y) = R dxC(y, x) ∞ sinon )

Baldi-Piccioni. Stat Prob Lett 1999 Ergodic diffusion process : ...Maes, Wynants, Chertkov, Chernyak je

T(x) = 1 T

T

0 δ (Xt − x) ◦ dXt

I2,5 [ρ, j] =

• 1

2

• dx (ρd)−1 (j − jρ) (j − jρ)

si ∇.j = 0 ∞ sinon

• By contraction, the entropy creation ”possess” a (formal) large deviation

regime....but it is almost never explicit because it is contraction of the explicit expression. All information weaker that the explicit rate function is welcome : for example the symmetry of this function.

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 10 / 16

Ergodic Markov : the Level 2,5

Ergodic Markov Chains : The Large deviation becomes explicit for the pair empirical measure : ρe

2,N(x, y) = 1 N

N

i=1 δ (Xi − x) δ (Xi+1 − y)

Ergodic pure jump process : LD explicit for joint empirical density and ”empirical jump distribution” : ρe

2,T(x, y) ≡ 1 T

• 0≤s≤T,∆xs=0 δ (X −

s − x) δ (X + s − y) then

• δ
• ρe

1,T − ρ

• δ
• ρe

2,T − C

• ≍ exp (−TI2,5 [ρ, C]) with

I2,5 [ρ, C] = ( R dxdy “ −C(x, y) + ρ(x)W (x, y) + C(x, y) ln

C(x,y) ρ(x)W (x,y)

” si R dxC(x, y) = R dxC(y, x) ∞ sinon )

Baldi-Piccioni. Stat Prob Lett 1999 Ergodic diffusion process : ...Maes, Wynants, Chertkov, Chernyak je

T(x) = 1 T

T

0 δ (Xt − x) ◦ dXt

I2,5 [ρ, j] =

• 1

2

• dx (ρd)−1 (j − jρ) (j − jρ)

si ∇.j = 0 ∞ sinon

• By contraction, the entropy creation ”possess” a (formal) large deviation

regime....but it is almost never explicit because it is contraction of the explicit expression. All information weaker that the explicit rate function is welcome : for example the symmetry of this function.

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 10 / 16

Ergodic Markov : the Level 2,5

Ergodic Markov Chains : The Large deviation becomes explicit for the pair empirical measure : ρe

2,N(x, y) = 1 N

N

i=1 δ (Xi − x) δ (Xi+1 − y)

Ergodic pure jump process : LD explicit for joint empirical density and ”empirical jump distribution” : ρe

2,T(x, y) ≡ 1 T

• 0≤s≤T,∆xs=0 δ (X −

s − x) δ (X + s − y) then

• δ
• ρe

1,T − ρ

• δ
• ρe

2,T − C

• ≍ exp (−TI2,5 [ρ, C]) with

I2,5 [ρ, C] = ( R dxdy “ −C(x, y) + ρ(x)W (x, y) + C(x, y) ln

C(x,y) ρ(x)W (x,y)

” si R dxC(x, y) = R dxC(y, x) ∞ sinon )

Baldi-Piccioni. Stat Prob Lett 1999 Ergodic diffusion process : ...Maes, Wynants, Chertkov, Chernyak je

T(x) = 1 T

T

0 δ (Xt − x) ◦ dXt

I2,5 [ρ, j] =

• 1

2

• dx (ρd)−1 (j − jρ) (j − jρ)

si ∇.j = 0 ∞ sinon

• By contraction, the entropy creation ”possess” a (formal) large deviation

regime....but it is almost never explicit because it is contraction of the explicit expression. All information weaker that the explicit rate function is welcome : for example the symmetry of this function.

