RANDOM FIELDS AND THEIR GEOMETRY

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iv Contents 2.5 Orthogonal expansions . . . . . . . . . . . . . . . . . . . . . 77 2.5.1 Karhunen-Loève expansion . . . . . . . . . . . . . . 84 2.6 Majorising measures . . . . . . . . . . . . . . . . . . . . . . 87 3 Geometry 95 3.1 Excursion sets . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.2 Basic integral geometry . . . . . . . . . . . . . . . . . . . . 97 3.3 Excursion sets again . . . . . . . . . . . . . . . . . . . . . . 103 3.4 Intrinsic volumes . . . . . . . . . . . . . . . . . . . . . . . . 112 3.5 Manifolds and tensors . . . . . . . . . . . . . . . . . . . . . 117 3.5.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . 118 3.5.2 Tensors and exterior algebras . . . . . . . . . . . . . 122 3.5.3 Tensor bundles and differential forms . . . . . . . . . 128 3.6 Riemannian manifolds . . . . . . . . . . . . . . . . . . . . . 129 3.6.1 Riemannian metrics . . . . . . . . . . . . . . . . . . 129 3.6.2 Integration of differential forms . . . . . . . . . . . . 133 3.6.3 Curvature tensors and second fundamental forms . . 138 3.6.4 A Euclidean example . . . . . . . . . . . . . . . . . . 142 3.7 Piecewise smooth manifolds . . . . . . . . . . . . . . . . . . 146 3.7.1 Piecewise smooth spaces . . . . . . . . . . . . . . . . 148 3.7.2 Piecewise smooth submanifolds . . . . . . . . . . . . 154 3.8 Intrinsic volumes again . . . . . . . . . . . . . . . . . . . . . 157 3.9 Critical Point Theory . . . . . . . . . . . . . . . . . . . . . 160 3.9.1 Morse theory for piecewise smooth manifolds . . . . 161 3.9.2 The Euclidean case . . . . . . . . . . . . . . . . . . . 166 4 Gaussian random geometry 171 4.1 An expectation meta-theorem . . . . . . . . . . . . . . . . . 172 4.2 Suitable regularity and Morse functions . . . . . . . . . . . 186 4.3 An alternate proof of the meta-theorem . . . . . . . . . . . 190 4.4 Higher moments . . . . . . . . . . . . . . . . . . . . . . . . 191 4.5 Preliminary Gaussian computations . . . . . . . . . . . . . 193 4.6 Mean Euler characteristics: Euclidean case . . . . . . . . . . 197 4.7 The meta-theorem on manifolds . . . . . . . . . . . . . . . . 208 4.8 Riemannian structure induced by Gaussian fields . . . . . . 213 4.8.1 Connections and curvatures . . . . . . . . . . . . . . 214 4.8.2 Some covariances . . . . . . . . . . . . . . . . . . . . 216 4.8.3 Gaussian fields on R N . . . . . . . . . . . . . . . . . 218 4.9 Another Gaussian computation . . . . . . . . . . . . . . . . 220 4.10 Mean Euler characteristics: Manifolds . . . . . . . . . . . . 222 4.10.1 Manifolds without boundary . . . . . . . . . . . . . 223 4.10.2 Manifolds with boundary . . . . . . . . . . . . . . . 225 4.11 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 4.12 Chern-Gauss-Bonnet Theorem . . . . . . . . . . . . . . . . 235
0 Contents 5 Non-Gaussian geometry 237 5.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 5.2 Conditional expectations of double forms . . . . . . . . . . 237 6 Suprema distributions 243 6.1 The basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 6.2 Some Easy Bounds . . . . . . . . . . . . . . . . . . . . . . . 246 6.3 Processes with a Unique Point of Maximal Variance . . . . 248 6.4 General Bounds . . . . . . . . . . . . . . . . . . . . . . . . . 253 6.5 Local maxima . . . . . . . . . . . . . . . . . . . . . . . . . . 257 6.6 Local maxima above a level . . . . . . . . . . . . . . . . . . 258 7 Volume of tubes 263 7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 264 7.2 Volume of tubes for finite Karhunen-Loève Gaussian processes264 7.2.1 Local geometry of Tube(M, ρ) . . . . . . . . . . . . . 266 7.3 Computing F ∗ j,r (Ωr) . . . . . . . . . . . . . . . . . . . . . . 269 7.3.1 Case 1: � M = R l . . . . . . . . . . . . . . . . . . . . . 270 7.3.2 Case 2: � M = Sλ(R l ) . . . . . . . . . . . . . . . . . . 276 7.3.3 Volume of tubes for finite Karhunen-Loève Gaussian processes revisited . . . . . . . . . . . . . . . . . . . 278 7.4 Generalized Lipschitz-Killingcurvature measures . . . . . . 280 References 281
Page 1: RANDOM FIELDS AND THEIR GEOMETRY Ro
Page 6 and 7: 1 Random fields 1.1 Random fields a
Page 8 and 9: 1.2 Gaussian fields 3 • In applic
Page 10 and 11: 1.2 Gaussian fields 5 A judicious c
Page 12 and 13: 1.3 The Brownian family of processe
Page 14 and 15: 1.3 The Brownian family of processe
Page 16 and 17: 1.4 Stationarity 11 Then Aω and B
Page 18 and 19: 1.4 Stationarity 13 operation +, if
Page 20 and 21: 1.4 Stationarity 15 L 2 (P) ≡ L 2
Page 22 and 23: 1.4 Stationarity 17 additional defi
Page 24 and 25: s ∈ R N , set θ(g, t)(s) = As +
Page 26 and 27: 1.4 Stationarity 21 and the terms i
Page 28 and 29: W1 and W2, such that 26 (1.4.24) ft
Page 30 and 31: moments, when they exist, are zero;
Page 32 and 33: 1.4.5 Constant variance 1.4 Station
Page 34 and 35: 1.4 Stationarity 29 ∆ where δij
Page 36 and 37: Now use the spectral decomposition
Page 38 and 39: 1.4 Stationarity 33 If T = R N unde
Page 40 and 41: 1.5 Non-Gaussian fields 35 know the
Page 42 and 43: 1.5 Non-Gaussian fields 37 In other
Page 44 and 45: 2 Gaussian fields The aim of this C
Page 46 and 47: 2.1 Boundedness and continuity 41 b
Page 48 and 49: 2.1 Boundedness and continuity 43 T
Page 50 and 51: should be taken as the definition o
Page 52 and 53: 2.1 Boundedness and continuity 47 w
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2.2 Examples 2.2.1 Fields on R N 2.
