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Abstract

We consider the stochastic differential equation $$ du(t) = a(u(t), \xi (t))dt + \int _{{\mit \Theta }} \sigma (u(t), \theta ) \, {\cal N}_p(dt, d\theta ) \hskip 1em \hbox {for } t \ge 0\tag*{$ ({1} )$}$$ with the initial condition $u(0) = x_0$. We give sufficient conditions for the existence of an invariant measure for the semigroup $ \{ P^t \} _{t \ge 0} $ corresponding to (1). We show that the existence of an invariant measure for a Markov operator ${P} $ corresponding to the change of measures from jump to jump implies the existence of an invariant measure for the semigroup $ \{ P^t \} _{t \ge 0} $ describing the evolution of measures along trajectories and vice versa.