  We consider the problem of an overdamped Brownian particle moving in
multiscale potential with N + 1 characteristic length scales: the macroscale
and N separated microscales. We show that the coarse-grained dynamics is given
by an overdamped Langevin equation with respect to the free energy and with a
space dependent diffusion tensor, the calculation of which requires the
solution of N fully coupled Poisson equations. We study in detail the structure
of the bifurcation diagram for one-dimensional problems and we show that the
multiscale structure in the potential leads to hysteresis effects and to
noise-induced transitions. Furthermore, we obtain an explicit formula for the
effective diffusion coefficient for a self-similar separable potential and we
investigate the limit of infinitely many small scales.
