Qualitative Theory of Stochastic Systems

Abstract

This chapter presents the problems and results of the qualitative theory of stochastic dynamical systems. Qualitative theory studies the general nature of a solution in the entire time interval. Attention has been attracted mainly by the white noise case, in particular, by nondegenerate diffusion processes. The methods used were mainly Lyapunov function techniques and the interrelation between Markov processes and deterministic partial differential equations. White noise is a generalized Gaussian stationary process with constant spectral density on the whole real line. Qualitative theory studies the general nature of a solution on the entire time interval (asymptotic or long-term behavior) without solving the equation.