Dynamics of a planar Coulomb gas F . Bolley, D. Chafa , J. - PowerPoint PPT Presentation

Dynamics of a planar Coulomb gas Dynamics of a planar Coulomb gas F . Bolley, D. Chafa¨ ı, J. Fontbona Jussieu, Dauphine, Santiago Optimal Point Configurations and Orthogonal Polynomials April 19–22, 2017 Castro Urdiales, Cantabria, Spain 1/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions Outline Poincar´ e for diffusions Dyson Process Ginibre process 2/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions Diffusions � Markov process p X t q t ě 0 Stochastic Differential Equation ? d X t “ 2 d B t ´ ∇ H p X t q d t 3/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions Diffusions � Markov process p X t q t ě 0 Stochastic Differential Equation ? d X t “ 2 d B t ´ ∇ H p X t q d t � Energy H : x P R d ÞÑ H p x q P R with e ´ H P L 1 p dx q 3/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions Diffusions � Markov process p X t q t ě 0 Stochastic Differential Equation ? d X t “ 2 d B t ´ ∇ H p X t q d t � Energy H : x P R d ÞÑ H p x q P R with e ´ H P L 1 p dx q � Non-explosion: if ∇ 2 H ě c P R 3/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions Diffusions � Markov process p X t q t ě 0 Stochastic Differential Equation ? d X t “ 2 d B t ´ ∇ H p X t q d t � Energy H : x P R d ÞÑ H p x q P R with e ´ H P L 1 p dx q � Non-explosion: if ∇ 2 H ě c P R � Reversible (and thus invariant) Boltzmann-Gibbs measure µ p d x q “ e ´ H p x q d x Z p X 0 , X t q d X 0 „ µ ñ “ p X t , X 0 q @ t ě 0 3/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions Diffusions � Markov process p X t q t ě 0 Stochastic Differential Equation ? d X t “ 2 α d B t ´ α ∇ H p X t q d t � Energy H : x P R d ÞÑ H p x q P R with e ´ H P L 1 p dx q � Non-explosion: if ∇ 2 H ě c P R � Reversible (and thus invariant) Boltzmann-Gibbs measure µ p d x q “ e ´ H p x q d x Z p X 0 , X t q d X 0 „ µ ñ “ p X t , X 0 q @ t ě 0 3/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions Diffusions � Markov process p X t q t ě 0 Stochastic Differential Equation c 2 α d X t “ β d B t ´ α ∇ H p X t q d t � Energy H : x P R d ÞÑ H p x q P R with e ´ H P L 1 p dx q � Non-explosion: if ∇ 2 H ě c P R � Reversible (and thus invariant) Boltzmann-Gibbs measure µ β p d x q “ e ´ β H p x q d x Z β p X 0 , X t q d X 0 „ µ ñ “ p X t , X 0 q @ t ě 0 3/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions Inifinitesimal generator � Conditional laws: S t p¨qp x q “ Law p X t | X 0 “ x q 4/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions Inifinitesimal generator � Conditional laws: S t p¨qp x q “ Law p X t | X 0 “ x q � Markov semigroup: S t p f qp x q “ E p f p X t q | X 0 “ x q S 0 “ Identity , S t ˝ S t 1 “ S t ` t 1 4/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions Inifinitesimal generator � Conditional laws: S t p¨qp x q “ Law p X t | X 0 “ x q � Markov semigroup: S t p f qp x q “ E p f p X t q | X 0 “ x q S 0 “ Identity , S t ˝ S t 1 “ S t ` t 1 � Infinitesimal generator G “ ∆ ´ ∇ H ¨ ∇ ˇ d ˇ S t p f qp x q “ ∆ f p x q ´ x ∇ H p x q , ∇ f p x qy “ G p f qp x q ˇ dt ˇ t “ 0 4/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions Inifinitesimal generator � Conditional laws: S t p¨qp x q “ Law p X t | X 0 “ x q � Markov semigroup: S t p f qp x q “ E p f p X t q | X 0 “ x q S 0 “ Identity , S t ˝ S t 1 “ S t ` t 1 � Infinitesimal generator G “ ∆ ´ ∇ H ¨ ∇ ˇ d ˇ S t p f qp x q “ ∆ f p x q ´ x ∇ H p x q , ∇ f p x qy “ G p f qp x q ˇ dt ˇ t “ 0 � The operators G and S t “ e tG are symmetric in L 2 p µ q 4/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions Inifinitesimal generator � Conditional laws: S t p¨qp x q “ Law p X t | X 0 “ x q � Markov semigroup: S t p f qp x q “ E p f p X t q | X 0 “ x q S 0 “ Identity , S t ˝ S t 1 “ S t ` t 1 � Infinitesimal generator G “ ∆ ´ ∇ H ¨ ∇ ˇ d ˇ S t p f qp x q “ ∆ f p x q ´ x ∇ H p x q , ∇ f p x qy “ G p f qp x q ˇ dt ˇ t “ 0 � The operators G and S t “ e tG are symmetric in L 2 p µ q � Fokker-Planck equation if f t “ d µ t d µ with µ t “ Law p X t q then f t “ S t p f 0 q and B t f t “ Gf t 4/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions Exactly solvable model: Ornstein-Uhlenbeck process ? 