### Convex hulls, Sticky particle dynamics and Pressure-less gas system

We introduce a new condition which extends the definition of sticky particle dynamics to the case of discontinuous initial velocities ${u}_{0}$ with negative jumps. We show the existence of a stochastic process and a forward flow $\phi $ satisfying ${X}_{s+t}=\phi ({X}_{s},t,{P}_{s},{u}_{s})$ and $\mathrm{d}{X}_{t}=\mathrm{E}[{u}_{0}\left({X}_{0}\right)/{X}_{t}]\mathrm{d}t$, where ${P}_{s}=P{X}_{s}^{-1}$ is the law of ${X}_{s}$ and ${u}_{s}\left(x\right)=\mathrm{E}[{u}_{0}\left({X}_{0}\right)/{X}_{s}=x]$ is the velocity of particle $x$ at time $s\ge 0$. Results on the flow characterization and Lipschitz continuity are also given.Moreover, the map $(x,t)\mapsto M(x,t):=P({X}_{t}\le x)$ is the entropy solution of a scalar conservation law ${\partial}_{t}M+{\partial}_{x}\left(A\left(M\right)\right)=0$ where the flux $A$ represents the particles...