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Continuous time Markov Chains: construction and basic tools Conrado - PowerPoint PPT Presentation

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Continuous time Markov Chains: construction and basic tools Conrado da Costa Department of Mathematical Sciences (Durham University) email: conrado.da-costa@durham.ac.uk September, 2020 1 / 21 Outline Construction Finite Infinite

  1. Infinite state spaces Challenges of infinite state spaces: sup x r ( x ) = ∞ . Explosion, Implosions Solutions: Construction up to explosion, Construction via limits. 7 / 21

  2. Infinite state spaces Challenges of infinite state spaces: sup x r ( x ) = ∞ . Explosion, Implosions Solutions: Construction up to explosion, Construction via limits. This is the case of Interacting Particle Systems (IPS) 7 / 21

  3. Interacting Particle Systems S = X V , η ∈ S , x , y ∈ V 8 / 21

  4. Interacting Particle Systems S = X V , η ∈ S , x , y ∈ V Simple cases { 0 , 1 } N - not infinite, but very big and sparse. 8 / 21

  5. Interacting Particle Systems S = X V , η ∈ S , x , y ∈ V Simple cases { 0 , 1 } N - not infinite, but very big and sparse. Local interaction. Local transformation maps: Γ m : S → S , m ∈ M . 8 / 21

  6. Interacting Particle Systems S = X V , η ∈ S , x , y ∈ V Simple cases { 0 , 1 } N - not infinite, but very big and sparse. Local interaction. Local transformation maps: Γ m : S → S , m ∈ M . rates { r ( η, m ) , m ∈ S } . 8 / 21

  7. Interacting Particle Systems S = X V , η ∈ S , x , y ∈ V Simple cases { 0 , 1 } N - not infinite, but very big and sparse. Local interaction. Local transformation maps: Γ m : S → S , m ∈ M . rates { r ( η, m ) , m ∈ S } . Formal generators � Lf ( η ) = r ( η, m )[ f (Γ m η ) − f ( η )] m 8 / 21

  8. Interacting Particle Systems S = X V , η ∈ S , x , y ∈ V Simple cases { 0 , 1 } N - not infinite, but very big and sparse. Local interaction. Local transformation maps: Γ m : S → S , m ∈ M . rates { r ( η, m ) , m ∈ S } . Formal generators � Lf ( η ) = r ( η, m )[ f (Γ m η ) − f ( η )] m V = Z : 8 / 21

  9. Interacting Particle Systems S = X V , η ∈ S , x , y ∈ V Simple cases { 0 , 1 } N - not infinite, but very big and sparse. Local interaction. Local transformation maps: Γ m : S → S , m ∈ M . rates { r ( η, m ) , m ∈ S } . Formal generators � Lf ( η ) = r ( η, m )[ f (Γ m η ) − f ( η )] m V = Z : uncountable state space 8 / 21

  10. Overview What is the object of interest here? 9 / 21

  11. Overview What is the object of interest here? � d = L ∗ µ t dt µ t = δ x µ 0 9 / 21

  12. Overview What is the object of interest here? � d = L ∗ µ t dt µ t = δ x µ 0 Goal: To understand the behavior of µ t in the relevant scales. 9 / 21

  13. Questions The type of questions we look: 10 / 21

  14. Questions The type of questions we look: ◮ evolution of S t ( f ) for test functions f . 10 / 21

  15. Questions The type of questions we look: ◮ evolution of S t ( f ) for test functions f . ◮ invariant measures 10 / 21

  16. Questions The type of questions we look: ◮ evolution of S t ( f ) for test functions f . ◮ invariant measures ◮ convergence 10 / 21

  17. Questions The type of questions we look: ◮ evolution of S t ( f ) for test functions f . ◮ invariant measures ◮ convergence ◮ effects of space-time rescaling for S n · 10 / 21

  18. Questions The type of questions we look: ◮ evolution of S t ( f ) for test functions f . ◮ invariant measures ◮ convergence ◮ effects of space-time rescaling for S n · ◮ An attempt of explaining physical phenomena: 10 / 21

