Does there exist a Lévy-Process with normal increments but with paths that aren't even continuos when modified on null sets?
I'm asking because when defining Brownian motion as Lévy-Process, continuity is always required additionally. However, the only example of a noncontinuous "Brownian motion" I have encountered so far is where you take a continuous Brownian motion and set rational values (not rational points) to 0 (which I guess is modification on a null set?)
The conditions that a process $X$ is has independent increments and that these increments are normal uniquely determines the finite-dimensional distributions of the process, and so, in law, there exists only a single Levy process with normal increments, and that Levy process is the Brownian motion.
However, stochatic processes can exist relative to different sample spaces. The Kolmogorov extension theorem yields the existence of the Brownian motion on the space $\mathbb{R}^{[0,\infty)}$ of mappings from $[0,\infty)$ to $\mathbb{R}$, but not on the space $C[0,\infty)$ of conntinuous functions from $[0,\infty)$ to $\mathbb{R}$. The process on $\mathbb{R}^{[0,\infty)}$ will not have continuous sample paths, but will have a version with continuous sample paths.
The assumption that the Levy process with normal increments also should have continuous sample paths can be seen as a statement that when we talk about this Levy process, we always take a version with continuous sample paths.