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 Introduction 1 Large deviation for a Markovian process 2 Go Beyond the current 3 Generalization of Gallavotti-Cohen-Evans-Morriss symmetry 4 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 ou 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 A T which converges toward � a � Large deviation : How improbable for A T to converge towards a which is different from the typical value � a � (rare events) : LDP : � 1 A T = 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 A T which converges toward � a � Large deviation : How improbable for A T to converge towards a which is different from the typical value � a � (rare events) : LDP : � 1 A T = 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 A T which converges toward � a � Large deviation : How improbable for A T to converge towards a which is different from the typical value � a � (rare events) : LDP : � 1 A T = 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) 1 Λ( s ) ≡ lim T →∞ T ln � exp ( sTA T ) � Varadhan Theorem : If A T satisfies the large deviation principle then �� � Λ( s ) ≡ lim T →∞ 1 T ln da exp ( sTa ) exp ( − TI ( a )) ≡ �� � lim T →∞ 1 = sup a ∈ℜ ( sa − I ( a )) ≡ I ⋆ ( s ) T ln da exp ( T ( sa − I ( a )) Gartner-Ellis Theorem : If Λ( s ) exist and is differentiable, then the LDP exists, I ( a ) = sup s ∈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) 1 Λ( s ) ≡ lim T →∞ T ln � exp ( sTA T ) � Varadhan Theorem : If A T satisfies the large deviation principle then �� � Λ( s ) ≡ lim T →∞ 1 T ln da exp ( sTa ) exp ( − TI ( a )) ≡ �� � lim T →∞ 1 = sup a ∈ℜ ( sa − I ( a )) ≡ I ⋆ ( s ) T ln da exp ( T ( sa − I ( a )) Gartner-Ellis Theorem : If Λ( s ) exist and is differentiable, then the LDP exists, I ( a ) = sup s ∈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) 1 Λ( s ) ≡ lim T →∞ T ln � exp ( sTA T ) � Varadhan Theorem : If A T satisfies the large deviation principle then �� � Λ( s ) ≡ lim T →∞ 1 T ln da exp ( sTa ) exp ( − TI ( a )) ≡ �� � lim T →∞ 1 = sup a ∈ℜ ( sa − I ( a )) ≡ I ⋆ ( s ) T ln da exp ( T ( sa − I ( a )) Gartner-Ellis Theorem : If Λ( s ) exist and is differentiable, then the LDP exists, I ( a ) = sup s ∈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) 1 Λ( s ) ≡ lim T →∞ T ln � exp ( sTA T ) � Varadhan Theorem : If A T satisfies the large deviation principle then �� � Λ( s ) ≡ lim T →∞ 1 T ln da exp ( sTa ) exp ( − TI ( a )) ≡ �� � lim T →∞ 1 = sup a ∈ℜ ( sa − I ( a )) ≡ I ⋆ ( s ) T ln da exp ( T ( sa − I ( a )) Gartner-Ellis Theorem : If Λ( s ) exist and is differentiable, then the LDP exists, I ( a ) = sup s ∈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 R T Large deviation for empirical mean of type A e 1 T ≡ 0 A ( x t ) dt T Level 2 : Measure empiric R T Large deviation of local time spend in a state ρ e 1 1 , T ( x ) = 0 δ ( X t − x ) dt T Level 2 → Level 1 : contraction principle The level 1 can be obtain by contraction of Level 2 because A e dxA ( x ) ρ e R T = T ( x ) . Then � δ ( A e dxA ( x ) ρ ( x )= a d [ ρ ] � δ ( ρ e R T − a ) � = T − ρ ) � ≍ R R ` ` ´´ dxA ( x ) ρ ( x )= a d [ ρ ] exp ( − TI 2 [ ρ ]) ≍ exp − T inf ρ/ dxA ( x ) ρ ( x )= a ( I 2 [ ρ ]) R R Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 6 / 16
Pedestrian approach of Level in Large Deviation Level 1 : empirical mean R T Large deviation for empirical mean of type A e 1 T ≡ 0 A ( x t ) dt T Level 2 : Measure empiric R T Large deviation of local time spend in a state ρ e 1 1 , T ( x ) = 0 δ ( X t − x ) dt T Level 2 → Level 1 : contraction principle The level 1 can be obtain by contraction of Level 2 because A e dxA ( x ) ρ e R T = T ( x ) . Then � δ ( A e dxA ( x ) ρ ( x )= a d [ ρ ] � δ ( ρ e R T − a ) � = T − ρ ) � ≍ R R ` ` ´´ dxA ( x ) ρ ( x )= a d [ ρ ] exp ( − TI 2 [ ρ ]) ≍ exp − T inf ρ/ dxA ( x ) ρ ( x )= a ( I 2 [ ρ ]) R R Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 6 / 16
Pedestrian approach of Level in Large Deviation Level 1 : empirical mean R T Large deviation for empirical mean of type A e 1 T ≡ 0 A ( x t ) dt T Level 2 : Measure empiric R T Large deviation of local time spend in a state ρ e 1 1 , T ( x ) = 0 δ ( X t − x ) dt T Level 2 → Level 1 : contraction principle The level 1 can be obtain by contraction of Level 2 because A e dxA ( x ) ρ e R T = T ( x ) . Then � δ ( A e dxA ( x ) ρ ( x )= a d [ ρ ] � δ ( ρ e R T − a ) � = T − ρ ) � ≍ R R ` ` ´´ dxA ( x ) ρ ( x )= a d [ ρ ] exp ( − TI 2 [ ρ ]) ≍ exp − T inf ρ/ dxA ( x ) ρ ( x )= a ( I 2 [ ρ ]) R R Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 6 / 16
Pedestrian approach of Level in Large Deviation Level 1 : empirical mean R T Large deviation for empirical mean of type A e 1 T ≡ 0 A ( x t ) dt T Level 2 : Measure empiric R T Large deviation of local time spend in a state ρ e 1 1 , T ( x ) = 0 δ ( X t − x ) dt T Level 2 → Level 1 : contraction principle The level 1 can be obtain by contraction of Level 2 because A e dxA ( x ) ρ e R T = T ( x ) . Then � δ ( A e dxA ( x ) ρ ( x )= a d [ ρ ] � δ ( ρ e R T − a ) � = T − ρ ) � ≍ R R ` ` ´´ dxA ( x ) ρ ( x )= a d [ ρ ] exp ( − TI 2 [ ρ ]) ≍ exp − T inf ρ/ dxA ( x ) ρ ( x )= a ( I 2 [ ρ ]) R R Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 6 / 16
Pedestrian approach of Level in Large Deviation Level 1 : empirical mean R T Large deviation for empirical mean of type A e 1 T ≡ 0 A ( x t ) dt T Level 2 : Measure empiric R T Large deviation of local time spend in a state ρ e 1 1 , T ( x ) = 0 δ ( X t − x ) dt T Level 2 → Level 1 : contraction principle The level 1 can be obtain by contraction of Level 2 because A e dxA ( x ) ρ e R T = T ( x ) . Then � δ ( A e dxA ( x ) ρ ( x )= a d [ ρ ] � δ ( ρ e R T − a ) � = T − ρ ) � ≍ R R ` ` ´´ dxA ( x ) ρ ( x )= a d [ ρ ] exp ( − TI 2 [ ρ ]) ≍ exp − T inf ρ/ dxA ( x ) ρ ( x )= a ( I 2 [ ρ ]) R R Raphaël Chétrite (cnrs) Grandes déviation 30/05/2011 6 / 16
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