Motives of rigid analytic tubes and nearby motivic sheaves

Let k be a field of characteristic zero, R D k[[t]] the ring of formal power series and K D k((t)) its fraction field. Let X be a finite type R-scheme with smooth generic fiber. Let H be the t-adic completion of X and Hη the generic fiber of H. Let Z ⊂ Xσ bea locally closed subset of the special fiber of X. In this article, we establish a relation between the rigid motive of [Z] (the tube of Z in Hη) and the restriction to Z of the nearby motivic sheaf associated with the R-scheme X. Our main result, Theorem 7.1, can be interpreted as a motivic analog of a theorem of Berkovich. As an application, given a rational point x ∈ Xσ, we obtain an equality, in a suitable Grothendieck ring of motives, between the motivic Milnor fiber of Denef-Loeser at x and the class of the rigid motive of the analytic Milnor fiber of Nicaise-Sebag at x.