  We consider several critical wetting models. In the discrete case, these
probability laws are known to converge, after an appropriate rescaling, to the
law of a reflecting Brownian motion, or of the modulus of a Brownian bridge,
according to the boundary conditions. In the continuous case, a corresponding
convergence result is proven in this paper, which allows to approximate the law
of a reflecting Brownian motion by the law of Brownian meander tilted by its
local time near the origin. On the other hand, these laws can be seen as the
reversible probability measure of some Markov processes, namely, the dynamics
which are encoded by integration by parts formulae. After proving the tightness
of the associated reversible dynamics in the discrete case, based on heuristic
considerations on the integration by parts formulae, we provide a conjecture on
the limiting process, which we believe to satisfy a Bessel SPDE as introduced
in a recent work by Elad Altman and Zambotti.
