Probability measures on path space for rectilinear damped pressureless Euler-Poisson equations

We show global existence of a measure-valued solution to the Cauchy problem for the rectilinear Euler-Poisson equations, which govern flow of pressureless dust experiencing linear damping in the gravitational field generated by its own mass, and a given background charge field modeling exterior gravitational force. The velocity of the dust is a Borel measurable function, and the distribution of mass is given by time-dependent probability measures on the real line. Upon adapting the method established in Hynd (2020) [25], we obtain the solution by construction of a probability measure on the path space, through approximation of finite sticky particles.