Modelling of high-dimensional diffusion stochastic process with nonlinear coefficients for engineering applications - part I: Approximations for expectation and variance of nonstationary process

This work is devoted to diffusion stochastic processes (DSPs) with nonlinea r coefficients in n-dimensional Euclidean space at high n (n is much greate r than a few units). It deals with expectation and variance of a nonstation ary process whereas our accompanying work deals with covariance and spectra l density of a stationary process. Combined, analytical-numerical approach is a reasonable and perhaps the only way to treat high-dimensional DSPs in practice. Each of the above works develops the corresponding parts of the a nalytical basis for this combined treatment. The present work proposes appr oximate analytical expressions for the expectation and variance in the form of two ordinary differential equation (ODE) systems. They are derived with in DSP theory without any techniques directly related to stochastic differe ntial equations. Both ODE systems allow for space nonhomogeneities of the d iffusion and damping matrixes and thereby do take nonlinearities of the DSP coefficients into account. Some related topics like invariant processes an d the aspects of practical implementation of the above expressions are disc ussed as well. A proper attention is paid to formulation of some features i mportant in applications of DSPs to the real-world problems. The results of this work can equally be used in various engineering fields.