Abstract It is argued that Kolmogorov cascades, including those affecting space weather in the solar wind plasma, can effectively be described by realizations of a continuous-time random walk. The nature of the random walk is ruled by the nonlinearity of the structure function scaling exponent characterizing intermittency and multifractality. The scale-dependent probability density functions satisfy the Montroll–Weiss equation in Fourier–Laplace space. They are solutions of a linear Boltzmann equation in the Markovian approximation for cascades subordinated to a renewal Poisson process. The cascade paths become continuous in a properly scaled transition to the diffusion limit of the random walk describing the change of the logarithm of the turbulent fluctuation amplitudes among scales.