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Invariant measures related with randomly connected Poisson driven differential equations

Katarzyna Horbacz (2002)

Annales Polonici Mathematici

We consider the stochastic differential equation (1) d u ( t ) = a ( u ( t ) , ξ ( t ) ) d t + Θ σ ( u ( t ) , θ ) p ( d t , d θ ) for t ≥ 0 with the initial condition u(0) = x₀. We give sufficient conditions for the existence of an invariant measure for the semigroup P t t 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 0 describing the evolution of measures along trajectories and vice versa.

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