DES - Definition of 'Homogeneity'

Homogeneity:

A stochastic process is defined as homogeneous if the following holds:

$$\begin{array}{rl}& P(X(s^{\prime }+\Delta )=x^{\prime }\mid X(s+\Delta )=x)=\\ & =P(X(s^{\prime })=x^{\prime }\mid X(s)=x)\end{array}$$

$\Rightarrow$ The probability $P(X(s^{\prime })=x^{\prime }\mid X(s=x)$, do NOT depend on $s$ and $s^{\prime }$, but they depend only on the difference: $t=s^{\prime }-s$

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~Example: