Differential Topology of Gaussian Random Fields

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) $X:M\to \mathbb{R}^k$, that we interpret as random variables with values in $\mathcal{C}^r(M, \mathbb{R}^k)$, 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 ($r=\infty$), 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 ($r\in\mathbb{N}$), although the convergence of the covariance structures in the $\mathcal{C}^{r+2}$ sense is a sufficient condition for the convergence in law of the corresponding GRFs in the $\mathcal{C}^r$ 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.