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Velocity increments over a distance r and turbulent energy dissipation on abox of size r are well described by the multifractal models of fully developedturbulence. These quantities and models however, do not involvetime-correlations and therefore are not a detailed test of the dynamics of theturbulent cascade. If the time development of the turbulent cascade, in theinertial range, is related to the lifetime of the eddies at different lengthscales, the time correlations may be described by a stochastic process on atree with jumping kernels which are a function of the ultrametric (tree)distance. We obtain the solutions of the Chapman-Kolmogorov equation for such astochastic process, with jumping kernels depending on the ultrametric distance,but with an arbitrarily specified invariant probability measure. We then showhow to use these solutions to compute the time correlations in the turbulentcascade.

Stochastic process for the dynamics of the turbulent cascade