How to Set Up Stochastic Equations for Real World Processes: Markov–Einstein Time Scale

Abstract

In Chaps. 16–21 we address a central question in the field of complex systems: Given a fluctuating (in time or space), sequentially uni- or multi-variant measured set of experimental data (even noisy data), how should one analyse the data non-parametrically, assess their underlying trends, discover the characteristics of the fluctuations, including diffusion and jump parts, and construct stochastic evolution equation for the data?

[Type][CrossLinking]The original version of this chapter was revised: Belated correction has been incorporated. The correction to this chapter is available at https://doi.org/10.1007/978-3-030-18472-8_24

Problems

16.1

Drift vector and diffusion matrix

Fill in the details in the derivation of Eqs. (16.14) and (16.21).

16.2

Markovian embedding of a non-Markov process

Suppose we have a time series with Markov–Einstein time scale $t_M=n \tau$, where n and $\tau$ are integer number and sampling interval, respectively. Argue that this non-Morkov process can be transformed to a Markov process in n-dimensions which is known as the Markovian embedding of a non-Markov process.

16.3

Estimation of Markov–Einstein time scale $t_M$

Consider following stochastic equations:

$$\begin{aligned} \frac{d}{dt} x(t)= & {} -x(t) + y(t) \\\nonumber \\ \frac{d}{dt} y(t)= & {} -\frac{1}{L_0} y(t) + \frac{1}{L_0} \varGamma (t) \end{aligned}$$

where $\varGamma (t)$ is a Gaussian white noise, for $L_0=1$ and $L_0=10$ with $dt=0.001$.

(a) Use the Euler-Maruyama scheme to integrate the coupled two-dimensional diffusion process and by checking the $\chi ^2$-test (16.12) for Chapman–Kolmogorov equation or Wilcoxon test (Appendix 1), in the stationary state estimate the Markov–Einstein time scale of the process x (compare estimated $t_M$ with $L_0$).

(b) Show that the correlation time scale of y is $\simeq L_0$ (see Problem 14.11).