The Kramers-Moyal expansion Taylor-expands the gain terms of a jump-process master equation in the jump size. Keeping its first two terms yields a Fokker-Planck equation whose drift and diffusion coefficients are the first two infinitesimal jump moments.
For jumps at rate and at rate , the diffusion approximation hasThe stable deterministic population is . Its linear-noise stationary approximation is normal with mean and variance .
For a jump process with rate from state to , probability balance givesThe first term is inflow from and the second is outflow from .
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The Kramers–Moyal expansion is a mathematical framework used in stochastic processes, particularly in the context of describing the dynamics of systems subjected to random influences. It provides a way to derive the Fokker-Planck equation, which governs the time evolution of the probability density function of a stochastic variable. **Key concepts of the Kramers-Moyal expansion:** 1.