A particle system is a family of i.i.d. stochastic processes with values translated by Poisson points. We obtain conditions that ensure the stationarity in time of the particle system in R and in some cases provide a full characterisation of the stationarity property. In particular, a full characterisation of stationary multivariate Brown–Resnick processes is given.

where G(u) is a bounded, nondecreasing function with G(— °o)=0 and where 7 is a real-valued constant. Below it is shown that for certain processes of this type the measure of the Hilbert neighborhood of the origin is related to the solution of a certain differential system. In fact, (A) if {x(t), 0fkt< °° } is a separable stochastic process with symmetric, stationary, and independent increments...

We study existence and uniqueness of the invariant measure for a stochastic process with degenerate diffusion, whose infinitesimal generator is a linear subelliptic operator in the whole space R with coefficients that may be unbounded. Such a measure together with a Liouville-type theorem will play a crucial role in two applications: the ergodic problem studied through stationary problems with ...

We prove a functional central limit theorem for integrals ∫ W f(X(t)) dt, where (X(t))t∈Rd is a stationary mixing random field and the stochastic process is indexed by the function f , as the integration domain W grows in Van Hove-sense. We discuss properties of the covariance function of the asymptotic Gaussian process.

We are going to give necessary and sufficient conditions for a multivari-ate stationary stochastic process to be completely regular. We also give the answer to a question of V.V. Peller concerning the spectral measure characterization of such processes.