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What is the connection between vibrating string and stochastic process ?

Pawel Zareba

pzareba@few.vu.nl www.few.vu.nl/˜ pzareba

Vrije Universiteit Amsterdam

What is the connection between vibrating string and stochastic process ? – p.1/62

Linear spaces of the process

Given stochastic process Xt on [0, T], we’d like to investigate the space HT = sp{Xt : t ∈ [0, T]} ⊂ L2(P)

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Linear spaces of the process

Given stochastic process Xt on [0, T], we’d like to investigate the space HT = sp{Xt : t ∈ [0, T]} ⊂ L2(P) Recall that for any second order, si-process ∃ measure µ such that EXsXt =

• R

ˆ 1(0,t](λ)ˆ 1(0,s](λ)dµ(λ)

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Linear spaces of the process

Given stochastic process Xt on [0, T], we’d like to investigate the space HT = sp{Xt : t ∈ [0, T]} ⊂ L2(P) Recall that for any second order, si-process ∃ measure µ such that EXsXt =

• R

ˆ 1(0,t](λ)ˆ 1(0,s](λ)dµ(λ) Above relation defines the isometry HT ∋ Xt ← → ˆ 1(0,t] ∈ LT

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Linear spaces of the process

Given stochastic process Xt on [0, T], we’d like to investigate the space HT = sp{Xt : t ∈ [0, T]} ⊂ L2(P) Recall that for any second order, si-process ∃ measure µ such that EXsXt =

• R

ˆ 1(0,t](λ)ˆ 1(0,s](λ)dµ(λ) Above relation defines the isometry HT ∋ Xt ← → ˆ 1(0,t] ∈ LT where LT = sp{ˆ 1(0,t] : t ∈ [0, T]} ⊂ L2(µ).

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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).

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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′ ∂2u ∂t2 = ∂2u ∂x2

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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′ ∂2u ∂t2 = ∂2u ∂x2 For given frequency λ, we look at the solutions of the form u(x, t) = A(x, λ)eiλt, with A(0, λ) = 1, A′(0, λ) = 0.

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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′ ∂2u ∂t2 = ∂2u ∂x2 For given frequency λ, we look at the solutions of the form u(x, t) = A(x, λ)eiλt, with A(0, λ) = 1, A′(0, λ) = 0. The function A(x, λ) satisfies then A′′(x, λ) = −λ2m′(x)A(x, λ).

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Fundamental theorem

THEOREM [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 + λ2)−1dµ(λ) < ∞, there exists a unique string for which (1)

holds true. rλ(x, y) = 1 π

• R

A(x, ω)A(y, ω) ω2 − λ2 µ(dω) (1)

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Fundamental theorem

THEOREM [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 + λ2)−1dµ(λ) < ∞, there exists a unique string for which (1)

holds true. rλ(x, y) = 1 π

• R

A(x, ω)A(y, ω) ω2 − λ2 µ(dω) (1) with

rλ(x, y) = A(x, λ)D(y, λ)1{x≤y} + A(y, λ)D(x, λ)1{x≥y}, D(x, λ) = A(x, λ) l

x

A−2(y, λ)dy.

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Examples

Known "string"-"spectral measure" correspondences:

• ordinary Brownian motion

m(x) = x ← → µ(dλ) = dλ

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Examples

Known "string"-"spectral measure" correspondences:

• ordinary Brownian motion

m(x) = x ← → µ(dλ) = dλ

• power masses m(x) = cxp, including fractional Brownian

motion m(x) = cHx

1−H H

← → µ(dλ) = CH |λ|1−2H dλ

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Examples

Known "string"-"spectral measure" correspondences:

• ordinary Brownian motion

m(x) = x ← → µ(dλ) = dλ

• power masses m(x) = cxp, including fractional Brownian

motion m(x) = cHx

1−H H

← → µ(dλ) = CH |λ|1−2H dλ

• power masses with jumps (autoregression processes)

µ(dλ) = λ2p (λ2 + 1)r dλ, p, r = 0, 1, 2, ...

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Reproducing kernel

We’ll investigate the structure of the space LT using the reproducing kernel, i.e. the function

• ψ(λ)ST (ω, λ) µ(dλ) = ψ, ST (ω, ·)L2(µ) = ψ(ω),

ψ ∈ LT .

