What is the connection between vibrating string and stochastic - PowerPoint PPT Presentation

What is the connection between vibrating string and stochastic process ? Pawel Zareba www.few.vu.nl/ ˜ pzareba pzareba@few.vu.nl Vrije Universiteit Amsterdam What is the connection between vibrating string and stochastic process ? – p.1/62

Linear spaces of the process Given stochastic process X t on [0 , T ] , we’d like to investigate the space sp { X t : t ∈ [0 , T ] } ⊂ L 2 ( P ) = H T What is the connection between vibrating string and stochastic process ? – p.2/62

Linear spaces of the process Given stochastic process X t on [0 , T ] , we’d like to investigate the space sp { X t : t ∈ [0 , T ] } ⊂ L 2 ( P ) = H T Recall that for any second order, si-process ∃ measure µ such that � ˆ 1 (0 ,t ] ( λ )ˆ E X s X t = 1 (0 ,s ] ( λ ) dµ ( λ ) R What is the connection between vibrating string and stochastic process ? – p.3/62

Linear spaces of the process Given stochastic process X t on [0 , T ] , we’d like to investigate the space sp { X t : t ∈ [0 , T ] } ⊂ L 2 ( P ) = H T Recall that for any second order, si-process ∃ measure µ such that � ˆ 1 (0 ,t ] ( λ )ˆ E X s X t = 1 (0 ,s ] ( λ ) dµ ( λ ) R Above relation defines the isometry → ˆ H T ∋ X t ← 1 (0 ,t ] ∈ L T What is the connection between vibrating string and stochastic process ? – p.4/62

Linear spaces of the process Given stochastic process X t on [0 , T ] , we’d like to investigate the space sp { X t : t ∈ [0 , T ] } ⊂ L 2 ( P ) = H T Recall that for any second order, si-process ∃ measure µ such that � ˆ 1 (0 ,t ] ( λ )ˆ E X s X t = 1 (0 ,s ] ( λ ) dµ ( λ ) R Above relation defines the isometry → ˆ H T ∋ X t ← 1 (0 ,t ] ∈ L T where sp { ˆ 1 (0 ,t ] : t ∈ [0 , T ] } ⊂ L 2 ( µ ) . = L T What is the connection between vibrating string and stochastic process ? – p.5/62

Vibrating string Consider a string of the length l ≤ ∞ stretched between the points x = 0 and x = l and with the mass distribution m ( x ) . What is the connection between vibrating string and stochastic process ? – p.6/62

Vibrating string Consider a string of the length l ≤ ∞ stretched between the points x = 0 and x = l and with the mass distribution m ( x ) . Motion of such a string is described by the solutions u ( t, x ) of the wave equation m ′ ∂ 2 u ∂t 2 = ∂ 2 u ∂x 2 What is the connection between vibrating string and stochastic process ? – p.7/62

Vibrating string Consider a string of the length l ≤ ∞ stretched between the points x = 0 and x = l and with the mass distribution m ( x ) . Motion of such a string is described by the solutions u ( t, x ) of the wave equation m ′ ∂ 2 u ∂t 2 = ∂ 2 u ∂x 2 For given frequency λ , we look at the solutions of the form u ( x, t ) = A ( x, λ ) e iλt , with A (0 , λ ) = 1 , A ′ (0 , λ ) = 0 . What is the connection between vibrating string and stochastic process ? – p.8/62

Vibrating string Consider a string of the length l ≤ ∞ stretched between the points x = 0 and x = l and with the mass distribution m ( x ) . Motion of such a string is described by the solutions u ( t, x ) of the wave equation m ′ ∂ 2 u ∂t 2 = ∂ 2 u ∂x 2 For given frequency λ , we look at the solutions of the form u ( x, t ) = A ( x, λ ) e iλt , with A (0 , λ ) = 1 , A ′ (0 , λ ) = 0 . The function A ( x, λ ) satisfies then A ′′ ( x, λ ) = − λ 2 m ′ ( x ) A ( x, λ ) . What is the connection between vibrating string and stochastic process ? – p.9/62

Fundamental theorem T HEOREM [Krein (1950’s), Dym and McKean (1976)] For any given string, there exists a unique symmetric measure µ on R such that (1) holds. Conversely, given a symmetric measure µ on R such that (1 + λ 2 ) − 1 dµ ( λ ) < ∞ , there exists a unique string for which (1) � holds true. r λ ( x, y ) = 1 A ( x, ω ) A ( y, ω ) � µ ( dω ) (1) ω 2 − λ 2 π R What is the connection between vibrating string and stochastic process ? – p.10/62

Fundamental theorem T HEOREM [Krein (1950’s), Dym and McKean (1976)] For any given string, there exists a unique symmetric measure µ on R such that (1) holds. Conversely, given a symmetric measure µ on R such that (1 + λ 2 ) − 1 dµ ( λ ) < ∞ , there exists a unique string for which (1) � holds true. r λ ( x, y ) = 1 A ( x, ω ) A ( y, ω ) � µ ( dω ) (1) ω 2 − λ 2 π R with r λ ( x, y ) = A ( x, λ ) D ( y, λ )1 { x ≤ y } + A ( y, λ ) D ( x, λ )1 { x ≥ y } , � l A − 2 ( y, λ ) dy. D ( x, λ ) = A ( x, λ ) x What is the connection between vibrating string and stochastic process ? – p.11/62

Examples Known "string"-"spectral measure" correspondences: • ordinary Brownian motion m ( x ) = x ← → µ ( dλ ) = dλ What is the connection between vibrating string and stochastic process ? – p.12/62

