Randomly connected break differential equations with Poisson type perturbations

Abstract

We consider the stochastic differential equation $$ du = \alpha (u, \xi)dt + \sigma (u) dp \tag{1}$$ on a separable Banach space. We give sufficient conditions for the existence of an invariant measure for the semigroup $ \{ P^t \}_{t \ge 0} $ corresponding to the stochastic differential equation (1). We show that the existence of an invariant measure for a Markov operator $\overline{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.