Isotropic Ornstein-Uhlenbeck Flows Holger van Bargen (joint work - PowerPoint PPT Presentation

Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Isotropic Ornstein-Uhlenbeck Flows Holger van Bargen (joint work with Georgi Dimitroff) IRTG Summer School Disentis Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 1

Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Outline 1 Stochastic Flows And Stochastic Differential Equations Stochastic Flows: A First Example SDEs And Spatial Semimartingales Conclusion 2 IBFs and IOUFs Isotropic Brownian Flows Isotropic Ornstein-Uhlenbeck Flows 3 Spatial Regularity Statement Of The Result Sketch Of Proof Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 2

Stochastic Flows And Stochastic Differential Equations Stochastic Flows: A First Example IBFs and IOUFs SDEs And Spatial Semimartingales Spatial Regularity Conclusion Motivating Example Consider the following stochastic differential equation � X t � X t � 17 � � W (1) � � 0 � t d = A dt + d W (2) Y t Y t 0 42 t X s = x , Y s = y , A is a real matrix. The Solution to this equation is of course: � X t � x � 17 � t � � � = e ( t − s ) A + e ( t − s ) A e − ( u − s ) A dW u Y t y 42 0 Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 3

Stochastic Flows And Stochastic Differential Equations Stochastic Flows: A First Example IBFs and IOUFs SDEs And Spatial Semimartingales Spatial Regularity Conclusion Motivating Example Consider the following stochastic differential equation � X t � X t � 17 � � W (1) � � 0 � t d = A dt + d W (2) Y t Y t 0 42 t X s = x , Y s = y , A is a real matrix. The Solution to this equation is of course: � X t � x � 17 � t � � � = e ( t − s ) A + e ( t − s ) A e − ( u − s ) A dW u Y t y 42 0 Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 3

Stochastic Flows And Stochastic Differential Equations Stochastic Flows: A First Example IBFs and IOUFs SDEs And Spatial Semimartingales Spatial Regularity Conclusion The flow property Consider the solution as a function of the initial value. � x � x � 17 � t � � � �→ e ( t − s ) A + e ( t − s ) A e − ( u − s ) A Φ s , t : dW u y y 42 0 The function Φ = Φ s , t ( · , ω ) satisfies: it is a diffeomorphism for any ω, s , t Φ t , t ( · , ω ) is the identity for all ω and t Φ s , t ( · , ω ) = Φ u , t ( · , ω ) ◦ Φ s , u ( · , ω ) These properties state that Φ is a stochstic flow of diffeomorphisms. Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 4

Stochastic Flows And Stochastic Differential Equations Stochastic Flows: A First Example IBFs and IOUFs SDEs And Spatial Semimartingales Spatial Regularity Conclusion The flow property Consider the solution as a function of the initial value. � x � x � 17 � t � � � �→ e ( t − s ) A + e ( t − s ) A e − ( u − s ) A Φ s , t : dW u y y 42 0 The function Φ = Φ s , t ( · , ω ) satisfies: it is a diffeomorphism for any ω, s , t Φ t , t ( · , ω ) is the identity for all ω and t Φ s , t ( · , ω ) = Φ u , t ( · , ω ) ◦ Φ s , u ( · , ω ) These properties state that Φ is a stochstic flow of diffeomorphisms. Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 4

Stochastic Flows And Stochastic Differential Equations Stochastic Flows: A First Example IBFs and IOUFs SDEs And Spatial Semimartingales Spatial Regularity Conclusion The flow property Consider the solution as a function of the initial value. � x � x � 17 � t � � � �→ e ( t − s ) A + e ( t − s ) A e − ( u − s ) A Φ s , t : dW u y y 42 0 The function Φ = Φ s , t ( · , ω ) satisfies: it is a diffeomorphism for any ω, s , t Φ t , t ( · , ω ) is the identity for all ω and t Φ s , t ( · , ω ) = Φ u , t ( · , ω ) ◦ Φ s , u ( · , ω ) These properties state that Φ is a stochstic flow of diffeomorphisms. Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 4

Stochastic Flows And Stochastic Differential Equations Stochastic Flows: A First Example IBFs and IOUFs SDEs And Spatial Semimartingales Spatial Regularity Conclusion The flow property Consider the solution as a function of the initial value. � x � x � 17 � t � � � �→ e ( t − s ) A + e ( t − s ) A e − ( u − s ) A Φ s , t : dW u y y 42 0 The function Φ = Φ s , t ( · , ω ) satisfies: it is a diffeomorphism for any ω, s , t Φ t , t ( · , ω ) is the identity for all ω and t Φ s , t ( · , ω ) = Φ u , t ( · , ω ) ◦ Φ s , u ( · , ω ) These properties state that Φ is a stochstic flow of diffeomorphisms. Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 4

