Random Processes in Hyperbolic Spaces Hyperbolic Brownian ...

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2.3 Hyperbolic metric 22 We now consider the disk model (B 2 , η1). The map is a bijection of H 2 onto B 2 , in fact f(z) = z − i z + i |f(z)| 2 = x2 + (y − 1) 2 x2 4y = 1 − + (y + 1) 2 x2 + y2 + 1 + 2y . (2.30) Thus the metric ρ ∗ (u, v) = η(f −1 (u), f −1 (v)), u, v ∈ B 2 , is a metric on B 2 . However, since we have that 2|f ′ (z)| = 1 − |f(z)| 2 2 1 − 1 + 2 x2 +(y+1) 2 4y x2 +y2 +1+2y = 1 1 = y Im[z] , we can identify ρ ∗ with the metric η1 derived from the differential (2.17), then f is an isometry between (H 2 , η) and (B 2 , η1). We can derive an explicit expression for the hyperbolic metric η1 in B 2 by simply rewriting Theorem 2.3.2 by means of f. In particular for u = f(z) ∈ B 2 and v = f(w) ∈ B 2 we have that sinh η1(u, v) 2 = |u − v| (1 − |u| 2 ) 1/2 (1 − |v| 2 ) 1/2 in fact, in view of (2.26), we have, for u = a + ib and v = α + iβ, that sinh 2 η1(u, v) 2 = sinh 2 η(f −1 (z), f −1 (w)) = = = 4Im −i u+1 u−1 −i u+1 u−1 2 v+1 + i v−1 Im 2 −i v+1 v−1 |u−v| 2 [(a−1) 2 +b2 ][(α−1) 2 +β2 ] 1−a2−b2 (a−1) 2 +b2 1−α2−β2 (α−1) 2 +β2 |u − v| 2 (1 − |u| 2 )(1 − |v| 2 ) . Starting from formula (2.31) we obtain that in fact cosh 2 η1(u, v) 2 cosh η1(u, v) 2 = 1 + sinh 2 η1(u, v) 2 = |1 − u¯v| 2 (1 − |u| 2 )(1 − |v| 2 ) = = 1 + = η 2 = sinh = |1 − u¯v| (1 − |u| 2 ) 1/2 (1 − |v| 2 ) 1/2 −i u+1 u−1 v+1 , −i v−1 2 4|u−v| 2 |(u−1)(v−1)| 2 4 1−a2−b2 (a−1) 2 +b2 1−α2−β2 (α−1) 2 +β2 |u − v| 2 (1 − a 2 − b 2 )(1 − α 2 − β 2 ) (2.31) (2.32) |u − v| 2 (1 − |u| 2 )(1 − |v| 2 ) = (1 − |u|2 )(1 − |v| 2 ) + |u − v| 2 (1 − |u| 2 )(1 − |v| 2 )
23 Hyperbolic Geometry since |1 − u¯v| 2 = |1 − ūv| 2 = |ū| 2 2 1 − v ū = |u| 2 2 1 − v ū = |u| 2 1 ū = 1 + |u| 2 |v| 2 − 2|u| 2 1 , v = 1 + |u| ū 2 |v| 2 − 2 〈u, v〉 = 1 + |u| 2 |v| 2 − 2 〈u, v〉 ± |u| 2 ± |v| 2 = |u − v| 2 + (|u| 2 − 1)(|v| 2 − 1). From (2.31) and (2.32) it follows that and from (2.33) we obtain that since In the particular case u = 0 we have tanh η1(u, v) 2 η1(u, v) = log = |u − v| |1 − u¯v| |1 − u¯v| + |u − v| |1 − u¯v| − |u − v| 1 |1 − u¯v| + |u − v| tanh log = 2 |1 − u¯v| − |u − v| |u − v| |1 − u¯v| . η1(0, v) = 1 + |v| 1 − |v| . 2 + |v| 2 1 − 2 , v ū (2.33) It follows that an hyperbolic circle in B 2 with center 0 and radius r is a Euclidean circle with center 0 and radius R = r−1 r+1 , in fact v ∈ B 2 : η1(0, v) = r = v ∈ B 2 : 1 + |v| = r = v ∈ B 1 − |v| 2 r − 1 : |v| = . (2.34) r + 1 In the next theorem is obtained the length of an hyperbolic circle and the area of an hyperbolic disk. Theorem 2.3.3. The area of a hyperbolic disk of hyperbolic radius r is 4π sinh 2 (r/2). The length of a hyperbolic circle of hyperbolic radius r is 2π sinh r. Proof We use the model B2 . Let C = {v ∈ B2 : η1(0, v) = r} and D = {v ∈ B2 : η1(0, v) ≤ r} be the circle and disk with center in the origin and hyperbolic radius r. In view of (2.31) and (2.34) we have that sinh 2 r 2 = sinh2 η1(0, v) = 2 |v|2 R2 = . 1 − |v| 2 1 − R2 Then h − area(D) = 4 D = 4π sinh 2 r 2 . 1 (1 − |v| 2 ) 2 dv1dv2 = 4 2π 0 dθ R 0 s ds (1 − s2 = 4(2π) ) 2 2(1 − R2 ) R 2
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 32 and 33: 2.4 Geodesics 24 In view of (2.34)
Page 34 and 35: 2.5 Hyperbolic trigonometry 26 for
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
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4.3 About the higher moments of the
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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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