Intuitively, standard Brownian motion has infinite propagation speed, as it has a non-zero probability of reaching any point in any arbitrarily short time. This is due to the fact that the probability density of the normal distribution is strictly greater than zero over the entire space.
Naturally, one might wonder whether there exists an independent stationary increment process $X_t$ that has a finite propagation speed. Specifically, for a given $t$, there exists $R > 0$ such that $\mathbb{P}(|X_t| > R) = 0$. In particular, it seems intuitive to believe that there exists some $v > 0$ such that
$$ \mathbb{P}(|X_t| > vt) = 0, \quad \forall t > 0. $$
This would imply that the influence of $X_t$ is confined within a ball of radius $vt$.
Initially, I thought such a process should exist, as it appears to be a natural consideration. Moreover, the wave equation is known to exhibit finite propagation speed, which leads me to draw an analogy to a "random" process.
However, upon reviewing this Wikipedia page, particularly the Lévy–Khintchine representation, it presents the specific form of a Lévy process. It states that
a Lévy process can be seen as having three independent components: a linear drift, a Brownian motion, and a Lévy jump process. This immediately gives that the only (nondeterministic) continuous Lévy process is a Brownian motion with drift.
From a literal standpoint, I did not expect a Lévy jump process to possess "finite propagation speed".
Therefore, my question is whether, aside from deterministic drift processes, there exists no stochastic process that has "finite propagation speed". I consider this to be a rather counterintuitive matter and wonder if there has been historical commentary or explanation regarding this phenomenon. Is it possible to find stochastic processes that exhibit "finite propagation speed" in some weaker sense?
