It is shown that for systems with a periodic potential, the flux is very sensitive to the strength of additive and/or
multiplicative noise. Multiplicative noise becomes important when its strength is of order the barrier height, and
it provides a means of additional control of the flux (voltage-current characteristics for a Josephson junction). In
addition to a numerical analysis, the cases of weak and strong additive noise have been considered analytically.
This short review contains description of distinctive features of the one-dimensional classical harmonic and
nonlinear oscillator with additive and multiplicative noises. In addition to the old articles considered at great
details in the recently appeared book,1 we consider some new articles appeared after 2005.
Multiplicative noise in the equation of motion of an overdamped oscillator leads to a series of new effects. These include the shift of stable points and stochastic resonance in linear system, noise enhanced stability and stabilization of metastable states in the non-linear single well model, resonance activation and coherent resonant activation in the non-linear double well model. The classical theoty of Brownian motion is supplemented by its applications to anomalous diffusion and to the special case of Brownian motion near the critical point
We analyze the stability boundaries of the convective Ginzburg-Landau equation which describes the phase transitions in moving systems. The stabiity criteria are different for convective velocities which are constant, random or varying periodically with time or coordinate. For the vortices in superconductors subjected to a magnetic field and bias current, the instability will manifest itself in the drastic change in the distribution of order and disorder state along a sample.
Different ways to stabilize a classical particle located at metastable or unstable state include an application of a periodic field or noise. We consider discrete and space-extended double-well potentials. Exact calculations are performed for the piecewise potentials.