  19. Questions The type of questions we look: ◮ evolution of S t ( f ) for test functions f . ◮ invariant measures ◮ convergence ◮ effects of space-time rescaling for S n · ◮ An attempt of explaining physical phenomena: ◮ fluid equation 10 / 21

  20. Questions The type of questions we look: ◮ evolution of S t ( f ) for test functions f . ◮ invariant measures ◮ convergence ◮ effects of space-time rescaling for S n · ◮ An attempt of explaining physical phenomena: ◮ fluid equation ◮ state of matter 10 / 21

  21. Questions The type of questions we look: ◮ evolution of S t ( f ) for test functions f . ◮ invariant measures ◮ convergence ◮ effects of space-time rescaling for S n · ◮ An attempt of explaining physical phenomena: ◮ fluid equation ◮ state of matter ◮ phase transition 10 / 21

  22. Tools 11 / 21

  23. Tools ◮ Kolmogorov equations 11 / 21

  24. Tools ◮ Kolmogorov equations ◮ Dynkin Martingales 11 / 21

  25. Tools ◮ Kolmogorov equations ◮ Dynkin Martingales ◮ Tightness criteria 11 / 21

  26. Tools ◮ Kolmogorov equations ◮ Dynkin Martingales ◮ Tightness criteria ◮ Martingale problems 11 / 21

  27. Tools ◮ Kolmogorov equations ◮ Dynkin Martingales ◮ Tightness criteria ◮ Martingale problems ◮ Limits of IPS 11 / 21

  28. Forward Kolmogorov equations 12 / 21

  29. Forward Kolmogorov equations � ∂ t P ( s , x ; t , • ) = L ∗ P ( s , x ; t , • ) lim t ↓ s P ( s , x ; t , • ) = δ x ( • ) 12 / 21

  30. Forward Kolmogorov equations � ∂ t P ( s , x ; t , • ) = L ∗ P ( s , x ; t , • ) lim t ↓ s P ( s , x ; t , • ) = δ x ( • ) � Let v ( t , x ) := E [ ϕ ( X t ) | X s = x ] = P ( s , x ; t , dy ) ϕ ( y ) 12 / 21

  31. Forward Kolmogorov equations � ∂ t P ( s , x ; t , • ) = L ∗ P ( s , x ; t , • ) lim t ↓ s P ( s , x ; t , • ) = δ x ( • ) � Let v ( t , x ) := E [ ϕ ( X t ) | X s = x ] = P ( s , x ; t , dy ) ϕ ( y ) � ∂ t v ( t , x ) = ∂ t P ( s , x ; t , dy ) ϕ ( y ) 12 / 21

  32. Forward Kolmogorov equations � ∂ t P ( s , x ; t , • ) = L ∗ P ( s , x ; t , • ) lim t ↓ s P ( s , x ; t , • ) = δ x ( • ) � Let v ( t , x ) := E [ ϕ ( X t ) | X s = x ] = P ( s , x ; t , dy ) ϕ ( y ) � ∂ t v ( t , x ) = ∂ t P ( s , x ; t , dy ) ϕ ( y ) � � P ( s , x ; t , dy ) h − 1 ∂ t v ( t , x ) = lim P ( t , y ; t + h , dz )( ϕ ( z ) − ϕ ( y )) h 12 / 21

  33. Forward Kolmogorov equations � ∂ t P ( s , x ; t , • ) = L ∗ P ( s , x ; t , • ) lim t ↓ s P ( s , x ; t , • ) = δ x ( • ) � Let v ( t , x ) := E [ ϕ ( X t ) | X s = x ] = P ( s , x ; t , dy ) ϕ ( y ) � ∂ t v ( t , x ) = ∂ t P ( s , x ; t , dy ) ϕ ( y ) � � P ( s , x ; t , dy ) h − 1 ∂ t v ( t , x ) = lim P ( t , y ; t + h , dz )( ϕ ( z ) − ϕ ( y )) h � = P ( s , x ; t , dy ) L ϕ ( y ) 12 / 21

