Mass-stationarity means that the origin is at a typical location in the mass of a random measure. For a simple example, consider the stationary Poisson process on the line conditioned on having a point at the origin. The origin is then at a typical point (at a typical location in the mass) because shifting the origin to the n:th point on the right (or on the left) does not alter the fact that the inter-point distances are i.i.d. exponential. Another (less obvious) example is the local time at zero of a two-sided standard Brownian motion.

In this talk we will concentrate on mass-stationarity on the line with Brownian motion as the main example. If time allows we will briefly extend the view beyond the line, moving through the Poisson process in the plane towards general random measures on groups.