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 10 / 16

Ergodic Markov : the Level 2,5

Ergodic Markov Chains : The Large deviation becomes explicit for the pair empirical measure : ρe

2,N(x, y) = 1 N

N

i=1 δ (Xi − x) δ (Xi+1 − y)

Ergodic pure jump process : LD explicit for joint empirical density and ”empirical jump distribution” : ρe

2,T(x, y) ≡ 1 T

• 0≤s≤T,∆xs=0 δ (X −

s − x) δ (X + s − y) then

• δ
• ρe

1,T − ρ

• δ
• ρe

2,T − C

• ≍ exp (−TI2,5 [ρ, C]) with

I2,5 [ρ, C] = ( R dxdy “ −C(x, y) + ρ(x)W (x, y) + C(x, y) ln

C(x,y) ρ(x)W (x,y)

” si R dxC(x, y) = R dxC(y, x) ∞ sinon )

Baldi-Piccioni. Stat Prob Lett 1999 Ergodic diffusion process : ...Maes, Wynants, Chertkov, Chernyak je

T(x) = 1 T

T

0 δ (Xt − x) ◦ dXt

I2,5 [ρ, j] =

• 1

2

• dx (ρd)−1 (j − jρ) (j − jρ)

si ∇.j = 0 ∞ sinon

• By contraction, the entropy creation ”possess” a (formal) large deviation

regime....but it is almost never explicit because it is contraction of the explicit expression. All information weaker that the explicit rate function is welcome : for example the symmetry of this function.

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 10 / 16

Ergodic Markov : the Level 2,5

Ergodic Markov Chains : The Large deviation becomes explicit for the pair empirical measure : ρe

2,N(x, y) = 1 N

N

i=1 δ (Xi − x) δ (Xi+1 − y)

Ergodic pure jump process : LD explicit for joint empirical density and ”empirical jump distribution” : ρe

2,T(x, y) ≡ 1 T

• 0≤s≤T,∆xs=0 δ (X −

s − x) δ (X + s − y) then

• δ
• ρe

1,T − ρ

• δ
• ρe

2,T − C

• ≍ exp (−TI2,5 [ρ, C]) with

I2,5 [ρ, C] = ( R dxdy “ −C(x, y) + ρ(x)W (x, y) + C(x, y) ln

C(x,y) ρ(x)W (x,y)

” si R dxC(x, y) = R dxC(y, x) ∞ sinon )

Baldi-Piccioni. Stat Prob Lett 1999 Ergodic diffusion process : ...Maes, Wynants, Chertkov, Chernyak je

T(x) = 1 T

T

0 δ (Xt − x) ◦ dXt

I2,5 [ρ, j] =

• 1

2

• dx (ρd)−1 (j − jρ) (j − jρ)

si ∇.j = 0 ∞ sinon

• By contraction, the entropy creation ”possess” a (formal) large deviation

regime....but it is almost never explicit because it is contraction of the explicit expression. All information weaker that the explicit rate function is welcome : for example the symmetry of this function.

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 10 / 16

Ergodic Markov : the Level 2,5

Ergodic Markov Chains : The Large deviation becomes explicit for the pair empirical measure : ρe

2,N(x, y) = 1 N

N

i=1 δ (Xi − x) δ (Xi+1 − y)

Ergodic pure jump process : LD explicit for joint empirical density and ”empirical jump distribution” : ρe

2,T(x, y) ≡ 1 T

• 0≤s≤T,∆xs=0 δ (X −

s − x) δ (X + s − y) then

• δ
• ρe

1,T − ρ

• δ
• ρe

2,T − C

• ≍ exp (−TI2,5 [ρ, C]) with

I2,5 [ρ, C] = ( R dxdy “ −C(x, y) + ρ(x)W (x, y) + C(x, y) ln

C(x,y) ρ(x)W (x,y)

” si R dxC(x, y) = R dxC(y, x) ∞ sinon )

Baldi-Piccioni. Stat Prob Lett 1999 Ergodic diffusion process : ...Maes, Wynants, Chertkov, Chernyak je

T(x) = 1 T

T

0 δ (Xt − x) ◦ dXt

I2,5 [ρ, j] =

• 1

2

• dx (ρd)−1 (j − jρ) (j − jρ)

si ∇.j = 0 ∞ sinon

• By contraction, the entropy creation ”possess” a (formal) large deviation

regime....but it is almost never explicit because it is contraction of the explicit expression. All information weaker that the explicit rate function is welcome : for example the symmetry of this function.