  34. Forward Kolmogorov equations � ∂ t P ( s , x ; t , • ) = L ∗ P ( s , x ; t , • ) lim t ↓ s P ( s , x ; t , • ) = δ x ( • ) � Let v ( t , x ) := E [ ϕ ( X t ) | X s = x ] = P ( s , x ; t , dy ) ϕ ( y ) � ∂ t v ( t , x ) = ∂ t P ( s , x ; t , dy ) ϕ ( y ) � � P ( s , x ; t , dy ) h − 1 ∂ t v ( t , x ) = lim P ( t , y ; t + h , dz )( ϕ ( z ) − ϕ ( y )) h � � = P ( s , x ; t , dy ) L ϕ ( y ) = L ∗ P ( s , x ; t , dy ) ϕ ( y ) 12 / 21

  35. Backward Kolmogorov equations � − ∂ s P ( s , x ; t , • ) = LP ( s , x ; t , • ) lim s ↑ t P ( s , x ; t , • ) = δ x ( • ) 13 / 21

  36. Backward Kolmogorov equations � − ∂ s P ( s , x ; t , • ) = LP ( s , x ; t , • ) lim s ↑ t P ( s , x ; t , • ) = δ x ( • ) � Let w ( s , x ) := E [ ϕ ( X t ) | X s = x ] = P ( s , x ; t , dy ) ϕ ( y ) 13 / 21

  37. Backward Kolmogorov equations � − ∂ s P ( s , x ; t , • ) = LP ( s , x ; t , • ) lim s ↑ t P ( s , x ; t , • ) = δ x ( • ) � Let w ( s , x ) := E [ ϕ ( X t ) | X s = x ] = P ( s , x ; t , dy ) ϕ ( y ) � ∂ s w ( s , x ) = − ∂ s P ( s , x ; t , dy ) ϕ ( y ) 13 / 21

  38. Backward Kolmogorov equations � − ∂ s P ( s , x ; t , • ) = LP ( s , x ; t , • ) lim s ↑ t P ( s , x ; t , • ) = δ x ( • ) � Let w ( s , x ) := E [ ϕ ( X t ) | X s = x ] = P ( s , x ; t , dy ) ϕ ( y ) � ∂ s w ( s , x ) = − ∂ s P ( s , x ; t , dy ) ϕ ( y ) � h h − 1 − ∂ s w ( s , x ) = lim P ( s − h , x ; s , dy )[ w ( s , y ) − w ( s , x )] 13 / 21

  39. Backward Kolmogorov equations � − ∂ s P ( s , x ; t , • ) = LP ( s , x ; t , • ) lim s ↑ t P ( s , x ; t , • ) = δ x ( • ) � Let w ( s , x ) := E [ ϕ ( X t ) | X s = x ] = P ( s , x ; t , dy ) ϕ ( y ) � ∂ s w ( s , x ) = − ∂ s P ( s , x ; t , dy ) ϕ ( y ) � h h − 1 − ∂ s w ( s , x ) = lim P ( s − h , x ; s , dy )[ w ( s , y ) − w ( s , x )] h h − 1 [ Lw ( s − h , x ) h + o ( h )] = lim 13 / 21

  40. Backward Kolmogorov equations � − ∂ s P ( s , x ; t , • ) = LP ( s , x ; t , • ) lim s ↑ t P ( s , x ; t , • ) = δ x ( • ) � Let w ( s , x ) := E [ ϕ ( X t ) | X s = x ] = P ( s , x ; t , dy ) ϕ ( y ) � ∂ s w ( s , x ) = − ∂ s P ( s , x ; t , dy ) ϕ ( y ) � h h − 1 − ∂ s w ( s , x ) = lim P ( s − h , x ; s , dy )[ w ( s , y ) − w ( s , x )] h h − 1 [ Lw ( s − h , x ) h + o ( h )] = Lw ( s , x ) = lim 13 / 21

  41. Dynkin Martingales If � < C � � ∂ j � F : R + × S → R sup s F ( s , x ) ( s , x ) 14 / 21

  42. Dynkin Martingales If � < C � � ∂ j � F : R + × S → R sup s F ( s , x ) ( s , x ) then � t � M F ( t ) := F ( t , X t ) − F (0 , X 0 ) − 0 ( ∂ s + L ) F ( s , X s ) ds � t N F ( t ) := [ M F ( t )] 2 − 0 ( QF )( s , X s ) ds are martingales, 14 / 21

