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

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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

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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

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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

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SLIDE 3

Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Stochastic Flows: A First Example SDEs And Spatial Semimartingales Conclusion

Motivating Example

Consider the following stochastic differential equation d Xt Yt

  • = A

Xt Yt

  • dt +

17 42

  • d
  • W (1)

t

W (2)

t

  • Xs = x, Ys = y, A is a real matrix.

The Solution to this equation is of course: Xt Yt

  • = e(t−s)A

x y

  • + e(t−s)A

t e−(u−s)A 17 42

  • dWu

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 3

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SLIDE 4

Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Stochastic Flows: A First Example SDEs And Spatial Semimartingales Conclusion

Motivating Example

Consider the following stochastic differential equation d Xt Yt

  • = A

Xt Yt

  • dt +

17 42

  • d
  • W (1)

t

W (2)

t

  • Xs = x, Ys = y, A is a real matrix.

The Solution to this equation is of course: Xt Yt

  • = e(t−s)A

x y

  • + e(t−s)A

t e−(u−s)A 17 42

  • dWu

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 3

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SLIDE 5

Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Stochastic Flows: A First Example SDEs And Spatial Semimartingales Conclusion

The flow property

Consider the solution as a function of the initial value. Φs,t : x y

  • → e(t−s)A

x y

  • +e(t−s)A

t e−(u−s)A 17 42

  • dWu

The function Φ = Φs,t(·, ω) satisfies: it is a diffeomorphism for any ω, s, t Φt,t(·, ω) is the identity for all ω and t Φs,t(·, ω) = Φu,t(·, ω) ◦ Φs,u(·, ω) These properties state that Φ is a stochstic flow of diffeomorphisms.

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 4

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Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Stochastic Flows: A First Example SDEs And Spatial Semimartingales Conclusion

The flow property

Consider the solution as a function of the initial value. Φs,t : x y

  • → e(t−s)A

x y

  • +e(t−s)A

t e−(u−s)A 17 42

  • dWu

The function Φ = Φs,t(·, ω) satisfies: it is a diffeomorphism for any ω, s, t Φt,t(·, ω) is the identity for all ω and t Φs,t(·, ω) = Φu,t(·, ω) ◦ Φs,u(·, ω) These properties state that Φ is a stochstic flow of diffeomorphisms.

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 4

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Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Stochastic Flows: A First Example SDEs And Spatial Semimartingales Conclusion

The flow property

Consider the solution as a function of the initial value. Φs,t : x y

  • → e(t−s)A

x y

  • +e(t−s)A

t e−(u−s)A 17 42

  • dWu

The function Φ = Φs,t(·, ω) satisfies: it is a diffeomorphism for any ω, s, t Φt,t(·, ω) is the identity for all ω and t Φs,t(·, ω) = Φu,t(·, ω) ◦ Φs,u(·, ω) These properties state that Φ is a stochstic flow of diffeomorphisms.

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 4

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Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Stochastic Flows: A First Example SDEs And Spatial Semimartingales Conclusion

The flow property

Consider the solution as a function of the initial value. Φs,t : x y

  • → e(t−s)A

x y

  • +e(t−s)A

t e−(u−s)A 17 42

  • dWu

The function Φ = Φs,t(·, ω) satisfies: it is a diffeomorphism for any ω, s, t Φt,t(·, ω) is the identity for all ω and t Φs,t(·, ω) = Φu,t(·, ω) ◦ Φs,u(·, ω) These properties state that Φ is a stochstic flow of diffeomorphisms.

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 4

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Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Stochastic Flows: A First Example SDEs And Spatial Semimartingales Conclusion

The flow property

Consider the solution as a function of the initial value. Φs,t : x y

  • → e(t−s)A

x y

  • +e(t−s)A

t e−(u−s)A 17 42

  • dWu

The function Φ = Φs,t(·, ω) satisfies: it is a diffeomorphism for any ω, s, t Φt,t(·, ω) is the identity for all ω and t Φs,t(·, ω) = Φu,t(·, ω) ◦ Φs,u(·, ω) These properties state that Φ is a stochstic flow of diffeomorphisms.

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 4

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SLIDE 10

Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Stochastic Flows: A First Example SDEs And Spatial Semimartingales Conclusion

The flow property

Consider the solution as a function of the initial value. Φs,t : x y

  • → e(t−s)A

x y

  • +e(t−s)A

t e−(u−s)A 17 42

  • dWu

The function Φ = Φs,t(·, ω) satisfies: it is a diffeomorphism for any ω, s, t Φt,t(·, ω) is the identity for all ω and t Φs,t(·, ω) = Φu,t(·, ω) ◦ Φs,u(·, ω) These properties state that Φ is a stochstic flow of diffeomorphisms.

