  We present a generalized picture of intermittency in turbulence that is based
on the theory of stochastic processes. To this end, we rely on the
experimentally and numerically verified finding by R.~Friedrich and J.~Peinke
[Phys. Rev. Lett. 78, 863 (1997)] that allows for an interpretation of the
turbulent energy cascade as a Markov process of the velocity increments in
scale. It is explicitly shown that all known phenomenological models of
turbulence can be reproduced by the Kramers-Moyal expansion of the velocity
increment probability density function that is associated to a Markov process.
We compare the different sets of Kramers-Moyal coefficients of each
phenomenology and deduce that an accurate description of intermittency should
take into account an infinite number of coefficients. This is demonstrated in
more detail for the case of Burgers turbulence that exhibits the strongest
intermittency effects. Moreover, the influence of nonlocality on the
Kramers-Moyal coefficients is investigated by direct numerical simulations of a
generalized Burgers equation. Depending on the balance between nonlinearity and
nonlocality, we encounter different intermittency behaviour that ranges from
self-similarity (purely nonlocal case) to intermittent behaviour (intermediate
case that agrees with Yakhot's mean field theory [Phys. Rev. E 63 026307
(2001)]) to shock-like behaviour (purely nonlinear Burgers case).
