RANDOM FIELDS AND THEIR GEOMETRY

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1.4 Stationarity 13 operation +, if its finite dimensional distributions are invariant under this operation. That is, for any k ≥ 1 and any set of points τ, t1, . . . , tk ∈ T (1.4.2) (f(t1), . . . , f(tk)) L = (f(t1 + τ), . . . , f(tk + τ)) , where L = indicates equivalence in distribution (law). An immediate consequence of strict stationarity is that m(t) is constant and C(s, t) is a function of the difference s − t only. Going the other way, if for a random field with E{�f(t)� 2 } < ∞ for all t ∈ T , we have that m(t) is constant and C(s, t) is a function of the difference s − t only, then we call f simply stationary or homogeneous, occassionally adding either of the two adjectives ‘weakly’ or ‘second-order’. Note that none of the above required the Gaussianity we have assumed up until now on f. If, however, we do add the assumption of Gaussianity, it immediately follows from the structure (1.2.3) of the multivariate Gaussian density 15 that a weakly stationary Gaussian field will also be strictly stationary if C ′ (s, t) = E{[f(s)−m(s)] [f(t)−m(t)]} is also a function only of s − t. If f is real valued, then since C ≡ C ′ it follows that all weakly stationary real valued Gaussian fields are also strictly stationary and the issue of qualifying adjectives is moot 16 . If T is not Abelian we must distinguish between left and right stationary. We say that a random field f on T is right-stationary if (1.4.2) holds and that f is left-stationary if f ′ (t) ∆ = f(−t) is right-stationary. The corresponding conditions on the covariance function change accordingly. In order to build examples of stationary processes, we need to make a brief excursion into (Gaussian) stochastic integration. 1.4.1 Stochastic integration We return to the setting of Section 1.3, so that we have a σ-finite 17 measure space (T, T ,ν), along with the Gaussian ν-noise W defined over T . Our aim 15 Formally, (1.2.3) is not quite enough. Since we are currently treating complex valued processes, the k-dimensional marginal distributions of f now involve 2k-dimensional Gaussian vectors. (The real and imaginary parts each require k dimensions.) 16 The reason for needing the additional condition in the complex case should be intuitively clear: C alone will not even determine the variance of each of fI and fR but only their sum. However, together with C ′ both of these parameters, as well as the covariance between fI and fR, can be computed. See [69] for further details. 17 Note that ‘σ-finite’ includes ‘countable’ and ‘finite’, in which case ν will be a discrete measure and all of the theory we are about to develop is really much simpler. As we shall see later, these are more than just important special cases and so should not be forgotten as you read this Section. Sometimes it it is all too easy to lose sight of the important simple cases in the generality of the treatment.
14 1. Random fields will be to establish the existence of integrals of the form � (1.4.3) f(t) W (dt), T for deterministic f ∈ L 2 (ν) and, eventually, complex W . Appropriate choices of f will give us examples of stationary Gaussian fields, many of which we shall meet in the following Subsections. Before starting, it is only fair to note that we shall be working with two and a bit assumptions. The ‘bit’ is that for the moment we shall only treat real f and W . We shall, painlessly, lift that assumption soon. Of the other two, one is rather restrictive and one not, but neither of importance to us. The non-restrictive assumption is the Gaussian nature of the process W . Indeed, since all of what follows is based only on L 2 theory, we can, and so shall, temporarily drop the assumption that W is a Gaussian noise and replace conditions (1.3.1)–(1.3.3) with the following three requirements for all A, B ∈ T . (1.4.4) (1.4.5) (1.4.6) E{W (A)} = 0, E{W (A)} 2 = ν(A). A ∩ B = ∅ ⇒ W (A ∪ B) = W (A) + W (B) a.s. A ∩ B = ∅ ⇒ E{W (A)W (B)} = 0. Note that in the Gaussian case (1.4.6) is really equivalent to the seemingly stronger (1.3.3), since zero covariance and independence are then equivalent. The second restriction is that the integrand f in (1.4.3) is deterministic. Removing this assumption would lead us to having to define the Itô integral which is a construction for which we shall have no need. Since, by (1.4.5), W is a finitely additive (signed) measure, (1.4.3) is evocative of Lebesgue integration. Consequently, we start by defining the the stochastic version for simple functions (1.4.7) f(t) = n� ai�Ai(t), where A1, . . . , An are disjoint, T measurable sets in T , by writing (1.4.8) W (f) ≡ � T 1 f(t) W (dt) ∆ = n� ai W (Ai). It follows immediately from (1.4.4) and (1.4.6) that in this case W (f) has zero mean and variance given by � a 2 i ν(Ai). Think of W (f) as a mapping from simple functions in L 2 (T, T , ν) to random variables 18 in 18 Note that if W is Gaussian, then so is W (f). 1
Page 1: RANDOM FIELDS AND THEIR GEOMETRY Ro
Page 4 and 5: iv Contents 2.5 Orthogonal expansio
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 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
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Page 54 and 55: 2.2 Examples 2.2.1 Fields on R N 2.
Page 56 and 57: 2.2 Examples 51 for |t| small enoug
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Page 60 and 61: 2.2 Examples 55 The proof requires
Page 62 and 63: canonical metric d, where (2.2.15)
Page 64 and 65: 2.2 Examples 59 Similarly, again re
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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
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References 283 [25] R. M. Dudley. L
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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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