Holes in the two-dimensional probability density of bistable systems driven by strongly colored noise

Stochastic relaxation in a one-dimensional, bistable potential driven by colored noise ε is investigated by means of both numerical codes and analog simulation. By embedding the process under study x into a suitable two-dimensional phase space (x,ε), a striking topological effect, observable in the two-dimensional probability density P(x,ε), appears at a critical value of the noise correlation time ${\mathrm{\tau }}_{\mathit{c}}$. The change of the topology in the neighborhood of ${\mathrm{\tau }}_{\mathit{c}}$ can be characterized as a critical transition. In particular, the top of the potential barrier is shown to give rise to a single saddle in the trajectory space of the embedding process so long as τ${\mathrm{\tau }}_{\mathit{c}}$. Where τ=${\mathrm{\tau }}_{\mathit{c}}$, however, the single saddle disappears and is replaced by a pair of saddles, which move away from the location of the top of the potential barrier when τ>${\mathrm{\tau }}_{\mathit{c}}$, leaving behind a concave depression or ‘‘hole’’ in the probability density. The relevance of this observation to the problem of calculating the mean first-passage time for particle escape from one well to the other at large τ is discussed in detail. A complete analytical description of this transition has defied all attempts thus far. Consequently, many interesting open questions remain.