טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentManos Nadav
SubjectThe Rate of Convergence of Markov Chains to the Steady State
DepartmentDepartment of Applied Mathematics
Supervisor Mr. Boris Granovsky


Abstract

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.