  We investigate asymptotic decay phenomenon towards the nonequilibrium steady
state of the thermal diffusion in the presence of a tilted periodic potential.
The parameter dependence of the decay rate is revealed by investigating the
Fokker-Planck (FP) equation in the low temperature case under the spatially
periodic boundary condition (PBC). We apply the WKB method to the associated
Schr\"odinger equation. While eigenvalues of the non-Hermitian FP operator are
complex in general, in a small tilting case the imaginary parts of the
eigenvalues are almost vanishing. Then the Schr\"odinger equation is solved
with PBC. The decay rate is analyzed in the context of quantum tunneling
through a triple-well effective periodic potential. In a large tilting case,
the imaginary parts of the eigenvalues of FP operator are crucial. We apply the
complex-valued WKB method to the Schr\"odinger equation with the absorbing
boundary condition, finding that the decay rate saturates and depends only on
the temperature, the potential periodicity and the viscous constant. The
intermediate tilting case is also explored. The analytic results agree well
with the numerical data for a wide range of tilting.
