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Eyring-Kramers formula for Poincar e and logarithmic Sobolev inequalities Andr e Schlichting Institute for Applied Mathematics, University of Bonn 5 th Workshop on Random Dynamical Systems, Bielefeld. October 5, 2012 Andr e

Eyring-Kramers formula for Poincar´ e and logarithmic Sobolev inequalities Andr´ e Schlichting Institute for Applied Mathematics, University of Bonn 5 th Workshop on Random Dynamical Systems, Bielefeld. October 5, 2012 Andr´ e Schlichting (IAM Bonn) Eyring-Kramers formula for PI and LSI October 5, 2012 1 / 10
Introduction Overdamped Langevin dynamics Hamiltonian H : R n → R energy landscape Dynamic at temperature ε ≪ 1 √ d X t = −∇ H ( X t )d t + 2 ε d W t � � 1 − H Gibbs measure µ (d x ) = Z µ exp d x , ε � e − H ε d x where Z µ = Generator law X t = f t µ evolves ∂ t f t = Lf t := ε ∆ f t − ∇ H · ∇ f t . � Dirichlet form E ( f ) := ( − Lf ) f d µ |∇ f | 2 d µ. � = ε Andr´ e Schlichting (IAM Bonn) Eyring-Kramers formula for PI and LSI October 5, 2012 2 / 10
Introduction Overdamped Langevin dynamics Hamiltonian H : R n → R energy landscape Dynamic at temperature ε ≪ 1 √ d X t = −∇ H ( X t )d t + 2 ε d W t � � 1 − H Gibbs measure µ (d x ) = Z µ exp d x , ε � e − H ε d x where Z µ = Generator law X t = f t µ evolves ∂ t f t = Lf t := ε ∆ f t − ∇ H · ∇ f t . � Dirichlet form E ( f ) := ( − Lf ) f d µ |∇ f | 2 d µ. � = ε Andr´ e Schlichting (IAM Bonn) Eyring-Kramers formula for PI and LSI October 5, 2012 2 / 10
Introduction Overdamped Langevin dynamics Hamiltonian H : R n → R energy landscape Dynamic at temperature ε ≪ 1 √ d X t = −∇ H ( X t )d t + 2 ε d W t � � 1 − H Gibbs measure µ (d x ) = Z µ exp d x , ε � e − H ε d x where Z µ = Generator law X t = f t µ evolves ∂ t f t = Lf t := ε ∆ f t − ∇ H · ∇ f t . � Dirichlet form E ( f ) := ( − Lf ) f d µ |∇ f | 2 d µ. � = ε Andr´ e Schlichting (IAM Bonn) Eyring-Kramers formula for PI and LSI October 5, 2012 2 / 10
Introduction Overdamped Langevin dynamics Hamiltonian H : R n → R energy landscape Dynamic at temperature ε ≪ 1 √ d X t = −∇ H ( X t )d t + 2 ε d W t � � 1 − H Gibbs measure µ (d x ) = Z µ exp d x , ε � e − H ε d x where Z µ = Generator law X t = f t µ evolves ∂ t f t = Lf t := ε ∆ f t − ∇ H · ∇ f t . � Dirichlet form E ( f ) := ( − Lf ) f d µ |∇ f | 2 d µ. � = ε Andr´ e Schlichting (IAM Bonn) Eyring-Kramers formula for PI and LSI October 5, 2012 2 / 10
Introduction Overdamped Langevin dynamics Hamiltonian H : R n → R energy landscape Dynamic at temperature ε ≪ 1 √ d X t = −∇ H ( X t )d t + 2 ε d W t � � 1 − H Gibbs measure µ (d x ) = Z µ exp d x , ε � e − H ε d x where Z µ = Generator law X t = f t µ evolves ∂ t f t = Lf t := ε ∆ f t − ∇ H · ∇ f t . � Dirichlet form E ( f ) := ( − Lf ) f d µ |∇ f | 2 d µ. � = ε Andr´ e Schlichting (IAM Bonn) Eyring-Kramers formula for PI and LSI October 5, 2012 2 / 10
