Kramers-Moyal Coefficients Research Articles

The Markovian metamorphosis of a simple turbulent cascade model

Markovian properties of a discrete random multiplicative cascade model of log-normal type are discussed. After taking small-scale resummation and observational breaking of the ultrametric hierarchy into account, qualitative agreement with Kramers–Moyal coefficients, recently deduced from a fully developed turbulent flow, is achieved.

How to quantify deterministic and random influences on the statistics of the foreign exchange market

It is shown that price changes of the U.S. dollar-German mark exchange rates upon different delay times can be regarded as a stochastic Marcovian process. Furthermore, we show how Kramers-Moyal coefficients can be estimated from the empirical data. Finally, we present an explicit Fokker-Planck equation which models very precisely the empirical probability distributions, in particular, their non-Gaussian heavy tails.

Theoretical Model for the Kramers-Moyal Description of Turbulence Cascades

We derive the Kramers-Moyal equation for the conditional probability density of velocity increments from the theoretical model recently proposed by V.Yakhot [Phys.Rev.E {\bf 57}, 1737 (1998)] in the limit of high Reynolds number limit. We show that the higher order (n>=3) Kramers-Moyal coefficients tends to zero and the velocity increments are evolved by the Fokker-Planck operator. Our result is compatible with the phenomenological descriptions by R.Friedrich and J.Peinke [Phys.Rev.Lett. {\bf 78}, 863 (1997)].

Stochastic inflation: scaling solution for N fields

The Fokker-Planck equation for the primordial inflationary fluctuations is solved in the scaling approximation, when the classical drift dominates over the quantum diffusion. The scaling procedure is generalized to the case in which several fields are involved. The conditions imposed on the Kramers-Moyal coefficients in order to derive the scaling variables turn out to be fulfilled in some simple multi-field case. An analytical solution in the case of a two-field inflationary model is worked out.

Bistability in active circuits: Application of a novel Fokker-Planck approach

The problem of metastability in electronic circuits with negative differential resistance, originally pioneered by Landauer in 1962, is reconsidered from the viewpoint of a Fokker-Planck modeling for nonlinear shot noise (master equation). A novel Fokker-Planck approximation scheme is presented that describes correctly the deterministic flow and the long-time dynamics of the master equation. It is demonstrated that the conventional scheme of a truncated Kramers-Moyal expansion at the second order overestimates the transition rates in leading exponential order. In order to obtain the correct relative stability, the novel scheme uses a diffusion coefficient which incorporates information about global nonlinear fluctuations characterized by the whole set of higher-order Kramers-Moyal transport coefficients.