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

• A stochastic process is generated through the path integral with a classical action.
• The transition probability per step is expressed as a perturbation series.
• Moment-generating function is expressed as a perturbation series.

The transition probability PVPV for a stochastic process generated by a conservative Lagrangian L=L0−εVL=L0−εV is obtained at first order from a perturbation series found using a path integral. This PVPV 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 PVPV 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.