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 10 / 16

Proof of the Level 2.5 ”a la” theoretical physicist by tilting method (in the language of jump process)

”Les preuves fatiguent la verité.” Goerges Braque. Le jour et la nuit. We perturb the forward jump process by replacing the jump process by another one W ′

ρ,C such that for this process the invariant density is ρ and the

invariant density of jump is C(x, y). We note Pρ,C its trajectorial measure. exp (−TI [ρ, C]) ≍

• δ
• ρe

1,T

• − [ρ]
• δ
• ρe

2,T

• − [C]
• =
• Pρ0,[0,T][dx]δ
• ρe

1,T

• − [ρ]
• δ
• ρe

2,T

• − [C]
• =

Pρ0,[0,T]

Pρ,C

µ0,T [x]Pρ,C

µ0,T[dx]δ

• ρe

1,T

• − [ρ]
• δ
• ρe

2,T

• − [C]
• =

Pµ0,T

Pρ,C

µ0,T

• ρe

T, ρe 2,T

• Pρ,C

µ0,T[dx]δ

• ρe

1,T

• − [ρ]
• δ
• ρe

2,T

• − [C]
• =

Pµ0,T Pρ,C

µ0,T [ρ, C]

• Pρ,C

µ0,T[dx]δ

• ρe

1,T

• − [ρ]
• δ
• ρe

2,T

• − [C]
• ≍ Pµ0,T

Pρ,C

µ0,T [ρ, C] .1 ≍

Pµ0,T Pρ,C

µ0,T [ρ, C] Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 11 / 16

2.5 < Level < 3 : go Beyond the current

Study of large deviation of observable with more that two consequential

• points. Can you learn something new on non equilibrium systems ?

For discrete time process, the Level 2,5 (resp 3) can be easily derived by applying the Sanov theorem (Level 2 ) for the coarse graine Markov chain Yn = (Xn, Xn+1) (resp Yn = (Xn, Xn+1, ..., Xn+m). Project : Can we find a similar enlargement for the continuous time process ? idea : Process with fictitious fermions. Stochastic flow, process on differential form (or with fermionnic degree of freedom.)

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 12 / 16

2.5 < Level < 3 : go Beyond the current

Study of large deviation of observable with more that two consequential

• points. Can you learn something new on non equilibrium systems ?

For discrete time process, the Level 2,5 (resp 3) can be easily derived by applying the Sanov theorem (Level 2 ) for the coarse graine Markov chain Yn = (Xn, Xn+1) (resp Yn = (Xn, Xn+1, ..., Xn+m). Project : Can we find a similar enlargement for the continuous time process ? idea : Process with fictitious fermions. Stochastic flow, process on differential form (or with fermionnic degree of freedom.)

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 12 / 16

Gallavotti-Cohen-Evans-Morriss (GC) symmetry

”Symmetry is like a disease. Or, perhaps more accurately, it is a disease at least in my case. I seem to have a bad case of it...I must always have had a tendency to

• symmetry. Moreover, far from being painful, these sever symptoms afford much

pleasure.” Joe Rosen. Symmetry Discovered I(a) = I(−a) − Ea ⇔ Λ(−E − s) = Λ(s) Where does it come from ? For what type of observables ? Gallavotti-Cohen relations (1995) : true with E = 1 for the fluctuating entropy production.

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 13 / 16

Gallavotti-Cohen-Evans-Morriss (GC) symmetry

”Symmetry is like a disease. Or, perhaps more accurately, it is a disease at least in my case. I seem to have a bad case of it...I must always have had a tendency to

• symmetry. Moreover, far from being painful, these sever symptoms afford much

pleasure.” Joe Rosen. Symmetry Discovered I(a) = I(−a) − Ea ⇔ Λ(−E − s) = Λ(s) Where does it come from ? For what type of observables ? Gallavotti-Cohen relations (1995) : true with E = 1 for the fluctuating entropy production.

