The aim of the paper is to a provide a link between two approaches that describe the cascade in fully developed turbulence. Precisely, we show that the propagator approach (B. Castaing) and the Markovian description (Friedrich & Peinke) are equivalent in the log-normal case. To prove this, we use an Itô stochastic differential equation in scale - Friedrich & Peinke approach - that can be explicitly integrated. The solution corresponds to the propagator approach. This is done for scale-invariant cascades as well as non scale-invariant cascade.