Exact cumulant Kramers–Moyal-like expansion

• Being an exact approach to derive a time-evolution equation for the PDF of a generic system.
• Having a distinct structure of jump-moments from the usual Fokker–Planck or Kramers–Moyal equations.
• Furnishing the time-evolution equation for systems that are not necessarily driven by Langevin-like equations.
• Allowing us to obtain the exact evolution for all the averages and cumulants of the system.

We derive an exact equation, a Cumulant Kramers–Moyal Equation (CKME), quite similar to the Kramers–Moyal Equation (KME), for the probability distribution of a Markovian dynamical system. It can be applied to any well behaved (converging cumulants) continuous time systems, such as Langevin equations or other models. An interesting but significant difference with respect to the KME is that their jump-moments are proportional to cumulants of the dynamical variables, but not proportional to central moments, as is the case for the KME. In fact, they still obey a weaker version of Pawula’s theorem, namely Marcinkiewicz’s theorem. We compare the results derived from the equations herein with the ones obtained by computing via Gaussian and biased, and unbiased, Poisson Langevin dynamics and a Poisson non-Langevin model. We obtain the exact CKME time-evolution equation for the systems, and in several cases, those are distinct from the Fokker–Planck equation or the KME.