2 | x | 2 , d X t “ � Gaussian model: H p x q “ 1 2 d B t ´ X t d t , µ “ N p 0 , I d q , Gf p x q “ ∆ f p x q ´ x x , ∇ f p x qy 5/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions Exactly solvable model: Ornstein-Uhlenbeck process ? 2 | x | 2 , d X t “ � Gaussian model: H p x q “ 1 2 d B t ´ X t d t , µ “ N p 0 , I d q , Gf p x q “ ∆ f p x q ´ x x , ∇ f p x qy � Mehler formula: S t p¨qp x q “ Law p X t | X 0 “ x q “ N p x e ´ t , 1 ´ e ´ 2 t q 5/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions Exactly solvable model: Ornstein-Uhlenbeck process ? 2 | x | 2 , d X t “ � Gaussian model: H p x q “ 1 2 d B t ´ X t d t , µ “ N p 0 , I d q , Gf p x q “ ∆ f p x q ´ x x , ∇ f p x qy � Mehler formula: S t p¨qp x q “ Law p X t | X 0 “ x q “ N p x e ´ t , 1 ´ e ´ 2 t q � Hermite polynomials: 5/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions Exactly solvable model: Ornstein-Uhlenbeck process ? 2 | x | 2 , d X t “ � Gaussian model: H p x q “ 1 2 d B t ´ X t d t , µ “ N p 0 , I d q , Gf p x q “ ∆ f p x q ´ x x , ∇ f p x qy � Mehler formula: S t p¨qp x q “ Law p X t | X 0 “ x q “ N p x e ´ t , 1 ´ e ´ 2 t q � Hermite polynomials: § L 2 p µ q “ k 8 n “ 0 vect p P n q 5/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions Exactly solvable model: Ornstein-Uhlenbeck process ? 2 | x | 2 , d X t “ � Gaussian model: H p x q “ 1 2 d B t ´ X t d t , µ “ N p 0 , I d q , Gf p x q “ ∆ f p x q ´ x x , ∇ f p x qy � Mehler formula: S t p¨qp x q “ Law p X t | X 0 “ x q “ N p x e ´ t , 1 ´ e ´ 2 t q � Hermite polynomials: § L 2 p µ q “ k 8 n “ 0 vect p P n q § GP n “ ´ nP n and S t p P n q “ e ´ nt P n 5/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions Exactly solvable model: Ornstein-Uhlenbeck process ? 2 | x | 2 , d X t “ � Gaussian model: H p x q “ 1 2 d B t ´ X t d t , µ “ N p 0 , I d q , Gf p x q “ ∆ f p x q ´ x x , ∇ f p x qy � Mehler formula: S t p¨qp x q “ Law p X t | X 0 “ x q “ N p x e ´ t , 1 ´ e ´ 2 t q � Hermite polynomials: § L 2 p µ q “ k 8 n “ 0 vect p P n q § GP n “ ´ nP n and S t p P n q “ e ´ nt P n § G “ ´ ř 8 n “ 0 n Π P n and S t “ ř 8 n “ 0 e ´ nt Π P n 5/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions Exactly solvable model: Ornstein-Uhlenbeck process ? 2 | x | 2 , d X t “ � Gaussian model: H p x q “ 1 2 d B t ´ X t d t , µ “ N p 0 , I d q , Gf p x q “ ∆ f p x q ´ x x , ∇ f p x qy � Mehler formula: S t p¨qp x q “ Law p X t | X 0 “ x q “ N p x e ´ t , 1 ´ e ´ 2 t q � Hermite polynomials: § L 2 p µ q “ k 8 n “ 0 vect p P n q § GP n “ ´ nP n and S t p P n q “ e ´ nt P n § G “ ´ ř 8 n “ 0 n Π P n and S t “ ř 8 n “ 0 e ´ nt Π P n � Exponential decay (spectral gap): 5/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions Exactly solvable model: Ornstein-Uhlenbeck process ? 2 | x | 2 , d X t “ � Gaussian model: H p x q “ 1 2 d B t ´ X t d t , µ “ N p 0 , I d q , Gf p x q “ ∆ f p x q ´ x x , ∇ f p x qy � Mehler formula: S t p¨qp x q “ Law p X t | X 0 “ x q “ N p x e ´ t , 1 ´ e ´ 2 t q � Hermite polynomials: § L 2 p µ q “ k 8 n “ 0 vect p P n q § GP n “ ´ nP n and S t p P n q “ e ´ nt P n § G “ ´ ř 8 n “ 0 n Π P n and S t “ ř 8 n “ 0 e ´ nt Π P n � Exponential decay (spectral gap): § } f t ´ 1 } L 2 p µ q ď e ´ t } f 0 ´ 1 } L 2 p µ q 5/20

Dynamics of a planar Coulomb gas Poincar´ e for diffusions Exactly solvable model: Ornstein-Uhlenbeck process ? 2 | x | 2 , d X t “ � Gaussian model: H p x q “ 1 2 d B t ´ X t d t , µ “ N p 0 , I d q , Gf p x q “ ∆ f p x q ´ x x , ∇ f p x qy � Mehler formula: S t p¨qp x q “ Law p X t | X 0 “ x q “ N p x e ´ t , 1 ´ e ´ 2 t q � Hermite polynomials: § L 2 p µ q “ k 8 n “ 0 vect p P n q § GP n “ ´ nP n and S t p P n q “ e ´ nt P n § G “ ´ ř 8 n “ 0 n Π P n and S t “ ř 8 n “ 0 e ´ nt Π P n � Exponential decay (spectral gap): § } f t ´ 1 } L 2 p µ q ď e ´ t } f 0 ´ 1 } L 2 p µ q § Var µ p S t f q ď e ´ t Var µ p f q 5/20