We know that a Markov process is a stochastic process in which the future evolution of the system depends solely on its current state and is independent of its past states given the present state. Also we know that the Brownian motion is a special case of the Markov process, which has other properties too. We know that a Lévy process is a stochastic process with stationary and independent increments, allowing for both continuous movements and jumps at random times. (Since I am a first-semester-stochastics student, I tried to write intuitive definition for the concepts above, so please correct me if they were wrong).
Up to this point, the definitions were clear. However, i cannot find an intuitive explanation for Feller processes. New concepts, i.e., transition functions/semigroups/generators are defined for this process, which seems to me another branch of stochastic processes. I mean, the new concepts are defined for this processes, and then this concepts are used for other processes (like Brownian motion). So I am confused, why we don't talk about those concepts when we are defining Brownian motions. Can anybody explain the Feller process and its relation/difference to other mentioned processes in simple words?
The terminology "Feller process" was introduced by E.B. Dynkin in 1956, in a series of papers. Dynkin was abstracting Feller's earlier work, on analytic properties of a transition semigroup permitting the construction of a one-dimensional diffusion process, to the construction of cadlag Markov processes with the strong Markov property in a general context. Independently and simultaneously, in the USA, R.M. Blumenthal (in his Ph.D. thesis) was proving essentially the same result. (Blumenthal's advisor G.A. Hunt had proved the strong Markov property of Brownian motion shortly before, using similar ideas.)