Stability of a random diffusion with nonlinear drift
The author considers a strong Markov process, \((X(t),Z(t))\), with state space \({\mathbb R}^d\times N\) (\(N\) is finite), such that \(X(t)\) satisfies a stochastic differential equation with a rather general nonlinear drift, and \(Z(t)\) is a continuous-time Markov chain with state space \(N\). The author shows how to construct the coupling of \((X(t),Z(t))\) and itself; with the help of these results, Feller continuity, as well as the existence and uniqueness of invariant measure, are proved for \((X(t),Z(t))\). By using the Foster-Lyapunov inequality and coupling, the author proves the convergence in the total variation norm of the transition probability of \((X(t),Z(t))\) to the invariant measure as \(t\to\infty\) (the stability of \((X(t),Z(t))\) in total variation).
scientific article; zbMATH DE number 2159990
scientific article