Ž . Journal of Mathematical Analysis and Applications 249, 668685 2000 doi:10.1006jmaa.2000.6968, available online at http:www.idealibrary.com on Continuous Markov Semigroups and Stability of Transport Equations 1 Katarzyna Pichor ´ Institute of Mathematics, Polish Academy of Sciences, Bankowa 14, 40-007 Katowice, Poland and Ryszard Rudnicki Institute of Mathematics, Polish Academy of Sciences, Bankowa 14, 40-007 Katowice, Poland; and Institute of Mathematics, Silesian Uni  ersity, Bankowa 14, 40-007 Katowice, Poland Submitted by Paul S. Muhly Received November 22, 1999 A new theorem for asymptotic stability of Markov semigroups is proved. This result is applied to transport equations connected with diffusion and jumping processes and randomly perturbed dynamical systems.  2000 Academic Press Key Words: Markov semigroup; asymptotic stability; transport equation; Markov process. 0. INTRODUCTION Markov operators and Markov semigroups are intensively studied be- cause they play a special role in applications. The book of Lasota and  Mackey 7 is an excellent survey of many results on this subject. The main problem of the theory of Markov operators is their asymptotic stability, i.e., when there exists an invariant density f such that P n f  f in L 1 for 1 Ž . This research was supported by the State Committee for Scientific Research Poland Grant 2 P03A 010 16 and by the Foundation for Polish Science. Part of this work was done during a visit at the University of Pau. The authors thank French Polish Integrated Action Programme Polonium for financial support of this visit. 668 0022-247X00 $35.00 Copyright  2000 by Academic Press All rights of reproduction in any form reserved.