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Reproducing kernel

We’ll investigate the structure of the space LT using the reproducing kernel, i.e. the function

• ψ(λ)ST (ω, λ) µ(dλ) = ψ, ST (ω, ·)L2(µ) = ψ(ω),

ψ ∈ LT . It can be shown that in LT, this kernel is given by

ST (ω, λ) = ei(λ−ω)T/2 A(x(T), ω)B(x(T), λ) − B(x(T), ω)A(x(T), λ) π(λ − ω) ,

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Reproducing kernel

We’ll investigate the structure of the space LT using the reproducing kernel, i.e. the function

• ψ(λ)ST (ω, λ) µ(dλ) = ψ, ST (ω, ·)L2(µ) = ψ(ω),

ψ ∈ LT . It can be shown that in LT, this kernel is given by

ST (ω, λ) = ei(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 t(x) = x

• m′(y)dy

and B(x, λ) = λ−1A′(x, λ).

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Reproducing kernel

We’ll investigate the structure of the space LT using the reproducing kernel, i.e. the function

• ψ(λ)ST (ω, λ) µ(dλ) = ψ, ST (ω, ·)L2(µ) = ψ(ω),

ψ ∈ LT . It can be shown that in LT, this kernel is given by

ST (ω, λ) = ei(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 t(x) = x

• m′(y)dy

and B(x, λ) = λ−1A′(x, λ).

BM case: A(x, λ) = cos λx , B(x, λ) = sin λx and x(t) = t.

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Reproducing kernel

We’ll investigate the structure of the space LT using the reproducing kernel, i.e. the function

• ψ(λ)ST (ω, λ) µ(dλ) = ψ, ST (ω, ·)L2(µ) = ψ(ω),

ψ ∈ LT . It can be shown that in LT, this kernel is given by

ST (ω, λ) = ei(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 t(x) = x

• m′(y)dy

and B(x, λ) = λ−1A′(x, λ).

fBM case: A(x(t), λ) = cH(λt)HJ−H(λt), B(x(t), λ) = cH(λt)HJ1−H(λt) and x(t) = dHt2−2H.

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Contributions of reproducing kernel: Basis

If · · · < ω−1 < ω0 = 0 < ω1 < · · · are real-valued zeros of the function B(x(T/2), ·), then the set

{ST(ωn, ·) : n ∈ Z}

is an orthogonal basis in LT .

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Contributions of reproducing kernel: Basis

If · · · < ω−1 < ω0 = 0 < ω1 < · · · are real-valued zeros of the function B(x(T/2), ·), then the set

{ST(ωn, ·) : n ∈ Z}

is an orthogonal basis in LT . Hence, any ψ ∈ LT can be expanded as

ψ(λ) =

• n∈Z

ψ, ST (ωn, ·)µ ST (ωn, λ) ST (ωn, ·)2

µ

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Contributions of reproducing kernel: Basis

If · · · < ω−1 < ω0 = 0 < ω1 < · · · are real-valued zeros of the function B(x(T/2), ·), then the set

{ST(ωn, ·) : n ∈ Z}

is an orthogonal basis in LT . Hence, any ψ ∈ LT can be expanded as

ψ(λ) =

• n∈Z

ψ, ST (ωn, ·)µ ST (ωn, λ) ST (ωn, ·)2

µ

=

• n∈Z

ψ(ωn) ST (ωn, λ) ST (ωn, ωn).

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Series expansion

We apply the expansion ψ(λ) =

• n∈Z

ψ(ωn) ST (ωn, λ) ST(ωn, ωn) to ψ = ˆ 1(0,t] and obtain ˆ 1(0,t](λ) =

• n∈Z

ˆ 1(0,t](ωn) ST (ωn, λ) ST (ωn, ωn).

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Series expansion

We apply the expansion ψ(λ) =

• n∈Z

ψ(ωn) ST (ωn, λ) ST(ωn, ωn) to ψ = ˆ 1(0,t] and obtain ˆ 1(0,t](λ) =

• n∈Z

ˆ 1(0,t](ωn) ST (ωn, λ) ST (ωn, ωn). Now, using the spectral correspondence Xt ↔ ˆ 1(0,t], we immediately have Xt =

• n∈Z

ˆ 1(0,t](ωn)Zn, t ≤ T,

Zn - centered, independent Gaussian r.v.’s with E|Zn|2 = S−1

T (2ωn/T, 2ωn/T)

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Contributions of reproducing kernel: Transform

If we define two functions

ϕ(2t, λ) = eiλt A(x(t), λ) + iB(x(t), λ)x′(t)

• and

V (2t) = 1 π m(x(t))

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Contributions of reproducing kernel: Transform

If we define two functions

ϕ(2t, λ) = eiλt A(x(t), λ) + iB(x(t), λ)x′(t)

• and

V (2t) = 1 π m(x(t))

and introduce Fourier-type transform Uf(λ) = T f(t)ϕ(t, λ)dV (t), f ∈ L2([0, T], dV ),

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Contributions of reproducing kernel: Transform

If we define two functions

ϕ(2t, λ) = eiλt A(x(t), λ) + iB(x(t), λ)x′(t)

• and

V (2t) = 1 π m(x(t))

and introduce Fourier-type transform Uf(λ) = T f(t)ϕ(t, λ)dV (t), f ∈ L2([0, T], dV ), then it can be proved that U : L2 ([0, T] , V ) → LT is an isometry between those spaces.