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 4

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SLIDE 11

Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Stochastic Flows: A First Example SDEs And Spatial Semimartingales Conclusion

Kunita-Type SDEs

Let is write M(t, x y

  • ) = A

x y

  • t +

17 42

  • Wt

Then the SDE becomes d Xt Yt

  • = M(dt,

Xt Yt

  • ), Xs = x, Ys = y

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 5

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Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Stochastic Flows: A First Example SDEs And Spatial Semimartingales Conclusion

Kunita-Type SDEs

Let is write M(t, x y

  • ) = A

x y

  • t +

17 42

  • Wt

Then the SDE becomes d Xt Yt

  • = M(dt,

Xt Yt

  • ), Xs = x, Ys = y

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 5

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Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Stochastic Flows: A First Example SDEs And Spatial Semimartingales Conclusion

Semimartingale Fields

M(t, x y

  • ) = A

x y

  • t +

17 42

  • Wt

M = M(t, x y

  • )) satisfies:

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

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Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Stochastic Flows: A First Example SDEs And Spatial Semimartingales Conclusion

Semimartingale Fields

M(t, x y

  • ) = A

x y

  • t +

17 42

  • Wt

M = M(t, x y

  • )) satisfies:

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

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SLIDE 15

Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Stochastic Flows: A First Example SDEs And Spatial Semimartingales Conclusion

Semimartingale Fields

M(t, x y

  • ) = A

x y

  • t +

17 42

  • Wt

M = M(t, x y

  • )) satisfies:

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

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Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Stochastic Flows: A First Example SDEs And Spatial Semimartingales Conclusion

Semimartingale Fields

M(t, x y

  • ) = A

x y

  • t +

17 42

  • Wt

M = M(t, x y

  • )) satisfies:

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

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SLIDE 17

Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Stochastic Flows: A First Example SDEs And Spatial Semimartingales Conclusion

Semimartingale Fields

M(t, x y

  • ) = A

x y

  • t +

17 42

  • Wt

M = M(t, x y

  • )) satisfies:

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

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Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Stochastic Flows: A First Example SDEs And Spatial Semimartingales Conclusion

Semimartingale Fields

M(t, x y

  • ) = A

x y

  • t +

17 42

  • Wt

M = M(t, x y

  • )) satisfies:

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

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Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Stochastic Flows: A First Example SDEs And Spatial Semimartingales Conclusion

Stochstic Flows And SDEs

φs,t(x) = x + t

s

M(du, φs,u(x)) The solution of an SDE driven by a sufficiently smooth semimartingale field generates a stochastic flow. For a sufficiently smooth stochastic flow there is a semimartingale field that generates it via an SDE.

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 7

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Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Stochastic Flows: A First Example SDEs And Spatial Semimartingales Conclusion

Stochstic Flows And SDEs

φs,t(x) = x + t

s

M(du, φs,u(x)) The solution of an SDE driven by a sufficiently smooth semimartingale field generates a stochastic flow. For a sufficiently smooth stochastic flow there is a semimartingale field that generates it via an SDE.

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 7

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Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Isotropic Brownian Flows Isotropic Ornstein-Uhlenbeck Flows

Isotropic Covariance Tensors, Isotropic Brownian Fields

Definition A function b : Rd → Rd×d is an isotropic covariance tensor if:

1

x → b(x) is smooth enough and the derivatives are bounded.

2

b(0) = Ed (the d-dimensional identity)

3

x → b(x) is not constant.

4

b(x) = O∗b(Ox)O for any x ∈ Rd and O ∈ O(d)

Definition Let b be as above.

  • M(t, x) : t ≥ 0, x ∈ Rd

is an isotropic Brownian field if:

1

(t, x) → M(t, x) is a centered Gaussian process.

2

cov(M(s, x), M(t, y)) = (s ∧ t)b(x − y)

3

(t, x) → M(t, x) is continuous for almost all ω.

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 8

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Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Isotropic Brownian Flows Isotropic Ornstein-Uhlenbeck Flows

Isotropic Covariance Tensors, Isotropic Brownian Fields

Definition A function b : Rd → Rd×d is an isotropic covariance tensor if:

1

x → b(x) is smooth enough and the derivatives are bounded.

2

b(0) = Ed (the d-dimensional identity)

3

x → b(x) is not constant.

