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We derive a covariance formula for the class of `topological events of smooth Gaussian fields on manifolds; these are events that depend only on the topology of the level sets of the field, for example (i) crossing events for level or excursion sets, (ii) events measurable with respect to the number of connected components of level or excursion sets of a given diffeomorphism class, and (iii) persistence events. As an application of the covariance formula, we derive strong mixing bounds for topological events, as well as lower concentration inequalities for additive topological functionals (e.g. the number of connected components) of the level sets that satisfy a law of large numbers. The covariance formula also gives an alternate justification of the Harris criterion, which conjecturally describes the boundary of the percolation university class for level sets of stationary Gaussian fields. Our work is inspired by a recent paper by Rivera and Vanneuville, in which a correlation inequality was derived for certain topological events on the plane, as well as by an old result of Piterbarg, in which a similar covariance formula was established for finite-dimensional Gaussian vectors.
Two types of Gaussian processes, namely the Gaussian field with generalized Cauchy covariance (GFGCC) and the Gaussian sheet with generalized Cauchy covariance (GSGCC) are considered. Some of the basic properties and the asymptotic properties of the
We study the small deviation probabilities of a family of very smooth self-similar Gaussian processes. The canonical process from the family has the same scaling property as standard Brownian motion and plays an important role in the study of zeros o
For the Bargmann-Fock field on R d with d $ge$ 3, we prove that the critical level c (d) of the percolation model formed by the excursion sets {f $ge$ } is strictly positive. This implies that for every sufficiently close to 0 (in particular for the
We consider Gaussian measures $mu, tilde{mu}$ on a separable Hilbert space, with fractional-order covariance operators $A^{-2beta}$ resp. $tilde{A}^{-2tilde{beta}}$, and derive necessary and sufficient conditions on $A, tilde{A}$ and $beta, tilde{bet
We use Yosida approximation to find an It^o formula for mild solutions $left{X^x(t), tgeq 0right}$ of SPDEs with Gaussian and non-Gaussian coloured noise, the non Gaussian noise being defined through compensated Poisson random measure associated to a