  The recent detection of a giant exosphere surrounding the warm Neptune GJ436
b has shed new light on the evaporation of close-in planets, revealing that
moderately irradiated, low-mass exoplanets could make exceptional targets for
studying this mechanism and its impact on the exoplanet population. Three
HST/STIS observations were performed in the Lyman-$\alpha$ line of GJ436 at
different epochs, showing repeatable transits with large depths and extended
durations. Here, we study the role played by stellar radiation pressure on the
structure of the exosphere and its transmission spectrum. We found that the
neutral hydrogen atoms in the exosphere of GJ436 b are not swept away by
radiation pressure as shown to be the case for evaporating hot Jupiters.
Instead, the low radiation pressure from the M-dwarf host star only brakes the
gravitational fall of the escaping hydrogen toward the star and allows its
dispersion within a large volume around the planet, yielding radial velocities
up to about -120 km s$^{-1}$ that match the observations. We performed
numerical simulations with the EVaporating Exoplanets code (EVE) to study the
influence of the escape rate, the planetary wind velocity, and the stellar
photoionization. While these parameters are instrumental in shaping the
exosphere and yield simulation results in general agreement with the
observations, the spectra observed at the different epochs show specific,
time-variable features that require additional physics.
