Spectral measure of Brownian field on hyperbolic plane
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1 Spectral measure of Brownian field on hyperbolic plane Serge Cohen and Michel Lifshits Institut de Mathématique de Toulouse Laboratoire de Statistique et de Probabilités Université Paul Sabatier and St.Petersburg State University Lille July the 3rd, 2008
2 Summary 1 Introduction 2 Spectral theory on hyperbolic plane 3 Spectral measure of Lévy Brownian field
3 Euclidean case Let (B x ) x R 2 be a real valued centered Gaussian field such that B 0 = 0 a.s. and E (B x B y ) 2 = Cst x y. (1)
4 Euclidean case Let (B x ) x R 2 be a real valued centered Gaussian field such that B 0 = 0 a.s. and E (B x B y ) 2 = Cst x y. (1) B is called a Lévy Brownian field on R 2.
5 Euclidean case Let (B x ) x R 2 be a real valued centered Gaussian field such that B 0 = 0 a.s. and E (B x B y ) 2 = Cst x y. (1) B is called a Lévy Brownian field on R 2. We have the following integral representation : e i x,ξ 1 B x = R 2 ξ 3/2 W (dξ), where W (dξ) is a Gaussian white noise.
6 Euclidean case E (B x B y ) 2 = R2 e i x,ξ 1 2 ξ 3 dξ
7 Euclidean case R2 E (B x B y ) 2 e i x,ξ 1 2 = ξ 3 dξ + = 2 (1 cos(r x y,θ )dθr 2 dr 0 S 1
8 Euclidean case R2 E (B x B y ) 2 e i x,ξ 1 2 = ξ 3 dξ + = 2 (1 cos(r x y,θ )dθr 2 dr 0 S 1 + ( ) = 2 1 cos(s u,θ )dθ s 2 x y ds 0 S 1 where u S 1.
9 Euclidean case R2 E (B x B y ) 2 e i x,ξ 1 2 = ξ 3 dξ + = 2 (1 cos(r x y,θ )dθr 2 dr 0 S 1 + ( ) = 2 1 cos(s u,θ )dθ s 2 x y ds 0 S 1 where u S 1. E (B x ) 2 = Cst x = + 0 ( 1 S 1 cos(s x,θ )dθ ) ds s 2.
10 Euclidean case R2 E (B x B y ) 2 e i x,ξ 1 2 = ξ 3 dξ + = 2 (1 cos(r x y,θ )dθr 2 dr 0 S 1 + ( ) = 2 1 cos(s u,θ )dθ s 2 x y ds 0 S 1 where u S 1. + ( ) E (B x ) 2 ds = Cst x = 1 cos(s x,θ )dθ 0 S 1 s 2. (2) φ s (x) = cos(s x,θ ) dθ S 1 2π is the spherical function in the Euclidean case.
11 Hyperbolic plane The disk model is a unit disk D = {z C, z < 1} on the complex plane. d(z 1,z 2 ) = log 1 z 1z 2 + z 2 z 1 1 z 1 z 2 z 2 z 1. (3)
12 Hyperbolic plane The disk model is a unit disk D = {z C, z < 1} on the complex plane. d(z 1,z 2 ) = log 1 z 1z 2 + z 2 z 1 1 z 1 z 2 z 2 z 1. (3) The group of isometries of D is the set {( ) } α β SU(1,1) = : α β 2 β 2 = 1. ᾱ If g SU(1,1) then g(z) = αz+β βz+ᾱ.
13 Hyperbolic plane The disk model is a unit disk D = {z C, z < 1} on the complex plane. d(z 1,z 2 ) = log 1 z 1z 2 + z 2 z 1 1 z 1 z 2 z 2 z 1. (3) The group of isometries of D is the set {( ) } α β SU(1,1) = : α β 2 β 2 = 1. ᾱ If g SU(1,1) then g(z) = αz+β. D is a Riemannian manifold and βz+ᾱ we have a Laplace Beltrami operator expressed in polar coordinate. f = 2 r 2 f + coth(r) r f + 4 sinh 2 (r) where r denotes hyperbolic distance from 0. 2 θ 2 f (4)
