## The local Feynman-Kac semigroup.(English)Zbl 0618.47038

The Feynman-Kac functionals with stopping time are studied in this paper. The concept of the local Feynman-Kac semigroup is given, and used to expand the results in [K. L. Chung, Lect. Notes Math. 784, 347-356 (1980; Zbl 0444.60061), K. L. Chung and K. M. Rao, Progr. Probab. Stat. 1, 1-29 (1981; Zbl 0492.60073), K. L. Chung and P. Li, Adv. Math., Suppl. Stud. 9, 25-34 (1986; Zbl 0608.60061)]. Another equivalence condition corresponding to the important results in the last two papers mentioned above are obtained in the form of the asymptotic property of the semigroup, and some conclusions in the last two papers mentioned above are proved in a different approach. Finally, a relationship between the F-K functionals and the potential operators of their semigroups, i.e. $E\cdot \exp \int^{\tau}_{0}q(W_ t)dt=1+(\int^{\infty}_{0}T_ tdt)q$ is found.

### MSC:

 47D03 Groups and semigroups of linear operators 28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) 60J45 Probabilistic potential theory 60G40 Stopping times; optimal stopping problems; gambling theory

### Citations:

Zbl 0444.60061; Zbl 0492.60073; Zbl 0608.60061