  43. Dynkin Martingales If � < C � � ∂ j � F : R + × S → R sup s F ( s , x ) ( s , x ) then � t � M F ( t ) := F ( t , X t ) − F (0 , X 0 ) − 0 ( ∂ s + L ) F ( s , X s ) ds � t N F ( t ) := [ M F ( t )] 2 − 0 ( QF )( s , X s ) ds are martingales, where QF ( s , x ) := LF 2 ( s , x ) − 2 F ( s , x ) LF ( s , x ) 14 / 21

  44. Dynkin Martingales If � < C � � ∂ j � F : R + × S → R sup s F ( s , x ) ( s , x ) then � t � M F ( t ) := F ( t , X t ) − F (0 , X 0 ) − 0 ( ∂ s + L ) F ( s , X s ) ds � t N F ( t ) := [ M F ( t )] 2 − 0 ( QF )( s , X s ) ds are martingales, where QF ( s , x ) := LF 2 ( s , x ) − 2 F ( s , x ) LF ( s , x ) = � r xy [ F ( s , y ) − F ( s , x )] 2 y 14 / 21

  45. Proof (Dynkin) � t M F ( t ) − M F ( s ) = F ( t , X t ) − F ( s , X s ) − ( ∂ r + L ) F ( r , X r ) dr s 15 / 21

  46. Proof (Dynkin) � t M F ( t ) − M F ( s ) = F ( t , X t ) − F ( s , X s ) − ( ∂ r + L ) F ( r , X r ) dr s E [ M F ( t ) − M F ( s ) |F s ] = 0 15 / 21

  47. Proof (Dynkin) � t M F ( t ) − M F ( s ) = F ( t , X t ) − F ( s , X s ) − ( ∂ r + L ) F ( r , X r ) dr s E [ M F ( t ) − M F ( s ) |F s ] = 0 ϕ ( r ) := E [ M F ( r ) − M F ( s ) |F s ] ϕ ′ ( r ) = 0 ϕ ( s ) = 0 15 / 21

  48. Proof (Dynkin) � t M F ( t ) − M F ( s ) = F ( t , X t ) − F ( s , X s ) − ( ∂ r + L ) F ( r , X r ) dr s E [ M F ( t ) − M F ( s ) |F s ] = 0 ϕ ( r ) := E [ M F ( r ) − M F ( s ) |F s ] ϕ ′ ( r ) = 0 ϕ ( s ) = 0 � r + h h − 1 E [ ( ∂ u + L ) F ( u , X u ) du |F s ] → E [( ∂ r + L ) F ( r , X r ) |F s ] r 15 / 21

  49. Proof (Dynkin) � t M F ( t ) − M F ( s ) = F ( t , X t ) − F ( s , X s ) − ( ∂ r + L ) F ( r , X r ) dr s E [ M F ( t ) − M F ( s ) |F s ] = 0 ϕ ( r ) := E [ M F ( r ) − M F ( s ) |F s ] ϕ ′ ( r ) = 0 ϕ ( s ) = 0 � r + h h − 1 E [ ( ∂ u + L ) F ( u , X u ) du |F s ] → E [( ∂ r + L ) F ( r , X r ) |F s ] r h − 1 E [ F ( r + h , X r + h ) − F ( r , X r ) |F s ] = 15 / 21

  50. Proof (Dynkin) � t M F ( t ) − M F ( s ) = F ( t , X t ) − F ( s , X s ) − ( ∂ r + L ) F ( r , X r ) dr s E [ M F ( t ) − M F ( s ) |F s ] = 0 ϕ ( r ) := E [ M F ( r ) − M F ( s ) |F s ] ϕ ′ ( r ) = 0 ϕ ( s ) = 0 � r + h h − 1 E [ ( ∂ u + L ) F ( u , X u ) du |F s ] → E [( ∂ r + L ) F ( r , X r ) |F s ] r h − 1 E [ F ( r + h , X r + h ) − F ( r , X r ) |F s ] = h − 1 E [ F ( r + h , X r + h ) − F ( r , X r ) |F s ] + h − 1 E [ F ( r , X r + h ) − F ( r , X r ) |F s ] 15 / 21