14 Fourier transform on hyperbolic plane There exists an invariant measure dz = sinh rdrdθ on D, if z = tanh(r)e iθ.
15 Fourier transform on hyperbolic plane There exists an invariant measure dz = sinh rdrdθ on D, if z = tanh(r)e iθ. The Fourier transform f f is an isometry ( L 2 (D,dz) L 2 R S 1, λ tanh(πλ/2) ) dλ db π
16 Fourier transform on hyperbolic plane There exists an invariant measure dz = sinh rdrdθ on D, if z = tanh(r)e iθ. The Fourier transform f f is an isometry ( L 2 (D,dz) L 2 R S 1, λ tanh(πλ/2) ) dλ db π given by f (λ,θ) = D f (z) (coshr sinhr cos θ) iλ+1 2 dz (5)
17 Fourier transform on hyperbolic plane There exists an invariant measure dz = sinh rdrdθ on D, if z = tanh(r)e iθ. The Fourier transform f f is an isometry ( L 2 (D,dz) L 2 R S 1, λ tanh(πλ/2) ) dλ db π given by f (λ,θ) = D f (z) (coshr sinhr cos θ) iλ+1 2 dz (5) and its inverse f (z) = 1 4π R iλ+1 f (λ,θ) (coshr sinhr cos θ) 2 S 1 λ tanh(πλ/2) dθ dλ. (6)
18 Spherical functions on hyperbolic plane Spherical functions are radial eigenfunctions of Laplace-Beltrami operator with eigenvalues λ s = s(1 s) 0.
19 Spherical functions on hyperbolic plane Spherical functions are radial eigenfunctions of Laplace-Beltrami operator with eigenvalues λ s = s(1 s) 0. Higher spectrum: s = 1+λi 2,λ R. λ, λ provide the same spherical function. Therefore, it is sufficient to keep λ 0.
20 Spherical functions on hyperbolic plane Spherical functions are radial eigenfunctions of Laplace-Beltrami operator with eigenvalues λ s = s(1 s) 0. Higher spectrum: s = 1+λi 2,λ R. λ, λ provide the same spherical function. Therefore, it is sufficient to keep λ 0. Lower spectrum: 0 < s 1. We keep the half-interval (0, 1 2 ].
21 Spherical functions on hyperbolic plane Spherical functions are radial eigenfunctions of Laplace-Beltrami operator with eigenvalues λ s = s(1 s) 0. Higher spectrum: s = 1+λi 2,λ R. λ, λ provide the same spherical function. Therefore, it is sufficient to keep λ 0. Lower spectrum: 0 < s 1. We keep the half-interval (0, 1 2 ]. ω s (z) = (coshr sinhr cos θ) s dθ. (7) S 1
22 Spherical functions on hyperbolic plane Spherical functions are radial eigenfunctions of Laplace-Beltrami operator with eigenvalues λ s = s(1 s) 0. Higher spectrum: s = 1+λi 2,λ R. λ, λ provide the same spherical function. Therefore, it is sufficient to keep λ 0. Lower spectrum: 0 < s 1. We keep the half-interval (0, 1 2 ]. ω s (z) = (coshr sinhr cos θ) s dθ. (7) S 1 We denote by S the spectrum of the Laplace Beltrami operator ( S = 0, 1 ) { } 1 + iλ,λ 0. (8) 2 2
23 Field with stationary increments on hyperbolic planes {X z,t D} is a Gaussian field with stationary increments if z 1,z 2 D,g SU(1,1) E X g(z1 ) X g(z2 ) 2 = E X z1 X z2 2. (9)
24 Field with stationary increments on hyperbolic planes {X z,t D} is a Gaussian field with stationary increments if z 1,z 2 D,g SU(1,1) E X g(z1 ) X g(z2 ) 2 = E X z1 X z2 2. (9) Theorem (Lévy Khintchine) Let V (z) = E X z 2 be a structure function of a field X with stationary increments. Then there exists a unique measure ν on the spectral set S and c 0 such that V (z) = c Q(z) + [1 ω s (η)]ν(ds), z D, (10) and S
25 Field with stationary increments on hyperbolic planes {X z,t D} is a Gaussian field with stationary increments if z 1,z 2 D,g SU(1,1) E X g(z1 ) X g(z2 ) 2 = E X z1 X z2 2. (9) Theorem (Lévy Khintchine) Let V (z) = E X z 2 be a structure function of a field X with stationary increments. Then there exists a unique measure ν on the spectral set S and c 0 such that V (z) = c Q(z) + [1 ω s (η)]ν(ds), z D, (10) S and ( s 1) ν(ds) <. (11) S
26 Generalized quadratic form Q(z) = 2 log cosh(r/2) is a generalized quadratic form on D :
27 Generalized quadratic form Q(z) = 2 log cosh(r/2) is a generalized quadratic form on D : (Q) = 1.
28 Generalized quadratic form Q(z) = 2 log cosh(r/2) is a generalized quadratic form on D : (Q) = 1. In the Euclidean case Q(x) = 1 2 x 2, (Q) = 1.
