Jump processes and boundary processes.(English)Zbl 0566.60057

Stochastic analysis, Proc. Taniguchi Int. Symp., Katata & Kyoto/Jap. 1982, North-Holland Math. Libr. 32, 53-104 (1984).
[For the entire collection see Zbl 0538.00017.]
The paper under review surveys the authors results in Z. Wahrscheinlichkeitstheor. Verw. Geb. 63, 147-235 (1983; Zbl 0494.60082) and Ann. Sci. Ec. Norm. Supér., IV. Sér. 17, 507-622 (1984; Zbl 0561.60081).
The first paper considers a Malliavin calculus for pure jump processes. An integration by parts formula on the path space is derived using a Girsanov transformation. Using this formula the author gives conditions on the Lévy kernel of an independent increment process which guarantee that the distribution of the process at time t has a density which is k times differentiable and bounded.
Nice applications of these results are made in the second paper to reflecting diffusion processes. The author gives conditions on a reflecting diffusion in a domain D implying the flow of the diffusion is a $$C^{\infty}$$ diffusion. Conditions are also given under which the joint distribution of the local time at the boundary and the process on the boundary has a $$C^{\infty}$$-density.
Reviewer: C.M.Cranston

MSC:

 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60J75 Jump processes (MSC2010) 60J60 Diffusion processes 60J99 Markov processes 60H99 Stochastic analysis