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 13 / 16

Gallavotti-Cohen-Evans-Morriss (GC) symmetry

”Symmetry is like a disease. Or, perhaps more accurately, it is a disease at least in my case. I seem to have a bad case of it...I must always have had a tendency to

• symmetry. Moreover, far from being painful, these sever symptoms afford much

pleasure.” Joe Rosen. Symmetry Discovered I(a) = I(−a) − Ea ⇔ Λ(−E − s) = Λ(s) Where does it come from ? For what type of observables ? Gallavotti-Cohen relations (1995) : true with E = 1 for the fluctuating entropy production.

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 13 / 16

Gallavotti-Cohen-Evans-Morriss (GC) symmetry

”Symmetry is like a disease. Or, perhaps more accurately, it is a disease at least in my case. I seem to have a bad case of it...I must always have had a tendency to

• symmetry. Moreover, far from being painful, these sever symptoms afford much

pleasure.” Joe Rosen. Symmetry Discovered I(a) = I(−a) − Ea ⇔ Λ(−E − s) = Λ(s) Where does it come from ? For what type of observables ? Gallavotti-Cohen relations (1995) : true with E = 1 for the fluctuating entropy production.

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 13 / 16

Gallavotti-Cohen-Evans-Morriss (GC) symmetry

”Symmetry is like a disease. Or, perhaps more accurately, it is a disease at least in my case. I seem to have a bad case of it...I must always have had a tendency to

• symmetry. Moreover, far from being painful, these sever symptoms afford much

pleasure.” Joe Rosen. Symmetry Discovered I(a) = I(−a) − Ea ⇔ Λ(−E − s) = Λ(s) Where does it come from ? For what type of observables ? Gallavotti-Cohen relations (1995) : true with E = 1 for the fluctuating entropy production.

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 13 / 16

Gallavotti-Cohen : Spectral characterization (for Pure Jump)

Be

T ≡ 1 T

P

0≤s≤T,∆xs=0 B(X − s , Xs) →

R dxdyB(x, y)ρinv(x)W (x, y) ”Feymann-Kac” formula : Ex (δ(xT − y) exp(sTBe

T)) = exp(T(W exp(sB) − λId))(x, y) then

Λ(s) = inf Spectrum(Hs) with the deformed generator Hs = −W exp(sB) + λId and the convention (W exp(sB))(x, y) = W (x, y) exp(sB(x, y)) We have

GC ≡ {inf Spectrum(Hs) = inf Spectrum(H−E−s)} ⇐ CS1 ≡ {Spectrum(Hs) = Spectrum(H−E−s)} ⇐ CS2 ≡  Hs(x, y) = f −1(x)H−E−s(y, x)f (y) ≡ „ W (x, y) = f −1(x)W (y, x) exp(EB(x, y))f (y)(Derrida :MDB B(x, y) = −B(y, x)

Entropy production is obtained for B(x, y) = ln “

W (x,y) W (y,x)

” , then Hs(x, y) = −W (x, y)1+sW (y, x)−s + λ(x)δ(x − y) ⇒ CS2 verified : Hs = HT

−1−s

In fact, CS2 is for an (antisymmetric) observable which is proportional to the entropy production in the large time limit : ∆Senv

T [x] = − ln(f (x0))+ln(f (xT )) T

− EBe

T[x]

Example of an observable for which CS2 is not verified, but CS1 or GC is ?

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 14 / 16

Gallavotti-Cohen : Spectral characterization (for Pure Jump)

Be

T ≡ 1 T

P

0≤s≤T,∆xs=0 B(X − s , Xs) →

R dxdyB(x, y)ρinv(x)W (x, y) ”Feymann-Kac” formula : Ex (δ(xT − y) exp(sTBe

T)) = exp(T(W exp(sB) − λId))(x, y) then

Λ(s) = inf Spectrum(Hs) with the deformed generator Hs = −W exp(sB) + λId and the convention (W exp(sB))(x, y) = W (x, y) exp(sB(x, y)) We have

