Decay of an unstable state.

The relaxation of a system from an initially unstable state towards a final stable state under the influence of random forces of microscopic origin is a fundamental problem in many branches of physics. ' A more diffi-' cult but related problem is that of the relaxation of a metastable state. These relaxation processes are examples of a stochastic process which eventually terminates when a certain variable characterizing the state of the system reaches a fixed threshold. The classic first-passage-time (FPT) problem arises naturally in the description of the dynamics of such processes. Although this problem has a long and distinguished history and has been studied in many papers in various specific contexts, the general problem has proved to be solvable for only the simplest of stochastic processes. In what follows, we wish to point out some general features of the FPT problem and show that for a large class of stochastic processes the FPT probability density can be represented by one or the other of the two analytic expressions given below. En the following we first present the results for one-dimensional problems and then generalize them to multidimensional processes with isotropic diffusion. G(x, tp+T;xp, tp) ~ „„=0.