Invariant measures related with randomly connected Poisson driven differential equations

Katarzyna Horbacz¹

¹ Institute of Mathematics Silesian University 40-007 Katowice, Poland

Annales Polonici Mathematici, Tome 79 (2002) no. 1, pp. 31-44.

Résumé

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.

Détail

DOI : 10.4064/ap79-1-3

Keywords: consider stochastic differential equation int mit theta sigma theta cal theta hskip hbox tag* initial condition sufficient conditions existence invariant measure semigroup corresponding existence invariant measure markov operator corresponding change measures jump jump implies existence invariant measure semigroup describing evolution measures along trajectories vice versa