  51. Proof (Dynkin) � t M F ( t ) − M F ( s ) = F ( t , X t ) − F ( s , X s ) − ( ∂ r + L ) F ( r , X r ) dr s E [ M F ( t ) − M F ( s ) |F s ] = 0 ϕ ( r ) := E [ M F ( r ) − M F ( s ) |F s ] ϕ ′ ( r ) = 0 ϕ ( s ) = 0 � r + h h − 1 E [ ( ∂ u + L ) F ( u , X u ) du |F s ] → E [( ∂ r + L ) F ( r , X r ) |F s ] r h − 1 E [ F ( r + h , X r + h ) − F ( r , X r ) |F s ] = h − 1 E [ F ( r + h , X r + h ) − F ( r , X r ) |F s ] + h − 1 E [ F ( r , X r + h ) − F ( r , X r ) |F s ] → E [ ∂ r F ( r , X r ) |F s ] + E [ LF ( r , X r ) |F s ] 15 / 21

  52. Proof (Dynkin) � t � t N F ( t ) := [ M F ( t )] 2 − M F ( t ) = F ( t , X t ) − F (0 , X 0 ) − ( QF )( s , X s ) ds ( ∂ s + L ) F ( s , X s ) ds 0 0 16 / 21

  53. Proof (Dynkin) � t � t N F ( t ) := [ M F ( t )] 2 − M F ( t ) = F ( t , X t ) − F (0 , X 0 ) − ( QF )( s , X s ) ds ( ∂ s + L ) F ( s , X s ) ds 0 0 � t � t 0 ( t ) := M F ( t ) + F (0 , X 0 ) = F ( t , X t ) − F 2 ( t , X t ) ∼ ( ∂ s + L ) F 2 ( s , X s ) ds M F ( ∂ s + L ) F ( s , X s ) ds 0 0 16 / 21

  54. Proof (Dynkin) � t � t N F ( t ) := [ M F ( t )] 2 − M F ( t ) = F ( t , X t ) − F (0 , X 0 ) − ( QF )( s , X s ) ds ( ∂ s + L ) F ( s , X s ) ds 0 0 � t � t 0 ( t ) := M F ( t ) + F (0 , X 0 ) = F ( t , X t ) − F 2 ( t , X t ) ∼ ( ∂ s + L ) F 2 ( s , X s ) ds M F ( ∂ s + L ) F ( s , X s ) ds 0 0 � t � t Integration by parts: G t M t = 0 G s dM s + 0 M s dG s 16 / 21

  55. Proof (Dynkin) � t � t N F ( t ) := [ M F ( t )] 2 − M F ( t ) = F ( t , X t ) − F (0 , X 0 ) − ( QF )( s , X s ) ds ( ∂ s + L ) F ( s , X s ) ds 0 0 � t � t 0 ( t ) := M F ( t ) + F (0 , X 0 ) = F ( t , X t ) − F 2 ( t , X t ) ∼ ( ∂ s + L ) F 2 ( s , X s ) ds M F ( ∂ s + L ) F ( s , X s ) ds 0 0 � t � t Integration by parts: G t M t = 0 G s dM s + 0 M s dG s � t � t � s M F � � 0 ( t ) ( ∂ s + L ) F ( s , X s ) ds ∼ F ( s , X s ) − ( ∂ r + L ) F ( r , X r ) dr ( ∂ s + L ) F ( s , X s ) ds 0 0 0 16 / 21

  56. Proof (Dynkin) � t � t N F ( t ) := [ M F ( t )] 2 − M F ( t ) = F ( t , X t ) − F (0 , X 0 ) − ( QF )( s , X s ) ds ( ∂ s + L ) F ( s , X s ) ds 0 0 � t � t 0 ( t ) := M F ( t ) + F (0 , X 0 ) = F ( t , X t ) − F 2 ( t , X t ) ∼ ( ∂ s + L ) F 2 ( s , X s ) ds M F ( ∂ s + L ) F ( s , X s ) ds 0 0 � t � t Integration by parts: G t M t = 0 G s dM s + 0 M s dG s � t � t � s M F � � 0 ( t ) ( ∂ s + L ) F ( s , X s ) ds ∼ F ( s , X s ) − ( ∂ r + L ) F ( r , X r ) dr ( ∂ s + L ) F ( s , X s ) ds 0 0 0 � t �� t � 2 [ M F ( t )] 2 = F 2 ( t , X t ) − 2 F ( t , X t ) ( ∂ s + L ) F ( s , X s ) ds + ( ∂ s + L ) F ( s , X s ) ds 0 0 16 / 21

