Continuous-time random walks in continuous spaces are generally defined as follows: sample a waiting time (from a continuous pdf), wait for this time, and then jump to a random position (from another continuous pdf).
This sounds like a very discrete process to me. For instance, the walker is at a countable number of positions over time, and moves a countable number of times.
This seems different from a walker continuously moving from its current position to the one after the jump, during the waiting time (which would then be a travel time).
This is visible in drawings of random walks on the plane: the consecutive walker positions are generally linked by a (straight) line:
This does not fit the "jump" definition, where the walker stays at one place until it jumps, and is never located at intermediate positions.
Is there something I misunderstand? May someone explain why the two approaches are equivalent, if they are? Or what are their differences, if any?
I am also interested in pointers to material on random walkers that would travel from places to places, rather than jump.
I have posted a related, more formal, question on MO but it was downvoted, probably because I do not correctly understand the points above.
There is indeed a difference between continuous-time random walks and the situation where a walker is moving continuously. For continuous-time random walks, the path of a walker is a series of points. For visualization purposes we can connect these to get the line segments which you mention. In contrast, a continuously moving walker's path can form any continuous line.
I'm not an expert, but an example of the latter is a Wiener process.