On the first integral of the Euler-Lagrange equation, boundary conditions and escape rates for stochastic processes

Following the path integral formalism for stochastic processes, we study formal properties of the extremal path, the Euler-Lagrange (E-L) equation and a first integral of it. A special interest is focused in the connection between this first integral and the boundary conditions. We are interested in the maximum probability for the variable to attain a given value, or escape rate process, so we analyze the variational problem of a mobile end point of the extremal path; this is a transversality condition, which in the present context means a stationary condition and gives a zero value for the first integral of the E-L equation. Under a certain initial condition, we can handle this differential equation as an initial value problem. This more rigorous approach to the variational problem is applied to some specific potentials, both for white and colored noise, in order to see clearly the qualitative difference introduced by the noise correlation time τ. The τ-dependence of the extremal action exhibits features that result from the nonlinearity of the integral of motion, even in the case of the quadratic potential. This behavior of S(τ)has been first reported recently by the author, for the bistable potential [1]. The present results can provide a better insight in the study of first-passage-time problems.