  We consider a particle living in $\mathbb{R}_+$, whose velocity is a positive
recurrent diffusion with heavy-tailed invariant distribution when the particle
lives in $(0,\infty)$. When it hits the boundary $x=0$, the particle restarts
with a random strictly positive velocity. We show that the properly rescaled
position process converges weakly to a stable process reflected on its infimum.
From a P.D.E. point of view, the time-marginals of $(X_t, V_t)_{t\geq0}$ solve
a kinetic Fokker-Planck equation on $(0,\infty)\times\mathbb{R}_+ \times
\mathbb{R}$ with diffusive boundary conditions. Properly rescaled, the
space-marginal converges to the solution of some fractional heat equation on
$(0,\infty)\times\mathbb{R}_+$.
