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Hamza-Klebaner posed the problem of constructing martingales with Brownianmarginals that differ from Brownian motion, so called fake Brownian motions.Besides its theoretical appeal, the problem represents the quintessentialversion of the ubiquitous fitting problem in mathematical finance where thetask is to construct martingales that satisfy marginal constraints imposed bymarket data.  Non-continuous solutions to this challenge were given by Madan-Yor,Hamza-Klebaner, Hobson, and Fan-Hamza-Klebaner whereas continuous (butnon-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 thatBrownian motion is the unique continuous strong Markov martingale with Brownianmarginals.  We took this as a challenge to construct examples of a "very fake'' Brownianmotion, that is, continuous Markov martingales with Brownian marginals thatmiss out only on the strong Markov property.

Faking Brownian motion with continuous Markov martingales