4

b(x) = O∗b(Ox)O for any x ∈ Rd and O ∈ O(d)

Definition Let b be as above.

  • M(t, x) : t ≥ 0, x ∈ Rd

is an isotropic Brownian field if:

1

(t, x) → M(t, x) is a centered Gaussian process.

2

cov(M(s, x), M(t, y)) = (s ∧ t)b(x − y)

3

(t, x) → M(t, x) is continuous for almost all ω.

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 8

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Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Isotropic Brownian Flows Isotropic Ornstein-Uhlenbeck Flows

Isotropic Brownian Flows (IBFs)

Properties translation invariance rotation invariance

  • ne-point motion is a d-dimensional standard Brownian

Motion SDEs for two-point-distance,. . . Lyapunov-Exponents are known, deterministic and constant Lebesgue measure is invariant No straightforward entropy definition

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 9

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SLIDE 24

Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Isotropic Brownian Flows Isotropic Ornstein-Uhlenbeck Flows

Isotropic Brownian Flows (IBFs)

Properties translation invariance rotation invariance

  • ne-point motion is a d-dimensional standard Brownian

Motion SDEs for two-point-distance,. . . Lyapunov-Exponents are known, deterministic and constant Lebesgue measure is invariant No straightforward entropy definition

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 9

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SLIDE 25

Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Isotropic Brownian Flows Isotropic Ornstein-Uhlenbeck Flows

Isotropic Brownian Flows (IBFs)

Properties translation invariance rotation invariance

  • ne-point motion is a d-dimensional standard Brownian

Motion SDEs for two-point-distance,. . . Lyapunov-Exponents are known, deterministic and constant Lebesgue measure is invariant No straightforward entropy definition

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 9

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SLIDE 26

Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Isotropic Brownian Flows Isotropic Ornstein-Uhlenbeck Flows

Isotropic Brownian Flows (IBFs)

Properties translation invariance rotation invariance

  • ne-point motion is a d-dimensional standard Brownian

Motion SDEs for two-point-distance,. . . Lyapunov-Exponents are known, deterministic and constant Lebesgue measure is invariant No straightforward entropy definition

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 9

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SLIDE 27

Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Isotropic Brownian Flows Isotropic Ornstein-Uhlenbeck Flows

Isotropic Brownian Flows (IBFs)

Properties translation invariance rotation invariance

  • ne-point motion is a d-dimensional standard Brownian

Motion SDEs for two-point-distance,. . . Lyapunov-Exponents are known, deterministic and constant Lebesgue measure is invariant No straightforward entropy definition

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 9

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SLIDE 28

Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Isotropic Brownian Flows Isotropic Ornstein-Uhlenbeck Flows

Isotropic Brownian Flows (IBFs)

Properties translation invariance rotation invariance

  • ne-point motion is a d-dimensional standard Brownian

Motion SDEs for two-point-distance,. . . Lyapunov-Exponents are known, deterministic and constant Lebesgue measure is invariant No straightforward entropy definition

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 9

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SLIDE 29

Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Isotropic Brownian Flows Isotropic Ornstein-Uhlenbeck Flows

Isotropic Brownian Flows (IBFs)

Properties translation invariance rotation invariance

  • ne-point motion is a d-dimensional standard Brownian

Motion SDEs for two-point-distance,. . . Lyapunov-Exponents are known, deterministic and constant Lebesgue measure is invariant No straightforward entropy definition

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 9

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SLIDE 30

Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Isotropic Brownian Flows Isotropic Ornstein-Uhlenbeck Flows

Isotropic Ornstein-Uhlenbeck Flows (IOUFs)

Introduce a drift into the SDE φs,t(x) = x + t

s

M(du, φs,u(x)) − c t

s

φs,u(x)du (1) NO translation invariance rotation invariance

  • ne-point motion is a d-dimensional std. OU-process

SDEs for two-point-distance,. . . Lyapunov-Exponents can be computed, are deterministic and constant a normal distribution is invariant standard definition of entropy applicable

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 10

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SLIDE 31

Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Isotropic Brownian Flows Isotropic Ornstein-Uhlenbeck Flows

Isotropic Ornstein-Uhlenbeck Flows (IOUFs)

Introduce a drift into the SDE φs,t(x) = x + t

s

M(du, φs,u(x)) − c t

s

φs,u(x)du (1) NO translation invariance rotation invariance

  • ne-point motion is a d-dimensional std. OU-process

SDEs for two-point-distance,. . . Lyapunov-Exponents can be computed, are deterministic and constant a normal distribution is invariant standard definition of entropy applicable

