Chapter 2 - Indicator function and Liouville equation

Abstract

This chapter discusses the indicator function and Liouville equation. Modern apparatus of the theory of random processes is able of constructing closed descriptions of dynamic systems if these systems meet the condition of dynamic causality and are formulated in terms of linear partial differential equations, or certain types of integral equations. One can use indicator functions to perform the transition from the initial, generally nonlinear system to the equivalent description in terms of the linear partial differential equations. It is found that if the initial value problem is formulated in terms of partial differential equations, one can always convert it to the equivalent formulation in terms of the linear variational differential equation in the infinite-dimensional space. It is found that if the initial-dynamic system includes higher order derivatives, derivation of a closed equation for the corresponding indicator function becomes impossible. It is observed that the variational-differential equation can be derived in the closed form for the functional whose average over an ensemble of realizations coincides with the characteristic functional of the solution to the corresponding dynamic equation.