GC ≡ {inf Spectrum(Hs) = inf Spectrum(H−E−s)} ⇐ CS1 ≡ {Spectrum(Hs) = Spectrum(H−E−s)} ⇐ CS2 ≡  Hs(x, y) = f −1(x)H−E−s(y, x)f (y) ≡ „ W (x, y) = f −1(x)W (y, x) exp(EB(x, y))f (y)(Derrida :MDB B(x, y) = −B(y, x)

Entropy production is obtained for B(x, y) = ln “

W (x,y) W (y,x)

” , then Hs(x, y) = −W (x, y)1+sW (y, x)−s + λ(x)δ(x − y) ⇒ CS2 verified : Hs = HT

−1−s

In fact, CS2 is for an (antisymmetric) observable which is proportional to the entropy production in the large time limit : ∆Senv

T [x] = − ln(f (x0))+ln(f (xT )) T

− EBe

T[x]

Example of an observable for which CS2 is not verified, but CS1 or GC is ?

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 14 / 16

Gallavotti-Cohen : Spectral characterization (for Pure Jump)

Be

T ≡ 1 T

P

0≤s≤T,∆xs=0 B(X − s , Xs) →

R dxdyB(x, y)ρinv(x)W (x, y) ”Feymann-Kac” formula : Ex (δ(xT − y) exp(sTBe

T)) = exp(T(W exp(sB) − λId))(x, y) then

Λ(s) = inf Spectrum(Hs) with the deformed generator Hs = −W exp(sB) + λId and the convention (W exp(sB))(x, y) = W (x, y) exp(sB(x, y)) We have

GC ≡ {inf Spectrum(Hs) = inf Spectrum(H−E−s)} ⇐ CS1 ≡ {Spectrum(Hs) = Spectrum(H−E−s)} ⇐ CS2 ≡  Hs(x, y) = f −1(x)H−E−s(y, x)f (y) ≡ „ W (x, y) = f −1(x)W (y, x) exp(EB(x, y))f (y)(Derrida :MDB B(x, y) = −B(y, x)

Entropy production is obtained for B(x, y) = ln “

W (x,y) W (y,x)

” , then Hs(x, y) = −W (x, y)1+sW (y, x)−s + λ(x)δ(x − y) ⇒ CS2 verified : Hs = HT

−1−s

In fact, CS2 is for an (antisymmetric) observable which is proportional to the entropy production in the large time limit : ∆Senv

T [x] = − ln(f (x0))+ln(f (xT )) T

− EBe

T[x]

Example of an observable for which CS2 is not verified, but CS1 or GC is ?

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 14 / 16

Gallavotti-Cohen : Spectral characterization (for Pure Jump)

Be

T ≡ 1 T

P

0≤s≤T,∆xs=0 B(X − s , Xs) →

R dxdyB(x, y)ρinv(x)W (x, y) ”Feymann-Kac” formula : Ex (δ(xT − y) exp(sTBe

T)) = exp(T(W exp(sB) − λId))(x, y) then

Λ(s) = inf Spectrum(Hs) with the deformed generator Hs = −W exp(sB) + λId and the convention (W exp(sB))(x, y) = W (x, y) exp(sB(x, y)) We have

GC ≡ {inf Spectrum(Hs) = inf Spectrum(H−E−s)} ⇐ CS1 ≡ {Spectrum(Hs) = Spectrum(H−E−s)} ⇐ CS2 ≡  Hs(x, y) = f −1(x)H−E−s(y, x)f (y) ≡ „ W (x, y) = f −1(x)W (y, x) exp(EB(x, y))f (y)(Derrida :MDB B(x, y) = −B(y, x)

Entropy production is obtained for B(x, y) = ln “

W (x,y) W (y,x)

” , then Hs(x, y) = −W (x, y)1+sW (y, x)−s + λ(x)δ(x − y) ⇒ CS2 verified : Hs = HT

−1−s

In fact, CS2 is for an (antisymmetric) observable which is proportional to the entropy production in the large time limit : ∆Senv

T [x] = − ln(f (x0))+ln(f (xT )) T

− EBe

T[x]

Example of an observable for which CS2 is not verified, but CS1 or GC is ?

