2019/02/11 by Antonio Lerario, Lerario, Antonio, Michele Stecconi +1
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Probability and Statistical Research
paper · pdf · doi:10.48550/arxiv.1902.03805
openalex publication_date 2019/02/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Motivated by numerous questions in random geometry, given a smooth manifold M, we approach a systematic study of the differential topology of Gaussian random fields (GRF) X:M→ ℝk, that we interpret as random variables with values in Cr(M, ℝk), inducing on it a Gaussian measure. When the latter is given the weak Whitney topology, the convergence in law of X allows to compute the limit probability of certain events in terms of the probability distribution of the limit. This is true, in particular, for the events of a geometric or topological nature, like: "X is transverse to W" or "X-1(0) is homeomorphic to Z". We relate the convergence in law of a sequence of GRFs with that of their covariance structures, proving that in the smooth case (r=∞), the two conditions coincide, in analogy with what happens for finite dimensional Gaussian measures. We also show that this is false in the case of finite regularity (r∈ℕ), although the convergence of the covariance structures in the Cr+2 sense is a sufficient condition for the convergence in law of the corresponding GRFs in the Cr sense. We complement this study by proving an important technical tools: an infinite dimensional, probabilistic version of the Thom transversality theorem, which ensures that, under some conditions on the support, the jet of a GRF is almost surely transverse to a given submanifold of the jet space.