What is a diffusion process? Wikipedia says "In probability theory, a branch of mathematics, a diffusion process is a solution to a stochastic differential equation. It is a continuous-time Markov process with almost surely continuous sample paths. Brownian motion, reflected Brownian motion and Ornstein–Uhlenbeck processes are examples of diffusion processes." What is continuous sample path?
If so, then
1- is a Brownian motion with drift a diffusion process? Why? I know it is a Markov process and I can prove it but I'm not sure if it has continuous sample path.
2- In fact, I am not quite sure if I got the meaning of "continuous sample path" right. Is it the same as continuous state space or continuous index set (i.e. parameter set)?
3- If a process is diffusion process then it can not be a Jump Process, right? (Perhaps I can answer this question by myself if I know what continuous sample time mean).