On the path structure of a semimartingale arising from monotone probability theory
Annales de l'I.H.P. Probabilités et statistiques, Volume 44 (2008) no. 2, pp. 258-279.

Let X be the unique normal martingale such that X 0 =0 and

dX t =(1-t-X t- )dX t +dt
and let Y t :=X t +t for all t0; the semimartingale Y arises in quantum probability, where it is the monotone-independent analogue of the Poisson process. The trajectories of Y are examined and various probabilistic properties are derived; in particular, the level set t0:Y t =1 is shown to be non-empty, compact, perfect and of zero Lebesgue measure. The local times of Y are found to be trivial except for that at level 1; consequently, the jumps of Y are not locally summable.

Soit X l’unique martingale normale telle que X 0 =0 et

dX t =(1-t-X t- )dX t +dt
et soit Y t :=X t +t pour tout t0; la semimartingale Y se manifeste dans la théorie des probabilités quantiques, où c’est analogue du processus de Poisson pour l’indépendance monotone. Les trajectoires de Y sont examinées et diverses propriétés probabilistes sont déduites; en particulier, l’ensemble de niveau t0:Y t =1 est montré être non vide, compact, parfait et de mesure de Lebesgue nulle. Les temps locaux de Y sont trouvés être triviaux sauf celui au niveau 1; par conséquent les sauts de Y ne sont pas localements sommables.

DOI: 10.1214/07-AIHP116
Classification: 60G44
Keywords: monotone independence, monotone Poisson process, non-commutative probability, quantum probability
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Belton, Alexander C. R. On the path structure of a semimartingale arising from monotone probability theory. Annales de l'I.H.P. Probabilités et statistiques, Volume 44 (2008) no. 2, pp. 258-279. doi : 10.1214/07-AIHP116. http://www.numdam.org/articles/10.1214/07-AIHP116/

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