On nearby cycles and D-modules of log schemes in characteristic p > 0

Let K be a complete discrete valuation field of mixed characteristic (0, p) with a perfect residue field k. For a semi-stable scheme over the ring of integers O(K) of K or, more generally, for a log smooth scheme of semi-stable type over k, we define nearby cycles as a single D-module endowed with a monodromy partial derivative(log)(t), whose cohomology should give the log crystalline cohomology. We also explicitly describe the monodromy filtration of the D-module with respect to the endomorphism partial derivative(log)(t), and construct a weight spectral sequence for the cohomology of the nearby cycles.