The Kolmogorov–Chentsov Theorem and Sample Path Regularity

Abstract

While constructions of probability distributions of stochastic processes indexed by uncountable parameter spaces, e.g., intervals, can be readily achieved via the Kolmogorov extension theorem, the regularity of the sample paths is not mathematically accessible in such constructions, for the simple reason that sample path properties that depend on uncountably many time points, e.g., continuity, do not define measurable events in the Kolmogorov model. In this chapter we consider a criterion, also due to Kolmogorov (Published in Slutsky (Giorn Ist Ital Attuari 8:183–199, 1937).) and to Chentsov (Chentsov (Theor Probab Appl 1:140–144, 1956)) for Holder continuous versions of processes and random fields. In addition to providing a tool for construction of k-dimensional Brownian motion, it yields the construction of continuous random fields such as the Brownian sheet. The chapter concludes with a demonstration of nowhere differentiability of the (continuous) Brownian paths.