  The voltage-conductance kinetic equation for an ensemble of neurons has been
studied by many scientists and mathematicians, while its rigorous analysis is
still at a premature stage. In this work, we obtain for the first time the
exponential convergence to the steady state of this kinetic model in the linear
setting. Our proof is based on a probabilistic reformulation, which allows us
to investigate microscopic trajectories and bypass the difficulties raised by
the special velocity field and boundary conditions in the macroscopic equation.
We construct an associated stochastic process, for which proving the
minorization condition becomes tractable, and the exponential ergodicity is
then proved using Harris' theorem.
