Lower and upper bounds for the eigenvalues of the Fokker-Planck equation in detailed balance

We derive a variational principle for the eigenvalues of the Fokker-Planck equation in detailed balance, which holds for additive as well as multiplicative stochastic processes and which is not restricted to one dimension. An intermediate theorem for the Fokker-Planck equation is presented for the first time from which lower bounds for the eigenvalues can be extracted. The same theorem can be applied to the master equation in detailed balance. We discuss as a first application the lower and upper bounds for the lowest eigenvalue of a quartic potential in one dimension. To show the applicability of our procedure to more complicated problems we give lower and upper bounds for the linewidth factor and the second eigenvalue of the Fokker-Planck equation for the single-mode laser. In addition, we discuss three models of multiplicative stochastic processes which represent three different classes of behavior. For each model we give stationary results, namely, the probability distribution and the first two nontrivial moments. To characterize the long-time behavior of the relaxation processes involved we calculate lower and upper bounds for the lowest eigenvalue. It is pointed out that, at least in the models presented, there does not exist critical slowing down as a function of the strength of the fluctuations. We conclude that one should be cautious when using the term noise-induced phase transition in connection with multiplicative stochastic processes.