Page 224 - Applied Probability

10. Molecular Phylogeny
210
where I is the identity matrix. It is easy to check that the solution of the
initial value problem (10.6) is furnished by the matrix exponential
∞
k
t Λ
tΛ
=
P(t)= e
k! k . (10.7)
k=0
Probabilists call Λ the infinitesimal generator or infinitesimal tran-
sition matrix of the process.
A probability distribution π =(π i ) on the states of a Markov chain is a
row vector whose components satisfy π i ≥ 0 for all i and π i =1. If
i
πP(t)= π (10.8)
holds for all t ≥ 0, then π is said tobean equilibrium distribution for
the chain. Written in components, the eigenvector equation (10.8) reduces
to π i p ij (t)= π j . Again, this is completely analogous to the discrete-
i
time theory described in Chapter 9. For small t, equation (10.8) can be
rewritten as
π(I + tΛ) + o(t)= π.
This approximate form makes it obvious that πΛ= 0 is a necessary condi-
tion for π to be an equilibrium distribution. Multiplying (10.7) on the left
by π shows that πΛ= 0 is also a sufficient condition for π to be an equi-
librium distribution. In components, this necessary and sufficient condition
amounts to

π j λ ji = π i λ ij (10.9)
j =i j =i
for all i. If all the states of a Markov chain communicate, then there is one
and only one equilibrium distribution π. Furthermore, each of the rows of
P(t) approaches π as t →∞. Lamperti [16] provides a clear exposition of
these facts.
Fortunately, the annoying feature of periodicity present in discrete-time
theory disappears in the continuous-time theory. The definition and proper-
ties of reversible chains carry over directly from discrete time to continuous
time provided we substitute infinitesimal transition probabilities for tran-
sition probabilities. For instance, the detailed balance condition becomes

π i λ ij = π j λ ji (10.10)
for all pairs i = j. Kolmogorov’s circulation criterion for reversibility contin-
ues to hold, and when it is true, the equilibrium distribution is constructed
from the infinitesimal transition probabilities exactly as in discrete time.