This monograph deals with the dynamical systems of two “particles”: a hard disk of radius \(r>1\) and mass \(M\gg 1\) and a point particle of mass \(m=1\). They move freely in a two-dimensional container \(D\) with concave smooth \(C^3\) boundaries and finite horizon. The initial position and velocity of the disk are \(Q(0)=Q_0\), \(V(0)=V_0\) and fixed while the initial state of the light particle \((q(0),v(0))\) is selected randomly, according to a smooth distribution. It is considered the trajectory of the disk \((Q(t),V(t))\) during a finite time interval. The random choice of \((q(0),v(0))\) induces a probability measure on the space of functions defining the disk trajectories. The paper deals with the convergence of this probability measure to a stochastic process as the existence interval of the state trajectory approaches \(\infty\). This convergence is proved in three different regimes in the dynamics of the disk. The monograph has 9 Chapters and two Appendices. The first two Chapters contain the problem statement and the basic results, what is left is dedicated to proofs and open problems (Chapter 9). There are given 97 references.