I've been working through this paper called "A First Look at First-Passage Processes" for my undergrad math research project. It's my first time looking at upper level statistics so cut me some slack.
I understand that P(r,t) is the occupation probability, meaning the probability that the walk is at site r at time t given that we start at some origin, and that F(r,t) is the first passage probability, the probability that the walk is at site r for the first time at time t with the same initial condition.
The paper expresses the relation between P(r,t) and F(r,t) through a convolution:
$$P(r,t) = \delta_{r,0}\delta_{t,0} + \sum_{t'\leq t}F(r,t')P(0,t-t')$$
And says that we can solve the convolution through the use of generating functions.
$$P(r,z) = \sum_{t=0}^\infty P(r,t)z^t$$ $$F(r,z) = \sum_{t=0}^\infty F(r,t)z^t$$
Question 1: How did we derive these generating functions? and what exactly is $z^t$?
The paper says that we can solve for the FP probability by multiplying the convolution by $z^t$ and sum over all t to get a relation between P(r,t) and F(r,t) by
$$P(r,z) = \delta_{r,0} + F(r,z)P(0,z)$$
Question 2: Can someone please show the derivation from the convolution to the relation between P(r,t) and F(r,t)?
Thanks in advance,