A functional method for calculating averages of the time-ordered exponential of a continuous isotropic random N × N matrix process is presented. The process is not assumed to be Gaussian. In particular, the Lyapunov exponents and higher correlation functions of the T-exponent are derived from the statistical properties of the process.The approach may be of use in a wide range of physical problems. For example, in theory of turbulence the account of non-gaussian statistics is very important since the non-Gaussian behavior is responsible for the time asymmetry of the energy flow.

Statistical moments of magnetic field in a viscous range of turbulence are calculated for arbitrary initial conditions. It is shown that the evolution of magnetic field in the case of finite initial distribution in a linear velocity field consists of two or three consecutive regimes: exponential growth is followed by exponential decay. This solves the apparent contradiction between "anti-dynamo" theorems and growth of magnetic field with statistically homogeneous initial conditions.

Statistical properties of a statistically homogeneous random magnetic field in a viscous diffusive fluid are derived from the evolution of a single blob of the magnetic field. It is shown that, although the magnetic field of a single blob decreases in time, the volume occupied by the magnetic field and its energy increase; this is the cause of the magnetic field growth in a homogeneous medium. We also get an exact expression for the increment of the magnetic field in the generalized (not time-inversible) Kraichnan model.

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