If I have a particle which move like a Brownian motion in the interval $[0,1]$ which is reflected at the boundary and after an exponential time of parameter $1$ it jumps on a site $y\in [0,1]$ with probability $J(x,y)$ where $x$ is the position of the particle before the jump and $J:[0,1]\times [0,1]\to \mathbb R$ is continuous. Is the generator of the Markov process that gives the position of the particle in $[0,1]$ the following: $$Lf(x)=\frac{1}{2}f''(x)+\int_0^1J(x,y)(f(y)-f(x))?$$
I have some problems about how to define the position of the particle before the jump since it is moving like a Brownian motion. Should I define this position like a limit? Could someone help me or suggest me some reference in which they consider Brownian motion with jumps with rate depending on the position of the BM?
Thanks a lot