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Contributions of reproducing kernel: Transform

If we define two functions

ϕ(2t, λ) = eiλt A(x(t), λ) + iB(x(t), λ)x′(t)

• and

V (2t) = 1 π m(x(t))

and introduce Fourier-type transform Uf(λ) = T f(t)ϕ(t, λ)dV (t), f ∈ L2([0, T], dV ), then it can be proved that U : L2 ([0, T] , V ) → LT is an isometry between those spaces. It has an inverse

U−1ψ(t) = d dV (t)

• ψ(λ)

t ϕ(u, λ) dV (u)

• µ(dλ).

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Contributions of reproducing kernel: Transform

If we define two functions

ϕ(2t, λ) = eiλt A(x(t), λ) + iB(x(t), λ)x′(t)

• and

V (2t) = 1 π m(x(t))

and introduce Fourier-type transform Uf(λ) = T f(t)ϕ(t, λ)dV (t), f ∈ L2([0, T], dV ), then it can be proved that U : L2 ([0, T] , V ) → LT is an isometry between those spaces. It has an inverse

U−1ψ(t) = d dV (t)

• ψ(λ)

t ϕ(u, λ) dV (u)

• µ(dλ).

BM case: ϕ(t, λ) = eiλt, dV (t)=dt hence, Uf = ˆ f.

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Wiener integral

Integrals of deterministic functions with respect to Xt. Space of integrands: IT = {f ∈ L2[0, T] : ˆ f ∈ L2(µ)}.

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Wiener integral

Integrals of deterministic functions with respect to Xt. Space of integrands: IT = {f ∈ L2[0, T] : ˆ f ∈ L2(µ)}. For f ∈ IT we have ˆ f ∈ LT , hence we can define T f(u) dXu = spec( ˆ f) ∈ HT .

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Wiener integral

Integrals of deterministic functions with respect to Xt. Space of integrands: IT = {f ∈ L2[0, T] : ˆ f ∈ L2(µ)}. For f ∈ IT we have ˆ f ∈ LT , hence we can define T f(u) dXu = spec( ˆ f) ∈ HT . Then the integral is linear in f, t dXu = spec(ˆ 1(0,t)) = Xt,

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Wiener integral

Integrals of deterministic functions with respect to Xt. Space of integrands: IT = {f ∈ L2[0, T] : ˆ f ∈ L2(µ)}. For f ∈ IT we have ˆ f ∈ LT , hence we can define T f(u) dXu = spec( ˆ f) ∈ HT . Then the integral is linear in f, t dXu = spec(ˆ 1(0,t)) = Xt, and for f, g ∈ IT ,

E T f(u) dXu T g(u) dXu

• = E
• spec( ˆ

f)spec(ˆ g)

• = ˆ

f, ˆ gL2(µ).

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Fundamental martingale

For every t ≥ 0, define random variable Ht ∋ Mt ← → St(0, ·) ∈ Lt, it is a martingale with variance EM 2

t = V(t).

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Fundamental martingale

For every t ≥ 0, define random variable Ht ∋ Mt ← → St(0, ·) ∈ Lt, it is a martingale with variance EM 2

t = V(t).

If we can find kt such that St(0, λ) = ˆ kt(λ)

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Fundamental martingale

For every t ≥ 0, define random variable Ht ∋ Mt ← → St(0, ·) ∈ Lt, it is a martingale with variance EM 2

t = V(t).

If we can find kt such that St(0, λ) = ˆ kt(λ) =

• kt(u)dˆ

1(0,u](λ)

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Fundamental martingale

For every t ≥ 0, define random variable Ht ∋ Mt ← → St(0, ·) ∈ Lt, it is a martingale with variance EM 2

t = V(t).

If we can find kt such that St(0, λ) = ˆ kt(λ) =

• kt(u)dˆ

1(0,u](λ), then we’ll have Mt = t kt(u)dXu.

