Functional-calculus approach to stochastic differential equations

The connection between stochastic differential equations and associated Fokker-Planck equations is elucidated by the full functional calculus. One-variable equations with either additive or multiplicative noise are considered. The central focus is on approximate Fokker-Planck equations which describe the consequences of using ``colored'' noise, which has an exponential correlation function and a correlation time \ensuremath{\tau}. To leading order in \ensuremath{\tau}, the functional-calculus approach generalizes the \ensuremath{\tau}-expansion result and produces an approximate Fokker-Planck equation free from certain difficulties which have plagued the less general approximations. Mean first-passage-time behavior for bistable potentials, an additive case, is discussed in detail. The new result presented here leads to a mean first-passage-time formula in quantitative agreement with the results of numerical simulation and in contrast with earlier theoretical conclusions. The theory provides new results for the multiplicative case as well.