  We consider a 1-dimensional Brownian motion whose diffusion coefficient
varies when it crosses the origin. We study the long time behavior and we
establish different regimes, depending on the variations of the diffusion
coefficient: emergence of a non-Gaussian mul-tipeaked probability distribution
and a dynamical transition to an absorbing static state. We compute the
generator and we study the partial differential equation which involves its
adjoint. We discuss global existence and blow-up of the solution to this latter
equation.
