This paper is devoted to the theoretical investigation of the probabilistic aspect of the kinetic theory of nonequilibrium processes in a correlated fluctuating gas. More specifically, a stochastic kinetic equation for the phase density of the system, together with the corresponding equations for its distribution and characteristic functional, as well as the equation for the generating functional, is consistently derived in a form equivalent to the irreversible kinetic equation for the distribution function. In deriving this equation, pair collisions of particles in a monoatomic gas and large-scale fluctuations in the system are considered. Within the Gibbs phase space, a multitude of dynamical trajectories in the statistical system are considered, along with non-averaged values of the single-particle state density. This spatial representation of system states reveals a functional–probabilistic formulation of classical statistical mechanics that forms the basis of the stochastic approach. The stochastic equation for the phase density is derived from the Liouville equation using a model of Markov jump processes. Although this equation is equivalent to the Boltzmann equation, it includes an additional term referred to as the nonlinear “fluctuation source.” The statistical characteristics of the resulting equation are established. The stochastic equation for the phase density of the system accounts for both dissipative and fluctuation properties and represents a generalized form of the Boltzmann–Langevin equation for unstable states of an equilibrium gas. Using hydrodynamic variables and stochastic equations, a closed hydrodynamic description of the correlation system is obtained. The relationship between the characteristic functional and the generating functional for the hierarchy of partial distribution functions is formulated through integral equations, and particular solutions corresponding to the initial conditions are presented.