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Relation between occupation probability and first passage probability

  1. Dec 8, 2009 #1
    Let P(r,t) define the occupation probability, the probability that a particle emulating a random walk will find itself at position r at time t if starts from the origin at time zero.

    Let F(r,t) define the first passage probability, the probability that a particle emulating a random walk will find itself at position r at time t FOR THE FIRST TIME if starts from the origin at time zero.

    I was reading a book which says this :
    "For a random walk to be at position r at time t, the walk must first reach r at some earlier time step t' and then return to r after t-t' additional steps. This connection between F(r,t) and P(r,t) can thus be expressed by the equation

    [tex] P(r,t) = \delta_{r0} \delta_{t0} + \sum_{t'\leqt}F(r,t')P(0,t-t') [/tex]

    "

    Can someone please explain how this is possible?
     
  2. jcsd
  3. Dec 8, 2009 #2
    For some reason, I cannot edit the above. It is supposed to read t' [tex]\leq[/tex] t below the summation sign.
     
  4. Dec 9, 2009 #3
    Can someone move this to the Classical Physics subforum? I might probably get more replies there.
     
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