  A Markovian bridge is a probability measure taken from a disintegration of
the law of an initial part of the path of a Markov process given its terminal
value. As such, Markovian bridges admit a natural parameterization in terms of
the state space of the process. In the context of Feller processes with
continuous transition densities, we construct by weak convergence
considerations the only versions of Markovian bridges which are weakly
continuous with respect to their parameter. We use this weakly continuous
construction to provide an extension of the strong Markov property in which the
flow of time is reversed. In the context of self-similar Feller process, the
last result is shown to be useful in the construction of Markovian bridges out
of the trajectories of the original process.

Probability