  57. Proof (Dynkin) � t � t N F ( t ) := [ M F ( t )] 2 − M F ( t ) = F ( t , X t ) − F (0 , X 0 ) − ( QF )( s , X s ) ds ( ∂ s + L ) F ( s , X s ) ds 0 0 � t � t 0 ( t ) := M F ( t ) + F (0 , X 0 ) = F ( t , X t ) − F 2 ( t , X t ) ∼ ( ∂ s + L ) F 2 ( s , X s ) ds M F ( ∂ s + L ) F ( s , X s ) ds 0 0 � t � t Integration by parts: G t M t = 0 G s dM s + 0 M s dG s � t � t � s M F � � 0 ( t ) ( ∂ s + L ) F ( s , X s ) ds ∼ F ( s , X s ) − ( ∂ r + L ) F ( r , X r ) dr ( ∂ s + L ) F ( s , X s ) ds 0 0 0 � t �� t � 2 [ M F ( t )] 2 = F 2 ( t , X t ) − 2 F ( t , X t ) ( ∂ s + L ) F ( s , X s ) ds + ( ∂ s + L ) F ( s , X s ) ds 0 0 � t � t �� t � 2 ( ∂ s + L ) F 2 ( s , X s ) ds − 2 M F ∼ 0 ( t ) ( ∂ s + L ) F ( s , X s ) ds − ( ∂ s + L ) F ( s , X s ) ds 0 0 0 16 / 21

  58. Proof (Dynkin) � t � t N F ( t ) := [ M F ( t )] 2 − M F ( t ) = F ( t , X t ) − F (0 , X 0 ) − ( QF )( s , X s ) ds ( ∂ s + L ) F ( s , X s ) ds 0 0 � t � t 0 ( t ) := M F ( t ) + F (0 , X 0 ) = F ( t , X t ) − F 2 ( t , X t ) ∼ ( ∂ s + L ) F 2 ( s , X s ) ds M F ( ∂ s + L ) F ( s , X s ) ds 0 0 � t � t Integration by parts: G t M t = 0 G s dM s + 0 M s dG s � t � t � s M F � � 0 ( t ) ( ∂ s + L ) F ( s , X s ) ds ∼ F ( s , X s ) − ( ∂ r + L ) F ( r , X r ) dr ( ∂ s + L ) F ( s , X s ) ds 0 0 0 � t �� t � 2 [ M F ( t )] 2 = F 2 ( t , X t ) − 2 F ( t , X t ) ( ∂ s + L ) F ( s , X s ) ds + ( ∂ s + L ) F ( s , X s ) ds 0 0 � t � t �� t � 2 ( ∂ s + L ) F 2 ( s , X s ) ds − 2 M F ∼ 0 ( t ) ( ∂ s + L ) F ( s , X s ) ds − ( ∂ s + L ) F ( s , X s ) ds 0 0 0 � t � t ( ∂ s + L ) F 2 ( s , X s ) ds − 2 F ( s , X s )( ∂ s + L ) F ( s , X s ) ds ∼ 0 0 � t � s �� t � 2 + 2 ( ∂ r + L ) F ( r , X r ) dr ( ∂ s + L ) F ( s , X s ) ds − ( ∂ s + L ) F ( s , X s ) ds 0 0 0 16 / 21

