Fundamentals of higher order stochastic equations (Q2878637)

The issue of the paper is the fundamental behavior of the higher-order stochastic processes that are equivalent to higher-order Kramers-Moyal equations, and methods to generate the processes that locally preserve the correct moments. The Wigner function time-evolution equation with anharmonic potentials is one of the examples of equations with higher-order derivative. The paper describes the methods for generating stochastic higher-order differential equations of any order, and methods for generating the relevant higher-order stochastic process. It is established that the Gaussian and polar approaches have similar behavior, but the polar method results in lower sampling error. Unlike conventional second-order Gaussian stochastic noise, sampling error increases as the step-size is reduced, so that convergence in stepsize does not commute with convergence in ensemble. It is demonstrated that the corresponding stochastic equations with constant coefficients are well-behaved apart from these unusual convergence properties. Numerical results obtained are in good agreement with the predicted moments for exactly solvable cases treated with constant coefficients.