LAWS OF EXCURSIONS ASSOCIATED TO ADDITIVE FUNCTIONALS (Sciences & Technology. A, exactes sciences, 2006)
Let (Pt) be a right borel semigroup and let (St) the right inverse of a continuous additive functional (Bt). Let ( )t R t Y∈be aright stationary process with random birth and death, Markov with semi group (Pt) under the Kuznetsov measure Qassociated to an excessive measure. We define, under the assumption that the characteristic measure{ } ( ) 1B 0 Yt υ Q I B dt ∈. = ∫ of (Bt) is purely excessive for the semigroup (Ps), an additive functional for ( )t R t Y∈in terms of(Bt) and we study the laws of excursions associated to the regenerative set which consists in times of discontinuity of theright inverse (Ut) of this additive functional. More precisely, if we note by ( ) t Φ the processUt Y ⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠and by H the σ -algebra generated by Ht (t∈R) where Ht is the Q-completion of 0t H + ( 0t ⎛H ⎞⎜ ⎟⎝ ⎠ is the natural filtration of ( ) t Φ ),then if T is a ( ) t H -stopping time such that T T U U − ≠ and T − T Φ ≠ Φ , the conditional law of the excursionstraddling T T U U −⎤ ⎡⎥⎦ ⎢⎣ , with respect to H depend only on T Φ and T − Φ . Conditional laws of pairs of excursionsare also considered.