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Fundamental martingale

For every t ≥ 0, define random variable Ht ∋ Mt ← → St(0, ·) ∈ Lt, it is a martingale with variance EM 2

t = V(t).

If we can find kt such that St(0, λ) = ˆ kt(λ) =

• kt(u)dˆ

1(0,u](λ), then we’ll have Mt = t kt(u)dXu.

fBM case: kt(u) =

u1/2−H (t−u)1/2−H 2HΓ(1/2+H)Γ(3/2−H) ,

u ≤ t.

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Moving average representation

We’d like to have the opposite, i.e. ∃ lt ∈ L2([0, t], V ) such that Xt = t lt(u) dMu .

What is the connection between vibrating string and stochastic process ? – p.39/62

Moving average representation

We’d like to have the opposite, i.e. ∃ lt ∈ L2([0, t], V ) such that Xt = t lt(u) dMu . Via spectral isometry it translates to ˆ 1(0,t](λ) = t lt(u) dSu(0, λ)

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Moving average representation

We’d like to have the opposite, i.e. ∃ lt ∈ L2([0, t], V ) such that Xt = t lt(u) dMu . Via spectral isometry it translates to ˆ 1(0,t](λ) = t lt(u) dSu(0, λ) = t lt(u)ϕ(u, λ) dV (u)

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Moving average representation

We’d like to have the opposite, i.e. ∃ lt ∈ L2([0, t], V ) such that Xt = t lt(u) dMu . Via spectral isometry it translates to ˆ 1(0,t](λ) = t lt(u) dSu(0, λ) = t lt(u)ϕ(u, λ) dV (u) = Ult(λ)

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Moving average representation

We’d like to have the opposite, i.e. ∃ lt ∈ L2([0, t], V ) such that Xt = t lt(u) dMu . Via spectral isometry it translates to ˆ 1(0,t](λ) = t lt(u) dSu(0, λ) = t lt(u)ϕ(u, λ) dV (u) = Ult(λ) Hence, lt(u) = U −1ˆ 1(0,t](u).

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Moving average representation

We’d like to have the opposite, i.e. ∃ lt ∈ L2([0, t], V ) such that Xt = t lt(u) dMu . Via spectral isometry it translates to ˆ 1(0,t](λ) = t lt(u) dSu(0, λ) = t lt(u)ϕ(u, λ) dV (u) = Ult(λ) Hence, lt(u) = U −1ˆ 1(0,t](u).

fBM case: lt(u) = tH−1/2(t − u)H−1/2 − t

u(t − v)H−1/2dvH−1/2, u ≤ t.

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Three isometries

Applying spectral isometry to ˆ 1(0,t] = Ult gives Xt = (spec ◦ U) lt,

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Three isometries

Applying spectral isometry to ˆ 1(0,t] = Ult gives Xt = (spec ◦ U) lt, and new isometry I :=spec ◦ U guarantees existence of the kernel lt.

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Three isometries

Applying spectral isometry to ˆ 1(0,t] = Ult gives Xt = (spec ◦ U) lt, and new isometry I :=spec ◦ U guarantees existence of the kernel lt. In other terms I(f) = t f(u)dMu , f ∈ L2([0, t], V ).

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Three isometries

Applying spectral isometry to ˆ 1(0,t] = Ult gives Xt = (spec ◦ U) lt, and new isometry I :=spec ◦ U guarantees existence of the kernel lt. In other terms I(f) = t f(u)dMu , f ∈ L2([0, t], V ). The complete picture is now:

Xt ∈ HT

spec ր

տ I ˆ 1(0,t] ∈ LT

U

← − lt ∈ L2([0, T], V )

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Further applications: Orthogonal functions on the line.

Discrete case: Given finite measure µ on (−π, π] use Gramm-Schmidt to construct polynomials pn(z), such that λ − → pn(eiλ) form an orthogonal basis of L2((−π, π], µ).

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Further applications: Orthogonal functions on the line.

Discrete case: Given finite measure µ on (−π, π] use Gramm-Schmidt to construct polynomials pn(z), such that λ − → pn(eiλ) form an orthogonal basis of L2((−π, π], µ). With p∗

n(z) = znpn(z−1), recurrence relations hold

pn+1(z) = zpn(z) − anp∗

n(z)

p∗

n+1(z)

= p∗

n(z) − anpn(z)

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Further applications: Orthogonal functions on the line.

