The Kolmogorov–Čentsov theorem and Brownian motion in vector lattices

The well known Kolmogorov–Čentsov theorem is proved in a Dedekind complete vector lattice (Riesz space) with weak order unit on which a strictly positive conditional expectation is defined. It gives conditions that guarantee the Hölder-continuity of a stochastic process in the space. We discuss the notion of independence of projections and elements in the vector lattice and use this together with the Kolmogorov–Čentsov theorem to give an abstract definition of Brownian motion in a vector lattice. This definition captures the fact that the increments in a Brownian motion are normally distributed and that the paths are continuous.