V: 29, 290-310, LNM 191 (1971)
WALSH, John B.
Some topologies connected with Lebesgue measure (
Markov processes,
General theory of processes,
Potential theory)
It is a recurrent theme in the theory of stochastic processes that time sets of measure $0$ should be ignored. Thus topologies on the line which ignore sets of measure $0$ are useful. The main topic here is the so-called
essential topology, used in the paper of Chung and Walsh
522 in the same volume
Comment: See Doob
Bull. Amer. Math. Soc.,
72, 1966. An important application in given by Walsh
623 in the next volume. See the paper
1025 of Benveniste. For the use of a different topology see Ito
J. Math. Soc. Japan, 20, 1968
Keywords: Essential topologyNature: Original
Retrieve article from Numdam
VI: 22, 233-242, LNM 258 (1972)
WALSH, John B.
The perfection of multiplicative functionals (
Markov processes)
In the definition of multiplicative functionals the problem arose from the beginning whether the exceptional null set in the relation $M_{s+t}=M_s\,M_t\circ\theta_s$ was allowed to depend on $s$ or not---in the latter case the functional is said to be perfect. C.~Doléans showed by a detailed analysis (see
203) that every functional has a perfect modification, see also Dellacherie
304. Here a perfect version is constructed directly as $\lim_{s\rightarrow 0} M_{t-s}\circ\theta_s$, the limit being taken in the essential topology of the line, which ignores sets of zero Lebesgue measure
Keywords: Multiplicative functionals,
Perfection,
Essential topologyNature: Original Retrieve article from Numdam
VI: 23, 243-252, LNM 258 (1972)
MEYER, Paul-André
Quelques autres applications de la méthode de Walsh (``La perfection en probabilités'') (
Markov processes)
This is but an exercise on using the method of the preceding paper
622 to reduce the exceptional sets in other situations: additive functionals, cooptional times and processes, etc
Comment: A correction to this paper is mentioned on the errata list of vol. VII
Keywords: Additive functionals,
Return times,
Essential topologyNature: Original Retrieve article from Numdam
