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Stochastic linearization: the theory

Very little is known about the quantitative behaviour of dynamical systems with random excitation, unless the system is linear. Known techniques imply the resolution of parabolic partial differential equations (Fokker–Planck–Kolmogorov equation), which are degenerate and of high dimension and for which there is no effective known method of resolution. Therefore, users (physicists, mechanical engineers) concerned with such systems have had to design global linearization techniques, known as equivalent statistical linearization (Roberts and Spanos (1990)). So far, there has been no rigorous justification of these techniques, with the notable exception of the paper by Kozin (1987). In this contribution, using large deviation principles, several mathematically founded linearization methods are proposed. These principles use relative entropy, or Kullback information, of two probability measures, and Donsker–Varadhan entropy of a Gaussian measure relatively to a Markov kernel. The method of ‘true linearization’ (Roberts and Spanos (1990)) is justified.