Getting a stochastic process from a conservative Lagrangian: A first approach

The transition probability PV for a stochastic process generated by a conservative Lagrangian L=L0−eV is obtained at first order from a perturbation series found using a path integral. This PV corresponds to the transition probability for a random walk with a probability density given by the sum of a normal distribution and a perturbation which may be understood as the contribution of the interaction of the random walk with the external field. It is also found that the moment-generating function for PV can be expressed as the generating function of a normal distribution modified by a perturbation. Applications of these results to a linear potential, a harmonic oscillator potential, and an exponentially decaying potential are shown.