  The linearized of the Poisson-Nernst-Planck (PNP) equation under closed ends
around a neutral state is studied. It is reduced to a damped heat equation
under non-local boundary conditions, which leads to a stochastic interpretation
of the linearized equation as a Brownian particle which jump and is reflected,
at Poisson distributed time, to one of the end points of the channel, with a
probability which is proportional to its distance from this end point. An
explicit expansion of the heat kernel reveals the eigenvalues and eigenstates
of both the PNP equation and its dual. For this, we take advantage of the
representation of the resulvent operator and recover the heat kernel by
applying the inverse Laplace transform.
