One of the most important concepts of relativistic physics is that of a
world line, i.e., the space-time trajectory of a point particle. In
Lorentzian geometry, it is modelled by the so-called causal curve, and
the questions concerning which spacetime points can be connected by
means of causal curves lead to a vast area of study known as causality
theory. In the talk, I will present how the basic notions of causality
theory can be naturally extended to probability measures on spacetimes
(what is motivated by both classical and quantum physics). In
particular, I will discuss the notion of a causal evolution of
probability measures, its deep connection with the continuity equation
(known from elementary physics) and a surprisingly nice topological
properties of the space of causal curves.