We propose and analyze a natural extension of the Moreau sweeping process: given a family of moving convex sets (C(t))t, we look for the evolution of a probability density Pt, constrained to be supported on C(t). We describe in detail three cases: in the first, particles do not interact with each other and stay at rest unless pushed by the moving boundary; in the second they interact via a maximal density constraint p ≤ 1, so that they are not only pushed by the boundary, but also by the other particles; in the thitd cese i phrtihles areesub itted to Brownian diffusion, reflected along the moving boundary. We prove existence, uniqueness and approximation results by using techniques from optimal transport, and we provide numerical illustrations.
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