Random Processes in Hyperbolic Spaces Hyperbolic Brownian ...

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3.2 Hyperbolic Brownian motion in H 2 36 Remark 3.2.1. In view of Theorem 3.2.1 we obtain that the probability density p2(η, t) to be at time t at a distance η, normalized with respect to the area element sinh η dη, is given by p2(η, t) = 2π G2(η, t) = t − ∞ e 4 √ π (2t) 3 η ϕ2 − ϕe 4t √ dϕ. (3.27) cosh ϕ − cosh η Remark 3.2.2. Sometime, in the literature, instead of equation (3.2), it is considered the equation ∂ 1 ∂t u = 2∆2u, in this case the probability density (3.27) must be replaced by p2(η, t) = t − ∞ e 8 √ √ 3 π t η ϕ2 − ϕe 2t √ dϕ. cosh ϕ − cosh η Remark 3.2.3. From (3.27) also the distribution function of η(t) can be obtained: where ft(ϕ) = Pr{η(t) > η} = = = t − e 4 √ π (2t) 3 t − e 4 √ π (2t) 3 t − 2e 4 √ π (2t) 3 ∞ η ∞ η ∞ η = 2 √ ∞ t − 2e 4 η ∞ dη η ϕ2 − ϕe 4t dϕ ϕe ϕ2 − 4t ϕ2 − ϕe 4t sinh η √ dϕ cosh ϕ − cosh η ϕ η sinh η √ cosh ϕ − cosh η dη cosh ϕ − cosh ηdϕ ft(ϕ) cosh ϕ − cosh ηdϕ, ϕ2 ϕe− 4t √ , t > 0, ϕ > 0, as a function of t, is the distribution of the first-passage 2π(2t) 3 time for the standard Brownian motion, with infinitesimal variance equal to 2, through level ϕ. 3.2.2 Hitting distribution In several papers (see Baldi et al. (1), Matsumoto and Yor (30)) it is considered the diffusion on H 2 associated to the operator L = y2 2 2 ∂ ∂2 + ∂x2 ∂y2 − νy ∂ ∂x − µ − 1 y 2 ∂ ∂y where ν ∈ R and µ > 0. The coefficients µ and ν measure respectively the vertical and the horizontal component of the drift. In view of (3.1) we obtain that, for µ = 1/2 and ν = 0, the operator L coincides with ∆2/2 and in this case we have the standard two dimensional hyperbolic Brownian motion. Unlike ∆2, that is invariant under hyperbolic isometries, the differential operator L is invariant under transformations of the form z → az + b with a > 0 and b ∈ R. The diffusion process {Z(t) = (X(t), Y (t)), t > 0} associated to L and such that Z(0) = z =
37 Hyperbolic Brownian Motion (x, y) ∈ H 2 may be realized as the solution of the stochastic differential equations ⎧ ⎨dX(t) = Y (t)dB1(t) − νY (t)dt, ⎩dY (t) = Y (t)dB2(t) − µ − 1 Y (t)dt, where B1(t) and B2(t) are two independent one-dimensional Brownian motions. Z(t) is represented as follows ⎧ ⎨X(t) = x + y ⎩ t 0 exp{Bµ 2 (s)} dBν 1 (s), Y (t) = y exp{B µ 2 (t)}, where B ν 1 (t) = B1(t)−νt and B µ 2 (t) = B2(t)−µt. Since µ is assumed to be positive, Y (t) converges to 0 as t tends to ∞. It is possible to show that the distribution of the horizontal component X(t) converges weakly as t tends to infinity to a given limiting distribution. Let ha = {z ∈ H2 : Im{z} = a}, with 0 ≤ a < y, be the horocycle through ∞. If a = 0 we have that h0 is the boundary of the hyperbolic half plane H2. The hitting distribution on ha, with a > 0, of the diffusion process associated to L is given by the law of the random variable τa Xτa = x + y exp{B 0 µ 2 (s)} dBν 1 (s) where τa = inf{t ≥ 0 : Y (t) = a} is the first hitting time on ha. With a slight abuse of terminology the law of the random variable is the hitting distribution on h0. Hitting distribution on h0 X∞ = x + y ∞ 0 exp{B µ 2 (s)} dBν 1 (s) Let pz(ξ) be the density of the hitting distribution on h0 for the process associated to the operator L and starting at z = (x, y). We point out that pz(ξ) is the Poisson kernel of L in the domain H 2 and satisfies the following conditions: ⎧ Lpz(ξ) = 0, ∀ξ ∈ h0, ⎪⎨ pz(ξ) > 0, ∀ξ ∈ h0, ∞ −∞ ⎪⎩ pz(ξ)dξ = 1, limy→0 + pz(ξ) = 0, if ξ = x. Since L is invariant under the action of the map z → az + b then so is pz(ξ)dξ on R. That is pz(ξ) dξ = paz+b(aξ + b) d(aξ + b) = a paz+b(aξ + b) dξ. Setting a = 1 x y , b = − y and f(x) = pi(x) we have pz(ξ) = px+iy(ξ) = 1 y pi ξ − x y 2 = 1 y f ξ − x y (3.28) (3.29)
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 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
Page 78 and 79: 4.3 About the higher moments of the
Page 80 and 81: 4.3 About the higher moments of the
Page 82 and 83: 4.4 Motions with jumps backwards to
Page 88 and 89: 4.5 Motion at finite velocity on th
Page 90 and 91: 4.5 Motion at finite velocity on th
Page 92 and 93: 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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