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We define the log motivic nearby cycles functor. We show that this sends the motive of a proper smooth scheme over the fraction field of a DVR to the motive of the boundary of a log smooth model assuming absolute purity, which is unconditional in the equal characteristic case. In characteristic 0, we show that the -categories of motives over the standard log point and rigid analytic motives are equivalent, and we relate log motivic nearby cycles functor with Ayoub's motivic nearby cycles functor.
Doosung Park (Wed,) studied this question.