  We extend the study of weak local conditional independence (WCLI) based on a
measurability condition made by Commenges and G\'egout-Petit (2009) to a larger
class of processes that we call D'. We also give a definition related to the
same concept based on certain likelihood processes, using the Girsanov theorem.
Under certain conditions, the two definitions coincide on D'. These results may
be used in causal models in that we define what may be the largest class of
processes in which influences of one component of a stochastic process on
another can be described without ambiguity. From WCLI we can contruct a concept
of strong local conditional independence (SCLI). When WCLI does not hold, there
is a direct influence while when SCLI does not hold there is direct or indirect
influence. We investigate whether WCLI and SCLI can be defined via conventional
independence conditions and find that this is the case for the latter but not
for the former. Finally we recall that causal interpretation does not follow
from mere mathematical definitions, but requires working with a good system and
with the true probability.