29 Generalized quadratic form Q(z) = 2 log cosh(r/2) is a generalized quadratic form on D : (Q) = 1. In the Euclidean case Q(x) = 1 2 x 2, (Q) = 1. The corresponding Gaussian field with stationary increments is X (x,y) = xξ 1 + yξ 2, where ξ i are independent standard Gaussian random variables.
30 Spectral measure of Lévy Brownian field For Lévy Brownian field one can write r = d(0,z) = c Q(z) + [1 ω s (z)]ν(ds), z D. (12) S
31 Spectral measure of Lévy Brownian field For Lévy Brownian field one can write r = d(0,z) = c Q(z) + [1 ω s (z)]ν(ds), z D. (12) S Proposition The spectral measure of Lévy Brownian field is vanishing on the lower spectrum and c = 1.
32 Spectral measure of Lévy Brownian field For Lévy Brownian field one can write r = d(0,z) = c Q(z) + [1 ω s (z)]ν(ds), z D. (12) S Proposition The spectral measure of Lévy Brownian field is vanishing on the lower spectrum and c = 1. When r = d(0,z), if ν satisfies (11), then lim r r 1 S [1 ω s(z)]ν(ds) = 0.
33 Spectral measure of Lévy Brownian field For Lévy Brownian field one can write r = d(0,z) = c Q(z) + [1 ω s (z)]ν(ds), z D. (12) S Proposition The spectral measure of Lévy Brownian field is vanishing on the lower spectrum and c = 1. When r = d(0,z), if ν satisfies (11), then lim r r 1 S [1 ω s(z)]ν(ds) = 0. Moreover Q(z) = r Cst + o(e r ).
34 Spectral measure of Lévy Brownian field For Lévy Brownian field one can write r = d(0,z) = c Q(z) + [1 ω s (z)]ν(ds), z D. (12) S Proposition The spectral measure of Lévy Brownian field is vanishing on the lower spectrum and c = 1. When r = d(0,z), if ν satisfies (11), then lim r r 1 S [1 ω s(z)]ν(ds) = 0. Moreover Q(z) = r Cst + o(e r ). For 0 < s < 1/2 ) ω s (z) = C(s)e sr (1 + O(e 2r ) + O(e 2r( 1 2 s) ).
35 Spectral measure of Lévy Brownian field: Higher spectrum Theorem The spectral decomposition of the Brownian field on the hyperbolic plane is r = Q(r) + 0 [1 ω 1 2 +i λ 2 where the spectral density p is given by (r)]p(λ)dλ, (13) p(λ) = ˆϕ(λ) λ tanh(πλ/2) 2π(λ 2 + 1) and ϕ(u) = π/2 0 2 sin 2 θ 1 cosh 2 u sin 2 θ dθ.
36 Spectral measure of Lévy Brownian field: Higher spectrum Hint of the proof Let apply on both hands of the following equations r Q(r) = 0 [1 ω 1 2 +i λ (z)]p(λ)dλ 2
37 Spectral measure of Lévy Brownian field: Higher spectrum Hint of the proof Let apply on both hands of the following equations r Q(r) = coth(r) 1 = 0 0 [1 ω 1 2 +i λ 2 (λ 2 + 1)ω 1 2 +i λ 2 (z)]p(λ)dλ (z)p(λ)d λ.