Poincar´ e and logarithmic Sobolev inequality Definition e inequality PI( ̺ ) if ∀ f : R n → R µ satisfies the Poincar´ �� 2 � � d µ ≤ 1 f 2 − |∇ f | 2 d µ. var µ ( f ) := f d µ PI( ̺ ) ̺ and the logarithmic Sobolev inequality LSI( α ) if ∀ f : R n → R � |∇ f | 2 � f d µ d µ ≤ 1 f Ent µ ( f ) := f log d µ. LSI( α ) � 2 f α PI( ̺ ) and LSI( α ) imply exponential convergence to µ : PI( ̺ ) ⇒ var µ ( f t ) ≤ var µ ( f 0 ) e − 2 ̺ε t LSI( α ) ⇒ Ent µ ( f t ) ≤ Ent µ ( f 0 ) e − 2 αε t . Andr´ e Schlichting (IAM Bonn) Eyring-Kramers formula for PI and LSI October 5, 2012 3 / 10
Poincar´ e and logarithmic Sobolev inequality Definition e inequality PI( ̺ ) if ∀ f : R n → R µ satisfies the Poincar´ �� 2 � � d µ ≤ 1 f 2 − |∇ f | 2 d µ. var µ ( f ) := f d µ PI( ̺ ) ̺ and the logarithmic Sobolev inequality LSI( α ) if ∀ f : R n → R � |∇ f | 2 � f d µ d µ ≤ 1 f Ent µ ( f ) := f log d µ. LSI( α ) � 2 f α PI( ̺ ) and LSI( α ) imply exponential convergence to µ : PI( ̺ ) ⇒ var µ ( f t ) ≤ var µ ( f 0 ) e − 2 ̺ε t LSI( α ) ⇒ Ent µ ( f t ) ≤ Ent µ ( f 0 ) e − 2 αε t . Andr´ e Schlichting (IAM Bonn) Eyring-Kramers formula for PI and LSI October 5, 2012 3 / 10
Poincar´ e and logarithmic Sobolev inequality Definition e inequality PI( ̺ ) if ∀ f : R n → R µ satisfies the Poincar´ �� 2 � � d µ ≤ 1 f 2 − |∇ f | 2 d µ. var µ ( f ) := f d µ PI( ̺ ) ̺ and the logarithmic Sobolev inequality LSI( α ) if ∀ f : R n → R � � f 2 f 2 d µ d µ ≤ 2 f 2 log |∇ f | 2 d µ. Ent µ ( f 2 ) := LSI( α ) � α PI( ̺ ) and LSI( α ) imply exponential convergence to µ : PI( ̺ ) ⇒ var µ ( f t ) ≤ var µ ( f 0 ) e − 2 ̺ε t LSI( α ) ⇒ Ent µ ( f t ) ≤ Ent µ ( f 0 ) e − 2 αε t . Andr´ e Schlichting (IAM Bonn) Eyring-Kramers formula for PI and LSI October 5, 2012 3 / 10
Poincar´ e and logarithmic Sobolev inequality Definition e inequality PI( ̺ ) if ∀ f : R n → R µ satisfies the Poincar´ �� 2 � � d µ ≤ 1 f 2 − |∇ f | 2 d µ. var µ ( f ) := f d µ PI( ̺ ) ̺ and the logarithmic Sobolev inequality LSI( α ) if ∀ f : R n → R � � f 2 f 2 d µ d µ ≤ 2 f 2 log |∇ f | 2 d µ. Ent µ ( f 2 ) := LSI( α ) � α PI( ̺ ) and LSI( α ) imply exponential convergence to µ : PI( ̺ ) ⇒ var µ ( f t ) ≤ var µ ( f 0 ) e − 2 ̺ε t LSI( α ) ⇒ Ent µ ( f t ) ≤ Ent µ ( f 0 ) e − 2 αε t . Andr´ e Schlichting (IAM Bonn) Eyring-Kramers formula for PI and LSI October 5, 2012 3 / 10
Partitions Basins of attraction Ω 0 ⊎ Ω 1 = R n of local minima m 0 , m 1 : Ω i := { y 0 ∈ R n : ˙ y t = −∇ H ( y t ) , y t → m i } . Restricted measures µ 0 , µ 1 : Ω 0 Ω 1 µ i := µ � Ω i , i = 0 , 1 . Mixture representation m 0 m 1 s 0 , 1 µ = Z 0 µ 0 + Z 1 µ 1 , Z i := µ (Ω i ) . Andr´ e Schlichting (IAM Bonn) Eyring-Kramers formula for PI and LSI October 5, 2012 4 / 10
