  In micro-fluidics not only does capillarity dominate but also thermal
fluctuations become important. On the level of the lubrication approximation,
this leads to a quasi-linear fourth-order parabolic equation for the film
height $h$ driven by space-time white noise. The gradient flow structure of its
deterministic counterpart, the thin-film equation, which encodes the balance
between driving capillary and limiting viscous forces, provides the guidance
for the thermodynamically consistent introduction of fluctuations. We follow
this route on the level of a spatial discretization of the gradient flow
structure. Starting from an energetically conformal finite-element (FE)
discretization, we point out that the numerical mobility function introduced by
Gr\"un and Rumpf can be interpreted as a discretization of the metric tensor in
the sense of a mixed FE method with lumping. While this discretization was
devised in order to preserve the so-called entropy estimate, we use this to
show that the resulting high-dimensional stochastic differential equation (SDE)
preserves pathwise and pointwise strict positivity, at least in case of the
physically relevant mobility function arising from the no-slip boundary
condition. As a consequence, this discretization gives rise to a consistent
invariant measure, namely a discretization of the Brownian excursion (up to the
volume constraint), and thus features an entropic repulsion. The price to pay
over more naive discretizations is that when writing the SDE in It\^o's form,
which is the basis for the Euler-Mayurama time discretization, a correction
term appears. To conclude, we perform various numerical experiments to compare
the behavior of our discretization to that of the more naive finite difference
discretization of the equation.
