Relation between occupation probability and first passage probability

[WiFO215]

I posted this in the probability sub-forum, but did not receive any replies, so am linking it here. Hopefully, someone here can answer it.

https://www.physicsforums.com/showthread.php?t=361593

 

[mathman]

All the expression is saying is that a particle arriving at r at time t will have had a first arrival at r at time t' (which may = t) ahd then arrive at a relative position of 0 in time t-t'. The delta function term simply says that P(0,0) reflects the fact that it was at 0 to start with.

 

[WiFO215]

All the expression is saying is that a particle arriving at r at time t will have had a first arrival at r at time t' (which may = t) ahd then arrive at a relative position of 0 in time t-t'. The delta function term simply says that P(0,0) reflects the fact that it was at 0 to start with.

He is describing relative position zero using the same probability distribution P? So he is making the assumption there that P doesn't change once the thing reaches r? He hasn't mentioned this explicitly.

 

[mathman]

There is an implicit assumption of stationarity, i.e. the distribution depends only on time difference, not absolute time.

 

[WiFO215]

Okay! Thanks.