Fundamentals of higher order stochastic equations

Stochastic processes that correspond to equations of arbitrary differential order are obtained. We show that stochastic paths in a complex extension of the original phase-space allow an implementation of such higher-order derivative terms. The resulting stochastic process is equivalent to the original partial differential equation in the sense of having equivalent analytic moments. However, the correspondence has unusual properties. Only the analytic moments are convergent, while non-analytic moments such as the complex moduli are non-convergent. These results unify previous approaches that transform higher-derivative equations into probabilistic stochastic equations. Larger ensembles are required as time-steps are reduced, giving these equations unusual convergence properties. This type of process is relevant to the question of how to obtain stochastic quantum simulations using the Wigner representation