Modelling of High-Dimensional Diffusion Stochastic Process With Nonlinear Coefficients for Engineering Applications — Part I: Approximations for Expectation and Variance of Nonstationary Process

Abstract

This work is devoted to diffusion stochastic processes (DSPs) with nonlinear coefficients in n-dimensional Euclidean space at high n (n is much greater than a few units). It deals with expectation and variance of a nonstationary process whereas our accompanying work deals with covariance and spectral 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 analytical basis for this combined treatment. The present work proposes approximate analytical expressions for the expectation and variance in the form of two ordinary differential equation (ODE) systems. They are derived within DSP theory without any techniques directly related to stochastic differential equations. Both ODE systems allow for space nonhomogeneities of the diffusion and damping matrixes and thereby do take nonlinearities of the DSP coefficients into account. Some related topics like invariant processes and the aspects of practical implementation of the above expressions are discussed as well. A proper attention is paid to formulation of some features important in applications of DSPs to the real-world problems. The results of this work can equally be used in various engineering fields.

Modelling of High-Dimensional Diffusion Stochastic Process With Nonlinear Coefficients for Engineering Applications — Part Ii: Approximations for Covariance and Spectral Density of Stationary Process

“…6 and assumes that ordinary differential equation (ODE) system (see (C.4) in Ref. 1) which, in the present, time-independent case (2.7), (2.8) is written as …”

“…6 of Ref. 1 and concentrates on covariance (e.g., (8.2.6.b) in Ref. 2) C(∆) = E{[χ(· , t)−E(χ(· , t))][χ(· , t+∆)−E(χ(· , t+∆))] T } , for all t, ∆ ≥ 0 , An example of stationary DSPs with coefficients (2.7), (2.8) is (see Sec.…”

“…1) solutions of Itô's stochastic differential equation (ISDE) system (see (1.10), (1.11) in Ref. 1) dx = g(x) dt + h(x) dW (ξ, t) , (2.9) where matrix h(x) is coupled with diffusion matrix H(x) as follows: Since process χ is stationary, it is described with stationary invariant probability density ρ inv (t, x) ≡ ρ inv (x) , for all x .…”

“…1 is fulfilled for the mentioned versions of ϕ and ψ. Thus, formula (D.9) in Ref 1. can be applied to the present case.…”

The deterministic circuit model for noise influence on the averaged transient responses of large-scale nonlinear ICs analyzed with Ito's stochastic differential equations

“…Moreover, the effects crucial for many noise-induced phenomena like stochastic resonance, stochastic linearization, stochastic self-oscillations (e.g., [lo]-[ 121) are completely ingnored in (2). Solution of this problem is addressed by the deterministic (like (2)) second-order ODE system [6, (4.7)] derived as one of the results of Theorem 4 in [6] (see also [7, Chapter 21). This system directly includes the term dependent on both noiserelated matrix h(t,x) and the x -nonlinearities of vector g(t,x) .…”