Causal evolution of probability measures and continuity equation

We study the notion of a causal time-evolution of a conserved nonlocal physical quantity in a globally hyperbolic spacetime $\mathcal{M}$. The role of the `global time' is played by a chosen Cauchy temporal function $\mathcal{T}$, whereas the instantaneous configurations of the nonlocal quantity are modeled by probability measures $\mu_t$ supported on the corresponding time slices $\mathcal{T}^{-1}(t)$. We show that the causality of such an evolution can be expressed in three equivalent ways: (i) via the causal precedence relation $\preceq$ extended to probability measures, (ii) with the help of a probability measure $\sigma$ on the space of future-directed continuous causal curves endowed with the compact-open topology and (iii) through a causal vector field $X$ of $L^\infty_{\textrm{loc}}$-regularity, with which the map $t \mapsto \mu_t$ satisfies the continuity equation in the distributional sense. In the course of the proof we find that the compact-open topology is sensitive to the differential properties of the causal curves, being equal to the topology induced from a suitable $H^1_{\textrm{loc}}$-Sobolev space. This enables us to construct $X$ as a vector field in a sense `tangent' to $\sigma$. In addition, we discuss the general covariance of descriptions (i)-(iii), unraveling the geometrical, observer-independent notions behind them.