Splitting Lemma Z 0 Z 1 ( E µ 0 ( f ) − E µ 1 ( f )) 2 var µ ( f ) = Z 0 var µ 0 ( f ) + Z 1 var µ 1 ( f ) + � �� local variances mean-difference local entropies � �� Ent µ ( f 2 ) ≤ Z 0 Ent µ 0 ( f 2 ) + Z 1 Ent µ 1 ( f 2 ) � var µ 0 ( f ) + var µ 1 ( f ) + ( E µ 0 ( f ) − E µ 1 ( f )) 2 � Z 0 Z 1 + , Λ( Z 0 , Z 1 ) Z 0 − Z 1 where Λ( Z 0 , Z 1 ) = log Z 0 − log Z 1 is the logarithmic mean. Expect from heuristics: good estimate for local variances/entropies exponential estimate for mean-difference Andr´ e Schlichting (IAM Bonn) Eyring-Kramers formula for PI and LSI October 5, 2012 5 / 10
Splitting Lemma Z 0 Z 1 ( E µ 0 ( f ) − E µ 1 ( f )) 2 var µ ( f ) = Z 0 var µ 0 ( f ) + Z 1 var µ 1 ( f ) + � �� local variances mean-difference local entropies � �� Ent µ ( f 2 ) ≤ Z 0 Ent µ 0 ( f 2 ) + Z 1 Ent µ 1 ( f 2 ) � var µ 0 ( f ) + var µ 1 ( f ) + ( E µ 0 ( f ) − E µ 1 ( f )) 2 � Z 0 Z 1 + , Λ( Z 0 , Z 1 ) Z 0 − Z 1 where Λ( Z 0 , Z 1 ) = log Z 0 − log Z 1 is the logarithmic mean. Expect from heuristics: good estimate for local variances/entropies exponential estimate for mean-difference Andr´ e Schlichting (IAM Bonn) Eyring-Kramers formula for PI and LSI October 5, 2012 5 / 10
Splitting Lemma Z 0 Z 1 ( E µ 0 ( f ) − E µ 1 ( f )) 2 var µ ( f ) = Z 0 var µ 0 ( f ) + Z 1 var µ 1 ( f ) + � �� local variances mean-difference local entropies � �� Ent µ ( f 2 ) ≤ Z 0 Ent µ 0 ( f 2 ) + Z 1 Ent µ 1 ( f 2 ) � var µ 0 ( f ) + var µ 1 ( f ) + ( E µ 0 ( f ) − E µ 1 ( f )) 2 � Z 0 Z 1 + , Λ( Z 0 , Z 1 ) Z 0 − Z 1 where Λ( Z 0 , Z 1 ) = log Z 0 − log Z 1 is the logarithmic mean. Expect from heuristics: good estimate for local variances/entropies exponential estimate for mean-difference Andr´ e Schlichting (IAM Bonn) Eyring-Kramers formula for PI and LSI October 5, 2012 5 / 10
Main results Theorem (Local PI and LSI) The measures µ 0 and µ 1 satisfy PI( ̺ loc ) and LSI( α loc ) with ̺ − 1 α − 1 loc = O ( ε ) and loc = O (1) . PI is as good as convex potential Non-convexity of potential worsens LSI Both results scale optimal in one dimension Theorem (Mean-difference estimate) � � | det ∇ 2 H ( s 0 , 1 ) | Z µ 2 πε |∇ f | 2 d µ. ( E µ 0 f − E µ 1 f ) 2 � e ε − 1 H ( s 0 , 1 ) n | λ − ( ∇ 2 H ( s 0 , 1 )) | (2 πε ) 2 “ � ”: up to multiplicative error 1 + o (1) as ε → 0 . Andr´ e Schlichting (IAM Bonn) Eyring-Kramers formula for PI and LSI October 5, 2012 6 / 10
Main results Theorem (Local PI and LSI) The measures µ 0 and µ 1 satisfy PI( ̺ loc ) and LSI( α loc ) with ̺ − 1 α − 1 loc = O ( ε ) and loc = O (1) . PI is as good as convex potential Non-convexity of potential worsens LSI Both results scale optimal in one dimension Theorem (Mean-difference estimate) � � | det ∇ 2 H ( s 0 , 1 ) | Z µ 2 πε |∇ f | 2 d µ. ( E µ 0 f − E µ 1 f ) 2 � e ε − 1 H ( s 0 , 1 ) n | λ − ( ∇ 2 H ( s 0 , 1 )) | (2 πε ) 2 “ � ”: up to multiplicative error 1 + o (1) as ε → 0 . Andr´ e Schlichting (IAM Bonn) Eyring-Kramers formula for PI and LSI October 5, 2012 6 / 10

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