  We compare different approaches towards an effective description of
multi-scale velocity field correlations in turbulence. Predictions made by the
operator product expansion, the so-called fusion rules, are placed in
juxtaposition to an approach that interprets the turbulent energy cascade in
terms of a Markov process of velocity increments in scale. We explicitly show
that the fusion rules are a direct consequence of the Markov property provided
that the structure functions exhibit scaling in the inertial range.
Furthermore, the limit case of joint velocity gradient and velocity increment
statistics is discussed and put into the context of the notion of dissipative
anomaly. We generalize a prediction made by the multifractal (MF) approach
derived in [Phys. Rev. Lett. 80, 3244 (1998)] to correlations among inertial
range velocity increment and velocity gradients of any order. We show that for
the case of squared velocity gradients such a relation can be derived from
"first principles" in the case of Burgers equation. Our results are benchmarked
by intensive direct numerical simulations of Burgers turbulence.