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 14 / 16

Gallavotti-Cohen : Spectral characterization (for Pure Jump)

Be

T ≡ 1 T

P

0≤s≤T,∆xs=0 B(X − s , Xs) →

R dxdyB(x, y)ρinv(x)W (x, y) ”Feymann-Kac” formula : Ex (δ(xT − y) exp(sTBe

T)) = exp(T(W exp(sB) − λId))(x, y) then

Λ(s) = inf Spectrum(Hs) with the deformed generator Hs = −W exp(sB) + λId and the convention (W exp(sB))(x, y) = W (x, y) exp(sB(x, y)) We have

GC ≡ {inf Spectrum(Hs) = inf Spectrum(H−E−s)} ⇐ CS1 ≡ {Spectrum(Hs) = Spectrum(H−E−s)} ⇐ CS2 ≡  Hs(x, y) = f −1(x)H−E−s(y, x)f (y) ≡ „ W (x, y) = f −1(x)W (y, x) exp(EB(x, y))f (y)(Derrida :MDB B(x, y) = −B(y, x)

Entropy production is obtained for B(x, y) = ln “

W (x,y) W (y,x)

” , then Hs(x, y) = −W (x, y)1+sW (y, x)−s + λ(x)δ(x − y) ⇒ CS2 verified : Hs = HT

−1−s

In fact, CS2 is for an (antisymmetric) observable which is proportional to the entropy production in the large time limit : ∆Senv

T [x] = − ln(f (x0))+ln(f (xT )) T

− EBe

T[x]

Example of an observable for which CS2 is not verified, but CS1 or GC is ?

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 14 / 16

Gallavotti-Cohen : Spectral characterization (for Pure Jump)

Be

T ≡ 1 T

P

0≤s≤T,∆xs=0 B(X − s , Xs) →

R dxdyB(x, y)ρinv(x)W (x, y) ”Feymann-Kac” formula : Ex (δ(xT − y) exp(sTBe

T)) = exp(T(W exp(sB) − λId))(x, y) then

Λ(s) = inf Spectrum(Hs) with the deformed generator Hs = −W exp(sB) + λId and the convention (W exp(sB))(x, y) = W (x, y) exp(sB(x, y)) We have

GC ≡ {inf Spectrum(Hs) = inf Spectrum(H−E−s)} ⇐ CS1 ≡ {Spectrum(Hs) = Spectrum(H−E−s)} ⇐ CS2 ≡  Hs(x, y) = f −1(x)H−E−s(y, x)f (y) ≡ „ W (x, y) = f −1(x)W (y, x) exp(EB(x, y))f (y)(Derrida :MDB B(x, y) = −B(y, x)

Entropy production is obtained for B(x, y) = ln “

W (x,y) W (y,x)

” , then Hs(x, y) = −W (x, y)1+sW (y, x)−s + λ(x)δ(x − y) ⇒ CS2 verified : Hs = HT

−1−s

In fact, CS2 is for an (antisymmetric) observable which is proportional to the entropy production in the large time limit : ∆Senv

T [x] = − ln(f (x0))+ln(f (xT )) T

− EBe

T[x]

Example of an observable for which CS2 is not verified, but CS1 or GC is ?

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 14 / 16

Gallavotti-Cohen : Spectral characterization (for Pure Jump)

Be

T ≡ 1 T

P

0≤s≤T,∆xs=0 B(X − s , Xs) →

R dxdyB(x, y)ρinv(x)W (x, y) ”Feymann-Kac” formula : Ex (δ(xT − y) exp(sTBe

T)) = exp(T(W exp(sB) − λId))(x, y) then

Λ(s) = inf Spectrum(Hs) with the deformed generator Hs = −W exp(sB) + λId and the convention (W exp(sB))(x, y) = W (x, y) exp(sB(x, y)) We have

GC ≡ {inf Spectrum(Hs) = inf Spectrum(H−E−s)} ⇐ CS1 ≡ {Spectrum(Hs) = Spectrum(H−E−s)} ⇐ CS2 ≡  Hs(x, y) = f −1(x)H−E−s(y, x)f (y) ≡ „ W (x, y) = f −1(x)W (y, x) exp(EB(x, y))f (y)(Derrida :MDB B(x, y) = −B(y, x)

