Stationary distribution  is a stationary distribution for P t if  =  P t for all t. Theorem:  is stationary for P t iff  G = 0 (under suitable regularity.

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Stationary distribution  is a stationary distribution for P t if  =  P t for all t. Theorem:  is stationary for P t iff  G = 0 (under suitable regularity conditions). Proof:

Death process As for discrete time, the stationary distribution can be thought of as the limit of P t as. Death process case:

Persistence A chain is irreducible if P ij (t) > 0 for some t and for all pairs i,j in S. Fact (Lévy dichotomy): Either P ij (t) > 0 for all t, or P ij (t) = 0 for all t. We call a state persistent if Let Y n = X(nh) be the discrete skeleton of X. Let Q be the transition matrix for Y so persistence in continuous time is the same as persistence for the discrete skeleton

Some further facts (i)j is persistent iff (ii)p ii (t)>0 for all t Proof: (i)For the discrete skeleton j is persistent iff i.e. iff (ii)so For any t pick n so large that t≤hn. By C-K

Birth, death, immigration and emigration Let The  n are called death rates, and the n are called birth rates. The process is a birth and death process. If n = n + we have linear birth with immigration. If  n = (  +  )n we have linear death and emigration.

Generator Stationary distribution  G = 0 yields whence When is this a stationary distribution?