  59. Proof (Dynkin) � t � t N F ( t ) := [ M F ( t )] 2 − M F ( t ) = F ( t , X t ) − F (0 , X 0 ) − ( QF )( s , X s ) ds ( ∂ s + L ) F ( s , X s ) ds 0 0 � t � t 0 ( t ) := M F ( t ) + F (0 , X 0 ) = F ( t , X t ) − F 2 ( t , X t ) ∼ ( ∂ s + L ) F 2 ( s , X s ) ds M F ( ∂ s + L ) F ( s , X s ) ds 0 0 � t � t Integration by parts: G t M t = 0 G s dM s + 0 M s dG s � t � t � s M F � � 0 ( t ) ( ∂ s + L ) F ( s , X s ) ds ∼ F ( s , X s ) − ( ∂ r + L ) F ( r , X r ) dr ( ∂ s + L ) F ( s , X s ) ds 0 0 0 � t �� t � 2 [ M F ( t )] 2 = F 2 ( t , X t ) − 2 F ( t , X t ) ( ∂ s + L ) F ( s , X s ) ds + ( ∂ s + L ) F ( s , X s ) ds 0 0 � t � t �� t � 2 ( ∂ s + L ) F 2 ( s , X s ) ds − 2 M F ∼ 0 ( t ) ( ∂ s + L ) F ( s , X s ) ds − ( ∂ s + L ) F ( s , X s ) ds 0 0 0 � t � t ( ∂ s + L ) F 2 ( s , X s ) ds − 2 F ( s , X s )( ∂ s + L ) F ( s , X s ) ds ∼ 0 0 16 / 21

  60. Proof (Dynkin) � t � t N F ( t ) := [ M F ( t )] 2 − M F ( t ) = F ( t , X t ) − F (0 , X 0 ) − ( QF )( s , X s ) ds ( ∂ s + L ) F ( s , X s ) ds 0 0 � t � t 0 ( t ) := M F ( t ) + F (0 , X 0 ) = F ( t , X t ) − F 2 ( t , X t ) ∼ ( ∂ s + L ) F 2 ( s , X s ) ds M F ( ∂ s + L ) F ( s , X s ) ds 0 0 � t � t Integration by parts: G t M t = 0 G s dM s + 0 M s dG s � t � t � s M F � � 0 ( t ) ( ∂ s + L ) F ( s , X s ) ds ∼ F ( s , X s ) − ( ∂ r + L ) F ( r , X r ) dr ( ∂ s + L ) F ( s , X s ) ds 0 0 0 � t �� t � 2 [ M F ( t )] 2 = F 2 ( t , X t ) − 2 F ( t , X t ) ( ∂ s + L ) F ( s , X s ) ds + ( ∂ s + L ) F ( s , X s ) ds 0 0 � t � t �� t � 2 ( ∂ s + L ) F 2 ( s , X s ) ds − 2 M F ∼ 0 ( t ) ( ∂ s + L ) F ( s , X s ) ds − ( ∂ s + L ) F ( s , X s ) ds 0 0 0 � t � t � t ( ∂ s + L ) F 2 ( s , X s ) ds − 2 F ( s , X s )( ∂ s + L ) F ( s , X s ) ds ∼ ( QF )( s , X s ) ds ∼ 0 0 0 16 / 21

  61. Tightness criterion Family of processes ( X N · , N ∈ N ) 17 / 21

  62. Tightness criterion Family of processes ( X N · , N ∈ N ) is tight if N P ( X N ∈ K ( ε )) > 1 − ε. ∀ ε > 0 , ∃K ( ε ); inf 17 / 21

  63. Tightness criterion Family of processes ( X N · , N ∈ N ) is tight if N P ( X N ∈ K ( ε )) > 1 − ε. ∀ ε > 0 , ∃K ( ε ); inf Aldous’s criterion. 17 / 21

  64. Tightness criterion Family of processes ( X N · , N ∈ N ) is tight if N P ( X N ∈ K ( ε )) > 1 − ε. ∀ ε > 0 , ∃K ( ε ); inf Aldous’s criterion. Probabilistic version of Arzel´ a-Ascoli: 17 / 21

  65. Tightness criterion Family of processes ( X N · , N ∈ N ) is tight if N P ( X N ∈ K ( ε )) > 1 − ε. ∀ ε > 0 , ∃K ( ε ); inf Aldous’s criterion. Probabilistic version of Arzel´ a-Ascoli: P [ X N 1) sup ∈ K ( t , ǫ )] ≤ ǫ / t N P [ X N : ω ′ ( X N , γ ) > ǫ ] = 0 2) γ → 0 lim sup lim N 17 / 21

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