Stochastic Flows And Stochastic Differential Equations Stochastic Flows: A First Example IBFs and IOUFs SDEs And Spatial Semimartingales Spatial Regularity Conclusion The flow property Consider the solution as a function of the initial value. � x � x � 17 � t � � � �→ e ( t − s ) A + e ( t − s ) A e − ( u − s ) A Φ s , t : dW u y y 42 0 The function Φ = Φ s , t ( · , ω ) satisfies: it is a diffeomorphism for any ω, s , t Φ t , t ( · , ω ) is the identity for all ω and t Φ s , t ( · , ω ) = Φ u , t ( · , ω ) ◦ Φ s , u ( · , ω ) These properties state that Φ is a stochstic flow of diffeomorphisms. Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 4

Stochastic Flows And Stochastic Differential Equations Stochastic Flows: A First Example IBFs and IOUFs SDEs And Spatial Semimartingales Spatial Regularity Conclusion The flow property Consider the solution as a function of the initial value. � x � x � 17 � t � � � �→ e ( t − s ) A + e ( t − s ) A e − ( u − s ) A Φ s , t : dW u y y 42 0 The function Φ = Φ s , t ( · , ω ) satisfies: it is a diffeomorphism for any ω, s , t Φ t , t ( · , ω ) is the identity for all ω and t Φ s , t ( · , ω ) = Φ u , t ( · , ω ) ◦ Φ s , u ( · , ω ) These properties state that Φ is a stochstic flow of diffeomorphisms. Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 4

Stochastic Flows And Stochastic Differential Equations Stochastic Flows: A First Example IBFs and IOUFs SDEs And Spatial Semimartingales Spatial Regularity Conclusion Kunita-Type SDEs Let is write � x � x � 17 � � � 0 M ( t , ) = A t + W t y y 0 42 Then the SDE becomes � X t � X t � � d = M ( dt , ) , X s = x , Y s = y Y t Y t Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 5

Stochastic Flows And Stochastic Differential Equations Stochastic Flows: A First Example IBFs and IOUFs SDEs And Spatial Semimartingales Spatial Regularity Conclusion Kunita-Type SDEs Let is write � x � x � 17 � � � 0 M ( t , ) = A t + W t y y 0 42 Then the SDE becomes � X t � X t � � d = M ( dt , ) , X s = x , Y s = y Y t Y t Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 5

Stochastic Flows And Stochastic Differential Equations Stochastic Flows: A First Example IBFs and IOUFs SDEs And Spatial Semimartingales Spatial Regularity Conclusion Semimartingale Fields � x � x � 17 � � � 0 M ( t , ) = A t + W t y y 0 42 � x � M = M ( t , )) satisfies: y it is a semimartingale for fixed x , y it has smooth covariations in x , y for fixed t the part of finite variaton is smooth This states that is a semimartingale field Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 6

Stochastic Flows And Stochastic Differential Equations Stochastic Flows: A First Example IBFs and IOUFs SDEs And Spatial Semimartingales Spatial Regularity Conclusion Semimartingale Fields � x � x � 17 � � � 0 M ( t , ) = A t + W t y y 0 42 � x � M = M ( t , )) satisfies: y it is a semimartingale for fixed x , y it has smooth covariations in x , y for fixed t the part of finite variaton is smooth This states that is a semimartingale field Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 6

Stochastic Flows And Stochastic Differential Equations Stochastic Flows: A First Example IBFs and IOUFs SDEs And Spatial Semimartingales Spatial Regularity Conclusion Semimartingale Fields � x � x � 17 � � � 0 M ( t , ) = A t + W t y y 0 42 � x � M = M ( t , )) satisfies: y it is a semimartingale for fixed x , y it has smooth covariations in x , y for fixed t the part of finite variaton is smooth This states that is a semimartingale field Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 6

Stochastic Flows And Stochastic Differential Equations Stochastic Flows: A First Example IBFs and IOUFs SDEs And Spatial Semimartingales Spatial Regularity Conclusion Semimartingale Fields � x � x � 17 � � � 0 M ( t , ) = A t + W t y y 0 42 � x � M = M ( t , )) satisfies: y it is a semimartingale for fixed x , y it has smooth covariations in x , y for fixed t the part of finite variaton is smooth This states that is a semimartingale field Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 6