On the Integrated Form of the BBGKY Hierarchy for Hard Spheres

In my book ``Large Scale Dynamics of Interacting Particles'' [S] I refer to an unpublished note from early 1985 on the BBGKY hierarchy for hard spheres. My main point there was to provide a direct probabilistic proof for the time-integrated version of the hierarchy. Over recent years there has been repeated interest in this derivation, which encourages me to make my note public. I decided to leave it in its original form including likely inaccuracies. The work of R. Illner and M. Pulvirenti [IP] appeared in September 1985, see also the book by C. Cercignani, R. Illner, and M. Pulvirenti [CIP], who prove the same result using special flow representation and methods from the theory of differential operators. [S] H. Spohn, Large Scale Dynamics of Interacting Particles, Texts and Monographs in Physics, Springer-Verlag, Heidelberg, 1991. [IP] R. Illner and M. Pulvirenti, A derivation of the BBGKY-hierarchy for hard sphere particle systems, Transport Theory and Stat. Phys. \textbf{16}, 997--1012 (1987), preprint DM-388-IR, September 1985. [CIP] C. Cercignani, R. Illner, and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences \textbf{106}, Springer-Verlag, New York, 1994.