For log smooth rigid analytic varieties, Kummer pro-étale B_dR cohomology is isomorphic to log de Rham cohomology, and logarithmic B_dR^+ cohomology gives degeneration of Hodge-Tate and Hodge-log de Rham spectral sequences when proper.
Crystalline cohomology of rigid analytic spaces
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In this article, we introduce infinitesimal cohomology for rigid analytic spaces that are not necessarily smooth, with coefficients in a p-adic field or Fontaine's de Rham period ring.
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On the Kummer pro-\'etale cohomology of $\mathbb B_{\operatorname{dR}}$
For log smooth rigid analytic varieties, Kummer pro-étale B_dR cohomology is isomorphic to log de Rham cohomology, and logarithmic B_dR^+ cohomology gives degeneration of Hodge-Tate and Hodge-log de Rham spectral sequences when proper.