In this talk, we will explain the construction of the marked infinitesimal site. This Grothendieck site is used to define variants of crystalline cohomology in which the poles at infinity are controlled by a decay function. The formalism goes beyond logarithmic geometry, which is in general too restrictive for p-adic cohomology theories. As a main result, we obtain an algebraic framework for studying rigid cohomology in positive characteristic. During the talk, we also discuss the connection of our construction with Deligne’s pole order filtration in Hodge II.