Let $k$ be a perfect field of characteristic $p>0$ and let $K$ be
a finite extension of
$K_0=W(k)[p^{-1}]$. For a proper (strictly) semi-stable (formal) scheme
$X$ over a
${\cal O}_K$ we have the Hyodo-Kato isomorphism $\rho$ between the
de Rham cohomology
$H_{dR}^*(X_K)$ of the generic fibre $X_K$ and the Hyodo-Kato cohomology
$H_{HK}^*(Y)$
of the special fibre $Y$ of $X$, endowing (a $K_0$-lattice of) $H_{dR}^*(X_K)$
with a
Frobenius operator $\phi$ and a monodromy operator $N$. On the other
hand, the components
of $Y$ give rise to a canonical admissible open covering of $X_K$,
viewed as a rigid
analytic space, and thus to a canonical Cech-filtration $(F_C^r)_{r\ge0}$
on
$H_{dR}^*(X_K)$. I want to talk about a rigid analytic construction
of $\rho$;
it can be used to relate $(F_C^r)_{r\ge0}$ with $N$. Then I specialize
to the case where
$[K:\mathbb{Q}_p]$ is finite and $X_K$ is the quotient of Drinfel'd's
$p$-adic symmetric space
$\Omega^{(d+1)}_K$ by a cocompact discrete torsionfree subgroup of
$\mbox{PGL}_{d+1}(K)$.
Here the analytic construction of $\rho$ allows us to identify $(F_C^r)_{r\ge0}$
with the slope and the weight filtration for $\phi$ on $H_{dR}^d(X_K)$.
Together
with the now proven $C_{st}$-conjecture we get a new proof for the
fact that $(F_C^r)_{r\ge0}$
is opposite to the Hodge filtration, as was shown earlier by Spiess
and Iovita.