Continuity of sample paths of Gaussian Markov processes

Abstract

This chapter deals with almost sure continuity of sample paths of multi-dimensional Gaussian Markov processes. The well-known Kinney-Dynkin criterion of continuity of Markov processes is complemented by an entropy criterion which gives evidence that the necessary conditions of continuity of Gaussian processes due to V.Sudakov are sufficient for continuity of Gaussian Markov processes. This problem is considered in Section 6.4 which logically accomplishes the investigation started in Section 5.2. In Section 6.3, the link is established between the points of discontinuity of sample paths of a multi-dimensional Gaussian Markov process and the points where the rank of variance matrice of this process varies. The statements of Sections 6.3 and 6.4 dwell on the claim on equivalence of sequential and sample almost sure continuity of Gaussian processes. This claim complements the well-known oscillation theorem due to K.Ito and M.Nisio and reduces the problem of sample continuity of almost all sample paths of Gaussian processes to that of the almost sure convergence to zero of Gaussian sequences. Even though the statement on reduction is proved under fairly general assumptions on the parametric set and the range of values of the processes considered, the simplest situations in applications already prove the utility of this statement.