Doubly-Feller process with multiplicative functional. (English) Zbl 0603.60066

Stochastic processes, 5th Semin., Gainesville/Fla. 1985, Prog. Probab. Stat. 12, 63-78 (1986).
[For the entire collection see Zbl 0591.00011.]
A doubly-Feller process is a Feller process whose transition function has also the strong Feller property (i.e. sends bounded measurable functions to bounded continuous functions). This paper contains the following results, loosely stated. Let X be a doubly Feller process then
1. If B is an open, regular subset of the state space, then X killed outside B is doubly Feller.
2. The transition function \(Q_ tf(x)=E^ x\{M_ tf(X_ t)\}\) is doubly Feller, whenever M is a multiplicative functional satisfying certain moment conditions.
These results apply, for example, to the case where X is Brownian motion, M is the Feynman-Kac functional, and B is taken to be relatively compact; the conclusion is that \[ x\to E^ x[M_ tf(X_ t);\quad t < \text{ first exit from }B] \] is a continuous function on B, vanishing at the boundary, for each bounded measurable function \(f: B\to {\mathbb{R}}\).
Reviewer: R.W.R.Darling


60J25 Continuous-time Markov processes on general state spaces
60J57 Multiplicative functionals and Markov processes
60J40 Right processes


Zbl 0591.00011