  Hamza-Klebaner posed the problem of constructing martingales with Brownian
marginals that differ from Brownian motion, so called fake Brownian motions.
Besides its theoretical appeal, the problem represents the quintessential
version of the ubiquitous fitting problem in mathematical finance where the
task is to construct martingales that satisfy marginal constraints imposed by
market data.
  Non-continuous solutions to this challenge were given by Madan-Yor,
Hamza-Klebaner, Hobson, and Fan-Hamza-Klebaner whereas continuous (but
non-Markovian) fake Brownian motions were constructed by Oleszkiewicz, Albin,
Baker-Donati-Yor, Hobson, Jourdain-Zhou.
  In contrast it is known from Gy\"ongy, Dupire, and ultimately Lowther that
Brownian motion is the unique continuous strong Markov martingale with Brownian
marginals.
  We took this as a challenge to construct examples of a "very fake'' Brownian
motion, that is, continuous Markov martingales with Brownian marginals that
miss out only on the strong Markov property.