38 Spectral measure of Lévy Brownian field: Higher spectrum Hint of the proof Let apply on both hands of the following equations r Q(r) = coth(r) 1 = 0 0 [1 ω 1 2 +i λ 2 (λ 2 + 1)ω 1 2 +i λ 2 (z)]p(λ)dλ (z)p(λ)d λ. If f (z) = coth(r) 1, the Fourier transform f does not depend on θ and the inversion formula (6) f (z) = 1 2π 0 f (λ) λ tanh(πλ/2) ω 1 2 +i λ (z)dλ 2
39 Spectral measure of Lévy Brownian field: Higher spectrum Hint of the proof Let apply on both hands of the following equations r Q(r) = coth(r) 1 = 0 0 [1 ω 1 2 +i λ 2 (λ 2 + 1)ω 1 2 +i λ 2 (z)]p(λ)dλ (z)p(λ)d λ. If f (z) = coth(r) 1, the Fourier transform f does not depend on θ and the inversion formula (6) f (z) = 1 2π 0 f (λ) λ tanh(πλ/2) ω 1 2 +i λ (z)dλ 2 Then (λ 2 + 1)p(λ) = 1 2π f (λ) λ tanh(πλ/2),
40 Spectral measure of Lévy Brownian field: Higher spectrum Hint of the proof Let apply on both hands of the following equations r Q(r) = coth(r) 1 = 0 0 [1 ω 1 2 +i λ 2 (λ 2 + 1)ω 1 2 +i λ 2 (z)]p(λ)dλ (z)p(λ)d λ. If f (z) = coth(r) 1, the Fourier transform f does not depend on θ and the inversion formula (6) f (z) = 1 2π 0 f (λ) λ tanh(πλ/2) ω 1 2 +i λ (z)dλ 2 Then (λ 2 + 1)p(λ) = 1 2π f (λ) λ tanh(πλ/2), hence p(λ) = f (λ) λ tanh(πλ/2) 2π(λ 2 +1).
41 Spectral measure of Lévy Brownian field: Abel transform Hence we have to compute the Fourier transform on hyperbolic space of coth(r) 1. Actually the Fourier transform on hyperbolic space can be computed with the help of the Abel transform and a classical Fourier transform on R.
42 Spectral measure of Lévy Brownian field: Abel transform Hence we have to compute the Fourier transform on hyperbolic space of coth(r) 1. Actually the Fourier transform on hyperbolic space can be computed with the help of the Abel transform and a classical Fourier transform on R. If we let F be such that coth(r) 1 = F (cosh 2 r), elementary computations leads to 2F (q) = q/(q 1) 2 + (q 1)/q.
43 Spectral measure of Lévy Brownian field: Abel transform Hence we have to compute the Fourier transform on hyperbolic space of coth(r) 1. Actually the Fourier transform on hyperbolic space can be computed with the help of the Abel transform and a classical Fourier transform on R. If we let F be such that coth(r) 1 = F (cosh 2 r), elementary computations leads to 2F (q) = q/(q 1) 2 + (q 1)/q.Then the Abel transform is given by ϕ(u) = F (cosh 2 u + y 2 )dy
44 Spectral measure of Lévy Brownian field: Abel transform Hence we have to compute the Fourier transform on hyperbolic space of coth(r) 1. Actually the Fourier transform on hyperbolic space can be computed with the help of the Abel transform and a classical Fourier transform on R. If we let F be such that coth(r) 1 = F (cosh 2 r), elementary computations leads to 2F (q) = q/(q 1) 2 + (q 1)/q.Then the Abel transform is given by and ϕ(u) = F (cosh 2 u + y 2 )dy f (λ) = 2π ˆϕ(λ).
45 Spectral measure of Lévy Brownian field: Abel transform Hence we have to compute the Fourier transform on hyperbolic space of coth(r) 1. Actually the Fourier transform on hyperbolic space can be computed with the help of the Abel transform and a classical Fourier transform on R. If we let F be such that coth(r) 1 = F (cosh 2 r), elementary computations leads to 2F (q) = q/(q 1) 2 + (q 1)/q.Then the Abel transform is given by and ϕ(u) = F (cosh 2 u + y 2 )dy f (λ) = 2π ˆϕ(λ). where ϕ(u) = π/2 0 2 sin 2 θ 1 cosh 2 u sin 2 θ dθ.
46 References Clerc J.L., Faraut J, Rais M., Eymard P., Takahashi R. (1982) Analyse harmonique. Ser. Les Cours du CIMPA, Faraut, J., Harzallah, K. (1974) Distances hilbertiennes invariantes sur un espace homogène. Ann. Inst. Fourier 24, no. 3, Helgason, S. (1978) Differential Geometry Lie Groups and Symmetric Spaces Academic Press, second edition, vol.80.
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