Discrete case: Given finite measure µ on (−π, π] use Gramm-Schmidt to construct polynomials pn(z), such that λ − → pn(eiλ) form an orthogonal basis of L2((−π, π], µ). With p∗

n(z) = znpn(z−1), recurrence relations hold

pn+1(z) = zpn(z) − anp∗

n(z)

p∗

n+1(z)

= p∗

n(z) − anpn(z)

• important role in analysis of discrete processes (stationary time

series): prediction, interpolation, likelihood, etc.

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Further applications: Orthogonal functions on the line.

Continuous case: Something is known only in cases of "signal plus noise" processes.

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Further applications: Orthogonal functions on the line.

Continuous case: Something is known only in cases of "signal plus noise" processes. It appears that if for given measure µ on the real line we define

P(2t, λ) = eiλt A(x(t), λ) + iB(x(t), λ)x′(t)

• 2πx′(t)

, P ∗(t, λ) = eiλtP(t, λ)

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Further applications: Orthogonal functions on the line.

Continuous case: Something is known only in cases of "signal plus noise" processes. It appears that if for given measure µ on the real line we define

P(2t, λ) = eiλt A(x(t), λ) + iB(x(t), λ)x′(t)

• 2πx′(t)

, P ∗(t, λ) = eiλtP(t, λ)

they’ll satisfy

∂ ∂tP(t, λ) = iλP(t, λ) − a(t)P ∗(t, λ) ∂ ∂t P ∗(t, λ) = −a(t)P(t, λ),

with 4a(2t) = x′′(t)/x′(t).

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Further applications: Orthogonal functions on the line.

Continuous case: Something is known only in cases of "signal plus noise" processes. It appears that if for given measure µ on the real line we define

P(2t, λ) = eiλt A(x(t), λ) + iB(x(t), λ)x′(t)

• 2πx′(t)

, P ∗(t, λ) = eiλtP(t, λ)

they’ll satisfy

∂ ∂tP(t, λ) = iλP(t, λ) − a(t)P ∗(t, λ) ∂ ∂t P ∗(t, λ) = −a(t)P(t, λ),

with 4a(2t) = x′′(t)/x′(t). Also

• R

P(s, λ)P(t, λ) µ(dλ) = δ(t − s).

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Other results

Using methods of the spectral theory of vibrating strings we∗ have already discovered

• moving average representation of multi-dimensional fractional

Brownian motion (yet unknown and thought of as to complicated to obtain);

∗ with Harry van Zanten and Kacha Dzhaparidze

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Other results

Using methods of the spectral theory of vibrating strings we∗ have already discovered

• moving average representation of multi-dimensional fractional

Brownian motion (yet unknown and thought of as to complicated to obtain);

• series expansion of general isotropic Gaussian random field with

stationary increments;

∗ with Harry van Zanten and Kacha Dzhaparidze

What is the connection between vibrating string and stochastic process ? – p.57/62

Other results

Using methods of the spectral theory of vibrating strings we∗ have already discovered

• moving average representation of multi-dimensional fractional

Brownian motion (yet unknown and thought of as to complicated to obtain);

• series expansion of general isotropic Gaussian random field with

stationary increments;

• spectral conditions for equivalence of Gaussian processes with

stationary increments.

∗ with Harry van Zanten and Kacha Dzhaparidze

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Concluding remarks

• spectral theory of vibrating strings is a powerful machinery

which can be used to easily obtain moving average/series expansion (among many other) results for a wide class of second

• rder si-processes;

What is the connection between vibrating string and stochastic process ? – p.59/62

Concluding remarks

• spectral theory of vibrating strings is a powerful machinery

which can be used to easily obtain moving average/series expansion (among many other) results for a wide class of second

• rder si-processes;
• unfortunately, in general it is very difficult to find a string

associated with given spectral measure; we know few explicit examples.

What is the connection between vibrating string and stochastic process ? – p.60/62

Concluding remarks

• spectral theory of vibrating strings is a powerful machinery

which can be used to easily obtain moving average/series expansion (among many other) results for a wide class of second

• rder si-processes;
• unfortunately, in general it is very difficult to find a string

associated with given spectral measure; we know few explicit examples. Thank you for attention.

What is the connection between vibrating string and stochastic process ? – p.61/62

Appendix: fBM

Fractional Brownian motion is a centered Gaussian process X with EXsXt = 1 2

• s2H + t2H − |t − s|2H

, where H ∈ (0, 1) is the Hurst parameter. Important properties: H-self similar, stationary increments, α-Hölder for every α < H. H = 1/2: ordinary Brownian motion. H = 1/2: not Markovian, not a semimartingale.

What is the connection between vibrating string and stochastic process ? – p.62/62