Surfaces aléatoires. Mesure géométrique des ensembles de niveau
This monograph is devoted to some problems connected with geometrical measure of level sets of d-dimensional stochastic processes \(\{\) X(t): \(t\in R^ d\}\). In the 1st chapter the author considers the perimeters of Borel sets in \(R^ d:\) \(Q_ O(B)=\sup \{\int_{B}div(v(t))dt: v\in (C_ k^{\infty}(O))^ d,\| v\| \leq 1,t\in R^ d\}\) where \(v:R^ d\to R^ d\), v has compact support contained in O. He gives upper and lower bounds for \(Q_ O(B)\), proves sub-\(\sigma\)-additivity of \(Q_ O(\cdot)\) and gives Green's formula for it: \(Q_ O(B)=\sigma_{d- 1}(\partial B\cap O)\), where B is a differentiable manifold in \(R^ d\) with the bound \(\partial B\), \(\sigma_{d-1}\) is the usual measure on \(\partial B\cap O\). The text is close to the book of \textit{M. Miranda}, ''Frontiere minime'', Mon. Mat., No.27, IMPA (1976). The 2nd chapter contains the conditions under which the random set \(E^ x_ u=\{t: X_ t=u\), x(\(\cdot)\) has local extremum at \(t\}\) is empty a.s. The results concern Gaussian processes and processes with C- differentiable trajectories and are due to Yulvisaker, Bulinskaia, Belyayev and the author. The 3rd chapter is devoted to generalisations of the Rice formula for the moments of r.v. \(Q_ T(A^ x_ u)\), where \(A^ x_ u=\{t: X(t)<u\}\). The cases \(d=1\) and \(d>1\) are considered separately. In the 4th chapter the author gives an application of the Rice formula for approximating the local times of the Wiener process.
scientific article; zbMATH DE number 3915304