stationary independent increments
Stationary independent increments is the single defining property that makes a continuous-time process behave like a sum of iid steps. It is the structural hypothesis that turns the multi-dimensional question of a whole trajectory's law into a one-parameter question about the increment law, and it is exactly what forces the one-dimensional marginals to be infinitely divisible.
Independence of increments means: for any finite set of disjoint time intervals, the increments of the process over those intervals are mutually independent random variables. Stationarity of increments means: the distribution of X_{t+s} - X_s does not depend on s, only on the length t. Together they say the process forgets its past and is statistically time-homogeneous in its steps. The consequence is the multiplicative rule for characteristic functions: writing phi_t(theta) = E[e^(i theta(X_{t+s} - X_s))], stationarity makes phi_t well defined and independence plus stationarity give phi_{t_1 + t_2} = phi_{t_1} phi_{t_2}, the Cauchy functional equation whose measurable solution is phi_t = e^(t psi). Hence the increment law is infinitely divisible and the process is a Lévy process (once a cadlag version is chosen).
This is the continuous-time embodiment of the iid paradigm, and it is why Lévy processes are the natural limits of suitably scaled random walks. The crucial caveat: independent increments is far stronger than the Markov property — the past does not influence the future even in distribution, not merely conditionally. Many natural processes (Ornstein-Uhlenbeck, fractional Brownian motion) are Markov or Gaussian but do NOT have independent increments, and so are not Lévy. Stationary increments must also not be confused with a stationary process: a Lévy process itself is generally non-stationary (its variance grows in t); it is the increments that are stationary.
For Brownian motion, the increments B_{t_2} - B_{t_1} and B_{t_4} - B_{t_3} over disjoint intervals are independent N(0, length) variables — independent (no overlap) and stationary (law depends only on the length). Contrast the Ornstein-Uhlenbeck process: its increments are neither independent nor stationary because the mean-reverting drift ties each step to the current level.
Disjoint-interval increments are the independent, time-homogeneous steps.
Independent increments imply the process is Markov, but the converse fails badly. And a Lévy process is almost never a stationary process — only its increments are stationary; the process drifts and spreads.