ترغب بنشر مسار تعليمي؟ اضغط هنا

A covariance formula for topological events of smooth Gaussian fields

103   0   0.0 ( 0 )
 نشر من قبل Stephen Muirhead
 تاريخ النشر 2018
  مجال البحث
والبحث باللغة English




اسأل ChatGPT حول البحث

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.



قيم البحث

اقرأ أيضاً

128 - S. C.Lim , L. P. Teo 2008
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 spectral densities of these random fields are studied. The associated self-similar random fields obtained by applying the Lamperti transformation to GFGCC and GSGCC are studied.
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 f random polynomials. Our estimates are based on the entropy method, discovered in Kuelbs and Li (1992) and developed further in Li and Linde (1999), Gao (2004), and Aurzada et al. (2009). While there are several ways to obtain the result w.r.t. the $L_2$ norm, the main contribution of this paper concerns the result w.r.t. the supremum norm. In this connection, we develop a tool that allows to translate upper estimates for the entropy of an operator mapping into $L_2[0,1]$ by those of the operator mapping into $C[0,1]$, if the image of the operator is in fact a Holder space. The results are further applied to the entropy of function classes, generalizing results of Gao et al. (2010).
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 nodal hypersurfaces corresponding to the case = 0), {f = } contains an unbounded connected component that visits most of the ambient space. Our findings actually hold for a more general class of positively correlated smooth Gaussian fields with rapid decay of correlations. The results of this paper show that the behaviour of nodal hypersurfaces of these Gaussian fields in R d for d $ge$ 3 is very different from the behaviour of nodal lines of their two-dimensional analogues. Contents
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 $beta, tilde{bet a} > 0$ for I. equivalence of the measures $mu$ and $tilde{mu}$, and II. uniform asymptotic optimality of linear predictions for $mu$ based on the misspecified measure $tilde{mu}$. These results hold, e.g., for Gaussian processes on compact metric spaces. As an important special case, we consider the class of generalized Whittle-Matern Gaussian random fields, where $A$ and $tilde{A}$ are elliptic second-order differential operators, formulated on a bounded Euclidean domain $mathcal{D}subsetmathbb{R}^d$ and augmented with homogeneous Dirichlet boundary conditions. Our outcomes explain why the predictive performances of stationary and non-stationary models in spatial statistics often are comparable, and provide a crucial first step in deriving consistency results for parameter estimation of generalized Whittle-Matern fields.
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 Levy process. The functions to which we apply such It^o formula are in $C^{1,2}([0,T]times H)$, as in the case considered for SDEs in [9]. Using this It^o formula we prove exponential stability and exponential ultimate boundedness properties in mean square sense for mild solutions. We also compare such It^o formula to an It^o formula for mild solutions introduced by Ichikawa in [8], and an It^o formula written in terms of the semigroup of the drift operator [11] which we extend before to the non Gaussian case.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا