Moments and Correlation Functions of Solutions of Some Stochastic Matrix Differential Equations

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Abstract
The (not necessarily linear) vector differential equationdudz=f(u(z),M(z),z),u(0)=g[M(0)]is first considered, where M(z) is a finite-state Markov process which has, in general, a nonstationary transition mechanism. The joint process {u(z), M(z)} is a Markov process, and forward and backward Kolmogorov equations are derived for the transition probability density functions. Attention is then turned to the linear matrix differential equationdWdz=A(M(z),z)W(z),W(0)=γ[M(0)],where W and γ are n×m matrices and A is an n×n matrix. The forward equations for the corresponding probability density functions are used to obtain two different, but equivalent, formulations for the calculation of the moments of any given order, and of the correlation functions, of the solution. The calculation of the moments and correlation functions is reduced to the solution of systems of linear ordinary differential equations, with prescribed initial conditions. The inhomogeneous matrix equation dYdz=A(M(z),z)Y(z)+B(M(z),z),Y(0)=γ[M(0)]is also considered. Some applications, in particular to the calculation of the average modal powers in randomly coupled transmission lines, will be given elsewhere.

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