|M.Sc Student||Manos Nadav|
|Subject||The Rate of Convergence of Markov Chains to the Steady State|
|Department||Department of Applied Mathematics||Supervisor||Professor Boris Granovsky|
This thesis deals with the exponential rate of convergence of non-homogeneous Birth Death processes (BDPs). 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.