Random Processes in Hyperbolic Spaces Hyperbolic Brownian ...

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2.5 Hyperbolic trigonometry 26 for ζ = x + iy. The equality holds if and only if p = |ζ|. Definition 2.4.3. A horocycle is a curve in the hyperbolic plane that cuts at right angles all the parallels passing through the same point at infinity. In H 2 and B 2 the horocycles are represented by Euclidean circles tangent to the circle of points at infinity. 2.5 Hyperbolic trigonometry In this section we focus on some fundamental results of the hyperbolic trigonometry. With za, zb and zc we denote the vertices of an hyperbolic triangle T , the sides opposite these vertices will have lengths a, b and c respectively and the interior angles at the vertices will be α, β and γ. If, for example, the vertex za is on the circle of points at infinity, then we have α = 0, b = c = ∞. As isometries preserve lengths and angles, we note that the trigonometric formulae remain invariant under isometries. In what follows we will use (H 2 , η) as model of the hyperbolic plane. Theorem 2.5.1. Let T be a hyperbolic triangle with vertices za, zb, zc and let l be the geodesic containing the longest side, say [zb, zc], of T . Then the geodesic l1 through za and orthogonal to l meets l at a point w ∈ [zb, zc]. Proof We may assume that l is the positive imaginary axis, it follows that w = i|za|. In view of Theorem 2.4.3 we have that η(za, zb) ≥ η(w, zb) and that η(za, zc) ≥ η(w, zc). As [zb, zc] is the longest side of T we deduce that max{η(za, w), η(zc, w)} ≤ max{η(za, zc), η(za, zb)} ≤ η(zb, zc). Since the points zb, zc and w are collinear, from Theorem 2.4.2, we have that w must lie between zb and zc or be equal to one of them. Theorem 2.5.2. Let T be a right triangle with angles α = 0, β = 0 and γ = π/2. Then cosh b sin α = 1, (2.35) sinh b tan α = 1, (2.36) tanh b sec α = 1. (2.37) Proof In H 2 we may assume that za = x + iy with x 2 + y 2 = 1 and zc = i. Since y = sin α, from Theorem 2.3.2 we obtain (2.35), in fact cosh b = cosh η(i, x + iy) = 1 + |i − x − iy|2 2y = x2 + y 2 + 1 2y = 1 1 = y sin α . Formulae (2.36) and (2.37) follow easily by applying sinh 2 x = cosh 2 x − 1 and tanh x = sinh x cosh x .
27 Hyperbolic Geometry Remark 2.5.1. The famous Lobachevsky’s formula cot α 2 follows easily from Theorem 2.5.2, in fact, in view of (2.37), we have tanh b = cos α, then = eb cot α 2 = 1 + cos α 1 − cos α = eb e−b = eb . Theorem 2.5.3 (Hyperbolic Pythagorean Theorem). For any triangle with angles α, β and γ = π/2, we have cosh c = cosh a cosh b. Proof For any right triangle with angles α, β and π/2, we may assume, by applying a suitable isometry, that za = x + iy with x 2 + y 2 = 1, zb = it for t > 0 and zc = i. Using Theorem 2.3.2 we have cosh c = cosh η(it, x + iy) = 1 + cosh b = cosh η(i, x + iy) = 1 + cosh a = cosh η(i, it) = 1 + t2 . 2t |it − x − iy|2 2yt |i − x − iy|2 2y = 1 + t2 = 1 y , 2yt ,
Page 1: Dottorato di Ricerca in Statistica
Page 4 and 5: CONTENTS CONTENTS 3.3.4 Gruet’s f
Page 6 and 7: (n + 1)-dimensional Euclidean space
Page 8 and 9: differential equation 3 − 22 λt
Page 10 and 11: 1.1 Settings 2 is C r differentiabl
Page 12 and 13: 1.1 Settings 4 satisfying the follo
Page 14 and 15: 1.2 Laplace operator 6 in fact g k
Page 16 and 17: 1.3 Sectional curvature 8 different
Page 18 and 19: 1.4 Hyperbolic space 10 for all σ
Page 20 and 21: 1.4 Hyperbolic space 12 open unit b
Page 22 and 23: 2.1 Möbius transformations 14 The
Page 24 and 25: 2.2 Poincaré extension of Möbius
Page 26 and 27: 2.3 Hyperbolic metric 18 Given a po
Page 28 and 29: 2.3 Hyperbolic metric 20 we have th
Page 30 and 31: 2.3 Hyperbolic metric 22 We now con
Page 32 and 33: 2.4 Geodesics 24 In view of (2.34)
Page 36 and 37: 2.5 Hyperbolic trigonometry 28
Page 38 and 39: 3.1 Settings 30 distance η = η(z)
Page 40 and 41: 3.2 Hyperbolic Brownian motion in H
Page 52 and 53: 3.3 Multidimensional hyperbolic Bro
Page 62 and 63: 3.A Appendix 54 since ∂ ∂y (y
Page 64 and 65: 3.A Appendix 56 substituting (3.49)
Page 66 and 67: 3.A Appendix 58
Page 68 and 69: 4.1 Description of the planar rando
Page 70 and 71: 4.2 Equations related to the mean h
Page 78 and 79: 4.3 About the higher moments of the
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Page 82 and 83: 4.4 Motions with jumps backwards to
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4.4 Motions with jumps backwards to
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4.4 Motions with jumps backwards to
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4.5 Motion at finite velocity on th
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4.5 Motion at finite velocity on th
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5.1 Description of the randomly mov
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5.2 Mean hyperbolic distance: the L
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5.3 Equation governing the hyperbol
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5.4 Mean hyperbolic distance of the
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5.4 Mean hyperbolic distance of the
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Bibliography [1] Baldi, P., Casadio
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105 BIBLIOGRAPHY [37] Rogers, L., C
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