Entropy production is obtained for B(x, y) = ln “

W (x,y) W (y,x)

” , then Hs(x, y) = −W (x, y)1+sW (y, x)−s + λ(x)δ(x − y) ⇒ CS2 verified : Hs = HT

−1−s

In fact, CS2 is for an (antisymmetric) observable which is proportional to the entropy production in the large time limit : ∆Senv

T [x] = − ln(f (x0))+ln(f (xT )) T

− EBe

T[x]

Example of an observable for which CS2 is not verified, but CS1 or GC is ?

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 14 / 16

Gallavotti-Cohen : Spectral characterization (for Pure Jump)

Be

T ≡ 1 T

P

0≤s≤T,∆xs=0 B(X − s , Xs) →

R dxdyB(x, y)ρinv(x)W (x, y) ”Feymann-Kac” formula : Ex (δ(xT − y) exp(sTBe

T)) = exp(T(W exp(sB) − λId))(x, y) then

Λ(s) = inf Spectrum(Hs) with the deformed generator Hs = −W exp(sB) + λId and the convention (W exp(sB))(x, y) = W (x, y) exp(sB(x, y)) We have

GC ≡ {inf Spectrum(Hs) = inf Spectrum(H−E−s)} ⇐ CS1 ≡ {Spectrum(Hs) = Spectrum(H−E−s)} ⇐ CS2 ≡  Hs(x, y) = f −1(x)H−E−s(y, x)f (y) ≡ „ W (x, y) = f −1(x)W (y, x) exp(EB(x, y))f (y)(Derrida :MDB B(x, y) = −B(y, x)

Entropy production is obtained for B(x, y) = ln “

W (x,y) W (y,x)

” , then Hs(x, y) = −W (x, y)1+sW (y, x)−s + λ(x)δ(x − y) ⇒ CS2 verified : Hs = HT

−1−s

In fact, CS2 is for an (antisymmetric) observable which is proportional to the entropy production in the large time limit : ∆Senv

T [x] = − ln(f (x0))+ln(f (xT )) T

− EBe

T[x]

Example of an observable for which CS2 is not verified, but CS1 or GC is ?

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 14 / 16

Restricted Solid On Solid model

A.C. Barato, R. Chetrite, H. Hinrichsen, D. Mukamel. J.Stat.Mech. (2010) P10008 Theorems come and go, but an application is forever. Marc Kac

Deposition and evaporation with the constraint |hi − hi±1| ≤ 1 and periodic BC. Height : HT = P

0≤s≤t,∆xs=0 B(X − s , Xs) with B(X − s , Xs) = ±1 for a deposition (evap).

If qa

pa = qb pb = qc pc ≡ r then ∆Senv T

= (ln r) HT and then we have GC symmetry for the large deviation of HT

T

: I(h) = I(−h) − (ln r) h Now we treat the famous case : qa = qb = qc = q, pb = pc = 1 and pa = p.

L=4 : GC with E = 1

4 ln

• 3q4

2p+p2

• and CS2 non verified

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 15 / 16

Restricted Solid On Solid model

A.C. Barato, R. Chetrite, H. Hinrichsen, D. Mukamel. J.Stat.Mech. (2010) P10008 Theorems come and go, but an application is forever. Marc Kac

Deposition and evaporation with the constraint |hi − hi±1| ≤ 1 and periodic BC. Height : HT = P

0≤s≤t,∆xs=0 B(X − s , Xs) with B(X − s , Xs) = ±1 for a deposition (evap).

If qa

pa = qb pb = qc pc ≡ r then ∆Senv T

= (ln r) HT and then we have GC symmetry for the large deviation of HT

T

: I(h) = I(−h) − (ln r) h Now we treat the famous case : qa = qb = qc = q, pb = pc = 1 and pa = p.