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 10

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SLIDE 32

Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Isotropic Brownian Flows Isotropic Ornstein-Uhlenbeck Flows

Isotropic Ornstein-Uhlenbeck Flows (IOUFs)

Introduce a drift into the SDE φs,t(x) = x + t

s

M(du, φs,u(x)) − c t

s

φs,u(x)du (1) NO translation invariance rotation invariance

  • ne-point motion is a d-dimensional std. OU-process

SDEs for two-point-distance,. . . Lyapunov-Exponents can be computed, are deterministic and constant a normal distribution is invariant standard definition of entropy applicable

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 10

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SLIDE 33

Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Isotropic Brownian Flows Isotropic Ornstein-Uhlenbeck Flows

Isotropic Ornstein-Uhlenbeck Flows (IOUFs)

Introduce a drift into the SDE φs,t(x) = x + t

s

M(du, φs,u(x)) − c t

s

φs,u(x)du (1) NO translation invariance rotation invariance

  • ne-point motion is a d-dimensional std. OU-process

SDEs for two-point-distance,. . . Lyapunov-Exponents can be computed, are deterministic and constant a normal distribution is invariant standard definition of entropy applicable

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 10

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SLIDE 34

Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Isotropic Brownian Flows Isotropic Ornstein-Uhlenbeck Flows

Isotropic Ornstein-Uhlenbeck Flows (IOUFs)

Introduce a drift into the SDE φs,t(x) = x + t

s

M(du, φs,u(x)) − c t

s

φs,u(x)du (1) NO translation invariance rotation invariance

  • ne-point motion is a d-dimensional std. OU-process

SDEs for two-point-distance,. . . Lyapunov-Exponents can be computed, are deterministic and constant a normal distribution is invariant standard definition of entropy applicable

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 10

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SLIDE 35

Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Isotropic Brownian Flows Isotropic Ornstein-Uhlenbeck Flows

Isotropic Ornstein-Uhlenbeck Flows (IOUFs)

Introduce a drift into the SDE φs,t(x) = x + t

s

M(du, φs,u(x)) − c t

s

φs,u(x)du (1) NO translation invariance rotation invariance

  • ne-point motion is a d-dimensional std. OU-process

SDEs for two-point-distance,. . . Lyapunov-Exponents can be computed, are deterministic and constant a normal distribution is invariant standard definition of entropy applicable

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 10

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SLIDE 36

Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Isotropic Brownian Flows Isotropic Ornstein-Uhlenbeck Flows

Isotropic Ornstein-Uhlenbeck Flows (IOUFs)

Introduce a drift into the SDE φs,t(x) = x + t

s

M(du, φs,u(x)) − c t

s

φs,u(x)du (1) NO translation invariance rotation invariance

  • ne-point motion is a d-dimensional std. OU-process

SDEs for two-point-distance,. . . Lyapunov-Exponents can be computed, are deterministic and constant a normal distribution is invariant standard definition of entropy applicable

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 10

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Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Statement Of The Result Sketch Of Proof

Spatial Regularity Lemma

Lemma Let φ = φ0,1 : Rd → Rd be as in (1) (with s = 0 and t = 1). Then we have a.s.

1

lim

R→∞ sup ||x||≥R

||φ(x) − e−cx|| ||x|| = 0 (2)

2

lim

R→∞ sup ||x||≥R

||φ(x)|| ||x|| = e−c (3)

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 11

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Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Statement Of The Result Sketch Of Proof

Reformulation To Compact State Space

It is sufficient to show lim

R→∞

sup

R≤||x||≤R+1

||φ(x) − e−cx|| ||x|| = 0 x → ||φ(x)−e−cx||

||x||

is continuous X := {x ∈ Rd : R ≤ ||x|| ≤ R + 1} is compact

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 12

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Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Statement Of The Result Sketch Of Proof

Reformulation To Compact State Space

It is sufficient to show lim

R→∞

sup

R≤||x||≤R+1

||φ(x) − e−cx|| ||x|| = 0 x → ||φ(x)−e−cx||

||x||

is continuous X := {x ∈ Rd : R ≤ ||x|| ≤ R + 1} is compact

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 12

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Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Statement Of The Result Sketch Of Proof

Reformulation To Compact State Space

It is sufficient to show lim

R→∞

sup

R≤||x||≤R+1

||φ(x) − e−cx|| ||x|| = 0 x → ||φ(x)−e−cx||

||x||

is continuous X := {x ∈ Rd : R ≤ ||x|| ≤ R + 1} is compact

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 12

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Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Statement Of The Result Sketch Of Proof

The Chaining Lemma

(X, d) compact metric space, φ: X → R+ be a.s. cont. func., (δi)i≥0 positive real numbers with

∞

  • i=0

δi < ∞, (χi)∞

i=0 δi-dense in X with χ0 = {x0}, with d(x, x0) ≤ δ0∀x ∈ X.

