Asymptotic stability in distribution of stochastic differential equations with Markovian switching

Stability of stochastic differential equations with Markovian switching has recently been discussed by many authors, for example, Basak et al. (J. Math. Anal. Appl. 202 (1996) 604), Ji and Chizeck (IEEE Trans. Automat. Control 35 (1990) 777), Mariton (Jump Linear System in Automatic Control, Marcel Dekker, New York), Mao (Stochastic Process. Appl. 79 (1999) 45), Mao et al. (Bernoulli 6 (2000) 73) and Shaikhet (Theory Stochastic Process. 2 (1996) 180), to name a few. The aim of this paper is to study the asymptotic stability in distribution of nonlinear stochastic differential equations with Markovian switching.