Examples Known "string"-"spectral measure" correspondences: • ordinary Brownian motion m ( x ) = x ← → µ ( dλ ) = dλ • power masses m ( x ) = cx p , including fractional Brownian motion µ ( dλ ) = C H | λ | 1 − 2 H dλ 1 − H m ( x ) = c H x ← → H What is the connection between vibrating string and stochastic process ? – p.13/62

Examples Known "string"-"spectral measure" correspondences: • ordinary Brownian motion m ( x ) = x ← → µ ( dλ ) = dλ • power masses m ( x ) = cx p , including fractional Brownian motion µ ( dλ ) = C H | λ | 1 − 2 H dλ 1 − H m ( x ) = c H x ← → H • power masses with jumps (autoregression processes) λ 2 p µ ( dλ ) = ( λ 2 + 1) r dλ, p, r = 0 , 1 , 2 , ... What is the connection between vibrating string and stochastic process ? – p.14/62

Reproducing kernel We’ll investigate the structure of the space L T using the reproducing kernel, i.e. the function � ψ ( λ ) S T ( ω, λ ) µ ( dλ ) = � ψ, S T ( ω, · ) � L 2 ( µ ) = ψ ( ω ) , ψ ∈ L T . What is the connection between vibrating string and stochastic process ? – p.15/62

Reproducing kernel We’ll investigate the structure of the space L T using the reproducing kernel, i.e. the function � ψ ( λ ) S T ( ω, λ ) µ ( dλ ) = � ψ, S T ( ω, · ) � L 2 ( µ ) = ψ ( ω ) , ψ ∈ L T . It can be shown that in L T , this kernel is given by S T ( ω, λ ) = e i ( λ − ω ) T/ 2 A ( x ( T ) , ω ) B ( x ( T ) , λ ) − B ( x ( T ) , ω ) A ( x ( T ) , λ ) , π ( λ − ω ) What is the connection between vibrating string and stochastic process ? – p.16/62

Reproducing kernel We’ll investigate the structure of the space L T using the reproducing kernel, i.e. the function � ψ ( λ ) S T ( ω, λ ) µ ( dλ ) = � ψ, S T ( ω, · ) � L 2 ( µ ) = ψ ( ω ) , ψ ∈ L T . It can be shown that in L T , this kernel is given by S T ( ω, λ ) = e i ( λ − ω ) T/ 2 A ( x ( T ) , ω ) B ( x ( T ) , λ ) − B ( x ( T ) , ω ) A ( x ( T ) , λ ) , π ( λ − ω ) with x ( t ) being inverse of � x � B ( x, λ ) = λ − 1 A ′ ( x, λ ) . m ′ ( y ) dy t ( x ) = and 0 What is the connection between vibrating string and stochastic process ? – p.17/62

Reproducing kernel We’ll investigate the structure of the space L T using the reproducing kernel, i.e. the function � ψ ( λ ) S T ( ω, λ ) µ ( dλ ) = � ψ, S T ( ω, · ) � L 2 ( µ ) = ψ ( ω ) , ψ ∈ L T . It can be shown that in L T , this kernel is given by S T ( ω, λ ) = e i ( λ − ω ) T/ 2 A ( x ( T ) , ω ) B ( x ( T ) , λ ) − B ( x ( T ) , ω ) A ( x ( T ) , λ ) , π ( λ − ω ) with x ( t ) being inverse of � x � B ( x, λ ) = λ − 1 A ′ ( x, λ ) . m ′ ( y ) dy t ( x ) = and 0 BM case: A ( x, λ ) = cos λx , B ( x, λ ) = sin λx and x ( t ) = t . What is the connection between vibrating string and stochastic process ? – p.18/62

Reproducing kernel We’ll investigate the structure of the space L T using the reproducing kernel, i.e. the function � ψ ( λ ) S T ( ω, λ ) µ ( dλ ) = � ψ, S T ( ω, · ) � L 2 ( µ ) = ψ ( ω ) , ψ ∈ L T . It can be shown that in L T , this kernel is given by S T ( ω, λ ) = e i ( λ − ω ) T/ 2 A ( x ( T ) , ω ) B ( x ( T ) , λ ) − B ( x ( T ) , ω ) A ( x ( T ) , λ ) , π ( λ − ω ) with x ( t ) being inverse of � x � B ( x, λ ) = λ − 1 A ′ ( x, λ ) . m ′ ( y ) dy t ( x ) = and 0 A ( x ( t ) , λ ) = c H ( λt ) H J − H ( λt ) , B ( x ( t ) , λ ) = c H ( λt ) H J 1 − H ( λt ) fBM case: and x ( t ) = d H t 2 − 2 H . What is the connection between vibrating string and stochastic process ? – p.19/62

Contributions of reproducing kernel: Basis If · · · < ω − 1 < ω 0 = 0 < ω 1 < · · · are real-valued zeros of the function B ( x ( T/ 2) , · ) , then the set { S T ( ω n , · ) : n ∈ Z } is an orthogonal basis in L T . What is the connection between vibrating string and stochastic process ? – p.20/62

Contributions of reproducing kernel: Basis If · · · < ω − 1 < ω 0 = 0 < ω 1 < · · · are real-valued zeros of the function B ( x ( T/ 2) , · ) , then the set { S T ( ω n , · ) : n ∈ Z } is an orthogonal basis in L T . Hence, any ψ ∈ L T can be expanded as S T ( ω n , λ ) � ψ ( λ ) = � ψ, S T ( ω n , · ) � µ � S T ( ω n , · ) � 2 µ n ∈ Z What is the connection between vibrating string and stochastic process ? – p.21/62