The eigenvalues of the master equation describing the motion on a nested hierarchy ofd-dimensional intervals with selfsimilar scaling of spatial extension as well as of the level dependent transition rates are derived. Based on this spectrum the diffusion behaviour is obtained, which is anomalous, either exponential or obeying a power law with various exponents. Emphasis is put on the insight into the mechanism of the anomalous diffusion, in particular the geometrical structure of the decay rate spectrum.

We calculate stationary state correlation functions of the anharmonic overdamped oscillator driven by multiplicative (white Gaussian) noise of strengthQ together with additive noise of relative strengthq. (i) We donot observe a particular slowing down at the so-called noise induced transition. But there is a region of the oscillator's stiffness parametera neara≈0 in which the decay time is typically enhanced. (ii) If the phase transition point is defined by the minimum of the decay rate, it lies within the ordered phasea>0 forq⋟1 and shifts down toa=0 forq→0. In our approximation it is even in the disordered regiona<0 for very smallq. (iii) As a function of the multiplicative noiseQ the decay ratedecreases with increasingQ if the system is well above or well below threshold. There seem to be experimental indications of this behavior. But within the proper threshold regime of smalla increasing noiseQ increases the decay rate. The valuesa c where the cross-over occurs depend on the fluctuating variable and onq.

We apply the renormalization group idea to a stationary probability distribution which is supposed to represent a turbulent fluid. In contrast to the common procedure the R.G.T. is defined by eliminating successivelylow wave numbers instead of integrating from largek. This means that instead of starting from the short distance fluctuations, as near phase transitions, the procedure corresponds to the von Weizsäcker-Heisenberg averaging over nestedr-space volumes of decreasing size.

With the use of the Herivel-Lin variational principle to describe incompressible inviscid fluids, a fluctuation Hamiltonian is derived in terms of a complex field $\ensuremath{\Psi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}, t)$ representing the fluid. The source-free part of the field current describes the velocity field. The field interaction is found to be of current-current type, i.e., ${({\ensuremath{\Psi}}^{*} \mathrm{grad} \ensuremath{\Psi}\ensuremath{-}\mathrm{c}.\mathrm{c}.)}^{2}$ instead of the commonly used ${|\ensuremath{\Psi}|}^{4}$ type. The pressure, together with a complete energy-momentum tensor of the $\ensuremath{\Psi}$ field, is introduced, depending on the boundary conditions. The latter imply a long-range interaction near the onset of turbulence. Viscosity is taken into account by proper extension of the equation of motion of $\ensuremath{\Psi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}, t)$, which turns out to be of the Landau type. This equation is solved exactly for laminar plane shear flow. The eddy interaction in $k$ space for fully developed turbulence is given together with a model Hamiltonian for the effects of viscosity. Finally the transient behavior of the $\ensuremath{\Psi}$-field amplitude of a fixed spatial mode near the onset of turbulence is compared with experiments.