Berkovich (Invent. Math. 125(2):367-390, 1996), Fujiwara (Duke Math. J. 80(1):15-57, 1995) and Huber (J. Algebraic Geom. 7(2):359-403, 1998) have proved that the fiber of the vanishing cycles at a point of the special fiber depends only on the formal completion at this point. We refine this result and prove the invariance under formal completion of the perverse monodromy filtration on the fiber of vanishing cycles. This result is used in an essential way by Boyer (Invent. Math. doi:10.1007/s00222-009-0183-9, 2009)
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