RANDOM FIELDS AND THEIR GEOMETRY

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1.2 Gaussian fields 5 A judicious choice of A then allows us to compute conditional distributions as well. If n < d make the partitions X = � X 1 , X 2� = ((X1, . . . , Xn), (Xn+1, . . . Xd)) , m = � m 1 , m 2� = ((m1, . . . , mn), (mn+1, . . . md)), � C11 C12 C = C21 C22 where C11 is an n × n matrix. Then each X i is N(m i , Cii) and the conditional distribution 7 of X i given X j is also Gaussian, with mean vector (1.2.7) and covariance matrix (1.2.8) � , m i|j = m i + CijC −1 jj (Xj − m j ) ′ C i|j = Cii − CijC −1 jj Cji. We can now define a real valued Gaussian random field to be a random field f on a parameter set T for which the (finite dimensional) distributions of (ft1, . . . , ftn) are multivariate Gaussian for each 1 ≤ n < ∞ and each (t1, . . . , tn) ∈ T n . The functions m(t) = E{f(t)} and C(s, t) = E {(fs − ms) (ft − mt)} are called the mean and covariance functions of f. Multivariate 8 Gaussian fields taking values in R d are fields for which 〈α, f ′ t〉 is a real valued Gaussian field for every α ∈ R d . In fact, one can also go in the other direction as well. Given any set T , a function m : T → R, and a non-negative definite function C : T × T → R there exists 9 a Gaussian process on T with mean function m and covariance function C. Putting all this together, we have the important principle that, for a Gaussian process, everything about it is determined by the mean and covariance functions. The fact that no real structure is required of the parameter 7 To prove this, take A = „ �n −C12C22 0 �d−n and define Y = (Y 1 , Y 2 ) = AX. Check using (1.2.6) that Y 1 and Y 2 ≡ X 2 are independent and use this to obtain (1.2.7) and (1.2.8) for i = 1, j = 2. 8 Similary, Gaussian fields taking values in a Banach space B are fields for which α(ft) is a real valued Gaussian field for every α in the topological dual B ∗ of B. The covariance function is then replaced by a family of operators Cst : B ∗ → B, for which Cov(α(ft), β(fs)) = β(Cstα), for α, β ∈ B ∗ . 9 This is a consequence of the Kolmogorov existence theorem, which, at this level of generality, can be found in Dudley [27]. Such a process is a random variable in R T and may have terrible properties, including lack of measurability in t. However, it will always exist. «
6 1. Random fields space T is what makes Gaussian fields such a useful model for random processes on general spaces. To build an appreciation for this, we need to look at some examples. The following Section looks at entire class of examples that are generated from something called ‘Gaussian noise’, that we shall also exploit in Section 1.4 to develop the notion of stationarity of random fields. Many more examples can be found in Chapter 2, which looks at Gaussian fields and their properties in far more depth. 1.3 The Brownian family of processes Perhaps the most basic of all random fields is a collection of independent Gaussian random variables. While it is simple to construct such random fields for finite and even countable parameter sets, deep technical difficulties obstruct the construction for uncountable parameter sets. The path that we shall take around these difficulties involves the introduction of random measures which, at least in the Gaussian case, are straightforward to formulate. Let (T, T ,ν) be a σ-finite measure space and denote by Tν the collection of sets of T of finite ν measure. A Gaussian noise based on ν, or ‘Gaussian ν-noise’ is a random field W : Tν → R such that, for all A, B ∈ Tν, (1.3.1) (1.3.2) (1.3.3) W (A) ∼ N(0, ν(A)). A ∩ B = ∅ ⇒ W (A ∪ B) = W (A) + W (B) a.s. A ∩ B = ∅ ⇒ W (A) and W (B) are independent. Property (1.3.2) encourages one to think of W 10 as a random (signed) measure, although it is not generally σ-finite. We describe (1.3.3) by saying that W has independent increments. Theorem 1.3.1 If (T, T , ν) is a measure space, then there exists a real valued Gaussian noise, defined for all A ∈ Tν, satisfying (1.3.1)–(1.3.3). Proof. In view of the closing remarks of the preceeding Section all we need do is provide an appropriate covariance function on Tν × Tν. Try (1.3.4) Cν(A, B) ∆ = ν(A ∩ B). 10 While the notation ‘W ’ is inconsistent our determination to use lower case Latin characters for random functions, we retain it as a tribute to Norbert Wiener, who is the mathematical father of these processes.
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 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
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Page 54 and 55: 2.2 Examples 2.2.1 Fields on R N 2.
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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
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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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