LAWS OF EXCURSIONS ASSOCIATED TO ADDITIVE FUNCTIONALS

Abstract

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 a
right stationary process with random birth and death, Markov with semi group (Pt) under the Kuznetsov measure Q
associated to an excessive measure. We define, under the assumption that the characteristic measure
{ } ( ) 1
B 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 the
right inverse (Ut) of this additive functional. More precisely, if we note by ( ) t Φ the process
Ut Y ⎛ ⎞
⎜ ⎟
⎜ ⎟
⎝ ⎠
and by H the σ -
algebra generated by Ht (t∈R) where Ht is the Q-completion of 0
t H + ( 0
t ⎛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 excursion
straddling T T U U −
⎤ ⎡
⎥⎦ ⎢⎣ , with respect to H depend only on T Φ and T − Φ . Conditional laws of pairs of excursions
are also considered.