Fractional diffusion limit for a kinetic Fokker-Planck equation with diffusive boundary conditions in the half-line (Q6618731)

The author considers a particle with position \((X_t)_{t\geq 0}\) in \(\mathbb R_+\), whose velocity \((V_t)_{t\geq 0}\) is a positive recurrent diffusion with heavy-tailed invariant distribution in \((0,\infty).\) When it hits the boundary \(x = 0,\) the particle restarts with a random strictly positive velocity.\N\NThe author shows that the properly rescaled position process converges weakly to a stable process reflected on its infimum.\N\NAlso, the time-marginals of \((X_t,V_t)_{t\geq 0}\) 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 \((0,\infty) \times \mathbb R_+.\)