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2.2 Examples 51 for |t| small enoug
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To see why, endow the space R N ×
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2.2 Examples 55 The proof requires
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canonical metric d, where (2.2.15)
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2.2 Examples 59 Similarly, again re
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of a measure µ on RN , then by ana
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2.2 Examples 63 These inequalities
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2.2 Examples 65 Fix ε > 0 and supp
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2.3 Borell-TIS inequality 2.3 Borel
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2.3 Borell-TIS inequality 69 But th
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2.3 Borell-TIS inequality 71 the se
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2.3 Borell-TIS inequality 73 so tha
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2.4 Comparison inequalities 75 g, a
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2.5 Orthogonal expansions 77 which
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versa. Start with � S = u : T →
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2.5 Orthogonal expansions 81 With S
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2.5 Orthogonal expansions 83 where
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2.5 Orthogonal expansions 85 where
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2.6 Majorising measures 87 is alway
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2.6 Majorising measures 89 replacin
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2.6 Majorising measures 91 This Lem
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Finally, set Λ = min(Dµ, Dν). Th
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3 Geometry For this Chapter we are
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3.2 Basic integral geometry 97 fiel
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3.2 Basic integral geometry 99 Now
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and define (3.2.9) ϕ(A) = � h(A,
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3.3 Excursion sets again 103 drille
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espect to T at the level u. (3.3.4)
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3.3 Excursion sets again 107 this p
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3.3 Excursion sets again 109 will c
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3.3 Excursion sets again 111 FIGURE
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3.4 Intrinsic volumes 113 suitable
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3.4 Intrinsic volumes 115 Comparing
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functional ˜ ψu for which � �
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3.5 Manifolds and tensors 119 If a
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3.5 Manifolds and tensors 121 for a
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3.5 Manifolds and tensors 123 up th
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y (3.5.8) α ∧ β = (r + s)! A(α
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3.5 Manifolds and tensors 127 and t
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3.6 Riemannian manifolds 129 Simila
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3.6 Riemannian manifolds 131 The ba
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3.6 Riemannian manifolds 133 If θ
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3.6 Riemannian manifolds 135 Given
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3.6 Riemannian manifolds 137 manifo
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3.6 Riemannian manifolds 139 FIGURE
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3.6 Riemannian manifolds 141 and we
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3.6 Riemannian manifolds 143 With t
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3.6 Riemannian manifolds 145 We are
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3.7 Piecewise smooth manifolds 147
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and the homeomorphisms ψt are such
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3.8 Intrinsic volumes again 157 We
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3.8 Intrinsic volumes again 159 to
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3.9 Critical Point Theory 161 C 2 f
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� k 3.9 Critical Point Theory 163
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3.9 Critical Point Theory 165 What
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3.9 Critical Point Theory 167 [0, 1
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3.9 Critical Point Theory 169 and t
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4 Gaussian random geometry With the
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4.1 An expectation meta-theorem 173
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have 4.1 An expectation meta-theore
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4.2 Suitable regularity and Morse f
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our manifold as usual as (4.2.2) 4.
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4.4 Higher moments 191 Take α(t) =
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4.5 Preliminary Gaussian computatio
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implying 4.5 Preliminary Gaussian c
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4.6 Mean Euler characteristics: Euc
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all out again. We start with 4.6 Me
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4.7 The meta-theorem on manifolds 2
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4.7 The meta-theorem on manifolds 2
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4.8 Riemannian structure induced by
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4.9 Another Gaussian computation 22
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4.10.1 Manifolds without boundary 4
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4.10 Mean Euler characteristics: Ma
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4.11 Examples 231 Stationary fields
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4.11 Examples 233 its own curvature
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4.12 Chern-Gauss-Bonnet Theorem 4.1
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References [1] R. J. Adler. The Geo
Page 244 and 245:
References 283 [25] R. M. Dudley. L
Page 246 and 247:
References 285 [58] John M. Lee. In
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References 287 [88] D. Slepian. On
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RANDOM FIELDS AND THEIR GEOMETRY

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