L=4 : GC with E = 1

4 ln

• 3q4

2p+p2

• and CS2 non verified

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 15 / 16

Restricted Solid On Solid model

A.C. Barato, R. Chetrite, H. Hinrichsen, D. Mukamel. J.Stat.Mech. (2010) P10008 Theorems come and go, but an application is forever. Marc Kac

Deposition and evaporation with the constraint |hi − hi±1| ≤ 1 and periodic BC. Height : HT = P

0≤s≤t,∆xs=0 B(X − s , Xs) with B(X − s , Xs) = ±1 for a deposition (evap).

If qa

pa = qb pb = qc pc ≡ r then ∆Senv T

= (ln r) HT and then we have GC symmetry for the large deviation of HT

T

: I(h) = I(−h) − (ln r) h Now we treat the famous case : qa = qb = qc = q, pb = pc = 1 and pa = p.

L=4 : GC with E = 1

4 ln

• 3q4

2p+p2

• and CS2 non verified

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 15 / 16

Restricted Solid On Solid model

A.C. Barato, R. Chetrite, H. Hinrichsen, D. Mukamel. J.Stat.Mech. (2010) P10008 Theorems come and go, but an application is forever. Marc Kac

Deposition and evaporation with the constraint |hi − hi±1| ≤ 1 and periodic BC. Height : HT = P

0≤s≤t,∆xs=0 B(X − s , Xs) with B(X − s , Xs) = ±1 for a deposition (evap).

If qa

pa = qb pb = qc pc ≡ r then ∆Senv T

= (ln r) HT and then we have GC symmetry for the large deviation of HT

T

: I(h) = I(−h) − (ln r) h Now we treat the famous case : qa = qb = qc = q, pb = pc = 1 and pa = p.

L=4 : GC with E = 1

4 ln

• 3q4

2p+p2

• and CS2 non verified

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 15 / 16

Restricted Solid On Solid model

A.C. Barato, R. Chetrite, H. Hinrichsen, D. Mukamel. J.Stat.Mech. (2010) P10008 Theorems come and go, but an application is forever. Marc Kac

Deposition and evaporation with the constraint |hi − hi±1| ≤ 1 and periodic BC. Height : HT = P

0≤s≤t,∆xs=0 B(X − s , Xs) with B(X − s , Xs) = ±1 for a deposition (evap).

If qa

pa = qb pb = qc pc ≡ r then ∆Senv T

= (ln r) HT and then we have GC symmetry for the large deviation of HT

T

: I(h) = I(−h) − (ln r) h Now we treat the famous case : qa = qb = qc = q, pb = pc = 1 and pa = p.

L=4 : GC with E = 1

4 ln

• 3q4

2p+p2

• and CS2 non verified

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 15 / 16

More general network of state

Work with A.C. Barato, H. Hinrichsen, D. Mukamel.

2 q−2

C

n

C C

2 1 q−1 q−1 q−1

q−1

C

n

C C

2 1 3 3 3

3

C

n

1

C C

1 1 2 1

C0 C

1

C

We can consider family of current which are different that the entropy production and possess the symmetry in the large deviation regime with : E ∼ ln P rate of forward Cycle

P rate of backward Cycle

• Open issue

Finite time fluctuation relation for open quantum systems : R.Chetrite, K. Mallick : Fluctuation Relations for Quantum Markovian Dynamical System arXiv :1002.0950 (2010) Large deviation for Open Quantum Systems....

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 16 / 16

More general network of state

Work with A.C. Barato, H. Hinrichsen, D. Mukamel.

2 q−2

C

n

C C

2 1 q−1 q−1 q−1

q−1

C

n

C C

2 1 3 3 3

3

C

n

1

C C

1 1 2 1

C0 C

1

C

We can consider family of current which are different that the entropy production and possess the symmetry in the large deviation regime with : E ∼ ln P rate of forward Cycle

P rate of backward Cycle

• Open issue

Finite time fluctuation relation for open quantum systems : R.Chetrite, K. Mallick : Fluctuation Relations for Quantum Markovian Dynamical System arXiv :1002.0950 (2010) Large deviation for Open Quantum Systems....

Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 16 / 16