By Robert J. Adler, Jonathan E. Taylor (auth.)
These notes, according to lectures introduced in Saint Flour, offer a simple creation to the authors’ 2007 Springer monograph “Random Fields and Geometry.” whereas now not as exhaustive because the complete monograph, also they are much less arduous, whereas nonetheless masking the elemental fabric, regularly at a extra intuitive and no more technical point. in addition they conceal a few newer fabric in relation to random algebraic topology and statistical functions. The notes comprise an creation to the final thought of Gaussian random fields, treating classical issues equivalent to continuity and boundedness. this is often by means of a short assessment of geometry, either fundamental and Riemannian, with an emphasis on tube formulae, to supply the reader with the fabric had to comprehend and use the Gaussian kinematic formulation, the most results of the notes. this is often through chapters on topological inference and random algebraic topology, either one of which offer purposes of the most results.
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Extra info for Topological Complexity of Smooth Random Functions: École d'Été de Probabilités de Saint-Flour XXXIX-2009
Example text
In the non-stationary case there is no such simple transformation available. However, there is a trick, based on Riemannian geometry, that allows one to change the Riemannian structure of the parameter space by introducing a Riemannian metric related to the covariance function that makes all first order Riemannian derivatives uncorrelated. It was this trick that, in many 34 2 Gaussian Processes ways, was one of the most important themes of RFG, and is what allows one to move from a theory of stationary random fields on subsets of RN to nonstationary fields on stratified manifolds.
As for Li (M ), we define the one parameter family of Lipschitz-Killing curvatures, or intrinsic volumes, Δ Lκi (M ) = Lκi (M, M ). 13), there is a simplification. 20) disappear. The result is Lκi (M, A) = N (2π)−(j−i)/2 C(N − 1 − j, j − i) j=i × ∂j M∩A S(Tt ∂j M ⊥ ) 1 TrTt ∂j M (Sηj−i ) (j − i)! × 11Nt M (−η)HN −j−2 (dη). 20) gives Lκi (·) ∞ = (−κ)n (i + 2n)! i! 23) and ∞ Li (·) = κn (i + 2n)! κ Li+2n (·). i! 24) Here now is Weyl’s tube formula on Sλ (Rl ). 2 (Weyl’s tube formula on Sλ (Rl )). Suppose M ⊂ Sλ (Rl ) is a regular stratified manifold.
The collection of all linear spaces in RN that do not necessarily pass through the origin. Noting that the affine Grassmanian is diffeomorphic to Gr(N, k)×RN , N it has a natural measure, λN k say, which factors as Haar measure νk on the Grassmanian Gr(N, k) of all linear subspaces of RN (which must pass through the origin) and Lebesque measure on RN . We normalize νkN so that νkN (Gr(N, k)) = N . k Crofton’s formula then states that, for a regular stratified manifold M ∈ RN , Graff(N,N −k) Lj (M ∩ V ) dλN N −k (V ) = k+j Lk+j (M ).