Suppose I have a particle which moves in the interval $[0,1]$. After an exponential time of rate $1$ the particle (which for instance is in the position $x\in [0,1]$ jumps to the site $y\in [0,1]$ with probability $J(x,y)$, where $J$ is such that $\int_0^1J(x,y)dy=1, \forall x\in [0,1]$. The process which identifies the position of the particle is a Markov process. Is its generator defined as
$$Lf(x)=\int_0^1J(x,y)(f(y)-f(x))dy?$$
Is there any reference in which they talk about Markov processes with continuous state space?
I would recommend chapters 7 then 17 of Kallenberg's textbook https://www.springer.com/gp/book/9780387953137
Your claim is correct, by Proposition 17.2 of the above book.