Lemma (Cranston, Scheutzow and Steinsaltz ’00) For arbitrary positive ǫ, z ≥ 0 and an arbitrary sequence of positive reals (ǫi)i≥0 such that ǫ +

∞

  • i=0

ǫi = 1 we have P

  • sup

x∈X

φ(x) > z

  • ≤

P

  • φ(x0) > ǫz
  • +

∞

  • i=0

|χi+1| sup

d(x,y)≤δi

P

  • |φ(x) − φ(y)| > ǫiz
  • .

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 13

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Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Statement Of The Result Sketch Of Proof

Estimates

P

  • ||φ(x0) − e−cx0|| > ˜

ǫR 2

  • ≤ c4e− ˜

ǫ2 8d2 R2

P

  • |||φ(x) − e−cx|| − ||φ(y) − e−cy||| > 2−j−2˜

ǫR

  • ≤

P

  • |||φ(x) − e−cx|| − ||φ(y) − e−cy||| > 2−j−2˜

ǫR3j|x − y|

  • ≤

P

  • |||φ(x) − φ(y)||| > 2−j−3˜

ǫR3j|x − y|

  • (4)

≤ P

  • B∗

1 ≥ log(2−3−j˜

ǫR3j) − λ σ

  • ≤

c5(2−j−3˜ ǫR3j)− log(2−3−j ˜

ǫR3j )−2λ 2σ2 Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 14

slide-43
SLIDE 43

Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Statement Of The Result Sketch Of Proof

Estimates

P

  • ||φ(x0) − e−cx0|| > ˜

ǫR 2

  • ≤ c4e− ˜

ǫ2 8d2 R2

P

  • |||φ(x) − e−cx|| − ||φ(y) − e−cy||| > 2−j−2˜

ǫR

  • ≤

P

  • |||φ(x) − e−cx|| − ||φ(y) − e−cy||| > 2−j−2˜

ǫR3j|x − y|

  • ≤

P

  • |||φ(x) − φ(y)||| > 2−j−3˜

ǫR3j|x − y|

  • (4)

≤ P

  • B∗

1 ≥ log(2−3−j˜

ǫR3j) − λ σ

  • ≤

c5(2−j−3˜ ǫR3j)− log(2−3−j ˜

ǫR3j )−2λ 2σ2 Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 14

slide-44
SLIDE 44

Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Statement Of The Result Sketch Of Proof

Open Problems - Directions Of Research

Pesin Formula for IOUFs Meaningful definition of the entropy for IBFs Pesin Theory for IBFs

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 15

slide-45
SLIDE 45

Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Statement Of The Result Sketch Of Proof

Open Problems - Directions Of Research

Pesin Formula for IOUFs Meaningful definition of the entropy for IBFs Pesin Theory for IBFs

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 15

slide-46
SLIDE 46

Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Statement Of The Result Sketch Of Proof

Open Problems - Directions Of Research

Pesin Formula for IOUFs Meaningful definition of the entropy for IBFs Pesin Theory for IBFs

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 15

slide-47
SLIDE 47

Stochastic Flows And Stochastic Differential Equations IBFs and IOUFs Spatial Regularity Statement Of The Result Sketch Of Proof

References

  • H. Kunita: Stochastic Flows and Stochastic Differential

Equations, Cambridge University Press, (1990)

  • P. Imkeller and M. Scheutzow: On The spatial Asymptotic

Behaviour Of Stochastic Flows In Euklidean Space, Ann. Probab 27, 109-129, (1999)

  • M. Cranston, M. Scheutzow and D. Steinsaltz: Linear Bounds

For Stochastic Dispersion, Ann. Probab., 28 4, 1852-1869, (2000)

  • G. Dimitrov: Dissertation, Technische Universit¨

at Berlin, (2006)

  • H. van Bargen and G. Dimitroff, Isotropic Ornstein-Uhlenbeck

Flows, submitted, (2008)

Holger van Bargen Isotropic Ornstein-Uhlenbeck Flows 24th July 2008 16