Random fields and their geometry

Random fields and excursion sets......Page 6
Gaussian fields......Page 7
The Brownian family of processes......Page 11
Stationarity......Page 16
Stochastic integration......Page 18
Moving averages......Page 21
Spectral representations on RN......Page 24
Spectral moments......Page 29
Isotropy......Page 32
Stationarity over groups......Page 37
Non-Gaussian fields......Page 39
Gaussian fields......Page 44
Boundedness and continuity......Page 45
Fields on RN......Page 54
Differentiability on RN......Page 57
Generalised fields......Page 59
Set indexed processes......Page 66
Non-Gaussian processes......Page 71
Borell-TIS inequality......Page 72
Comparison inequalities......Page 79
Orthogonal expansions......Page 82
Karhunen-Loève  expansion......Page 89
Majorising measures......Page 92
Excursion sets......Page 100
Basic integral geometry......Page 102
Excursion sets again......Page 108
Intrinsic volumes......Page 117
Manifolds and tensors......Page 122
Manifolds......Page 123
Tensors and exterior algebras......Page 127
Tensor bundles and differential forms......Page 133
Riemannian metrics......Page 134
Integration of differential forms......Page 138
Curvature tensors and second fundamental forms......Page 143
A Euclidean example......Page 147
Piecewise smooth manifolds......Page 151
Piecewise smooth spaces......Page 153
Piecewise smooth submanifolds......Page 159
Intrinsic volumes again......Page 162
Critical Point Theory......Page 165
Morse theory for piecewise smooth manifolds......Page 166
The Euclidean case......Page 171
Gaussian random geometry......Page 176
An expectation meta-theorem......Page 177
Suitable regularity and Morse functions......Page 191
An alternate proof of the meta-theorem......Page 195
Higher moments......Page 196
Preliminary Gaussian computations......Page 198
Mean Euler characteristics: Euclidean case......Page 202
The meta-theorem on manifolds......Page 213
Riemannian structure induced by Gaussian fields......Page 218
Connections and curvatures......Page 219
Some covariances......Page 221
Gaussian fields on RN......Page 223
Another Gaussian computation......Page 225
Mean Euler characteristics: Manifolds......Page 227
Manifolds without boundary......Page 228
Manifolds with boundary......Page 230
Examples......Page 235
Chern-Gauss-Bonnet Theorem......Page 240
References......Page 242