This thesis deals with the exponential rate of
convergence of *non-homogeneous* *Birth Death processes* (*BDP*s).
Let _{} be
a *BDP* on a discrete phase space _{} with intensity matrix _{}, and let for
each _{},
_{} be
the probability of the event _{}. The *forward Kolmogorov
equation *of the process is _{}, where _{}. We refer to that
equation* as a differential equation in the space *_{}, and we say that _{} is weakly
ergodic if for every two *probabilistic solutions*_{}, _{}. We are interested in
the exponential rate of decay of _{}. Following *A.I. Zeifman*
method, we pass from *Kolmogorov forward equation* to a related equation _{} which is also
a differential equation in the space _{}. As a generalization of *the
decay parameter* of *a* *homogeneous BDP*, we suggest to adopt *Liapunov-
exponent* _{},
and *Bohl- exponent* _{} of the last equation. Both of
them are concepts from the theory of *the* *stability of solutions of
differential equations in Banach spaces. *In general, _{}, but if the process is
homogeneous then _{} where _{} is the *decay
parameter* of *a* *homogeneous BDP*. *Bohl- exponent *allows
us to reduce a non-homogeneous problem to a homogeneous one based on theorems
by *J.L Dalecky & M.G Krein. * In order to calculate the decay
parameter of *a homogeneous BDP*, we apply the method of *B.L Granovsky
& A.I. Zeifman*.