Let $\mathcal{S}$ be the state space and $f$ be a function from $\mathcal{S}$ to $\mathbb{R}$. An interacting particle system is a continuous-time Markov processes with generator $$ Gf(x) = \sum_{m}r_m(f(m(x)) - f(x)), $$ where $m$ are some functions $m:\mathcal{S}\to\mathcal{S}$ and $r_m$ are some nonnegative constants.
Questions:
- How do I understand $Gf$ ? I know $G(x,y)$ is the rate of jumps from $x$ to $y$ but by definition $$Gf(x) := \sum_{y}G(x,y)f(y).$$
- How do I see that $r_mh + o(h)$ is the probability that the map $m$ is applied during the time $(t, t + h]$ from the expression of $Gf$ ?