Influence of Plasma and Electron Beam Noise on System Instability at Knudsen Numbers of the Order of 1

The work is devoted to the development of the kinetic theory of stability of the fast electron beam - gas-discharge plasma system taking into account the beam and plasma noise, which is important for optical radiation sources based on beam discharges, for example, a hollow cathode. It is shown that in this case the problem is of a mixed nature. Namely, when the shot noise of the beam and the intrinsic noise of the plasma are taken into account, stability problems arise with boundary conditions and initial conditions, respectively.