Generic behavior in linear systems with multiplicative noise

A classification of the probability distribution of linear multiplicative noise equations is developed which has a variety of physical applications, for example, to the temperature field of a turbulent fluid undergoing a chemical reaction. Without a reaction the tails of the distribution are shown generically to have exponential or stretched exponential behavior. With a reaction occurring, the tails cross over to power-law behavior. A simple criterion in terms of the generalized Lyapunov exponents for the system explains this kind of behavior. Rigorous results for the existence of power-law tails under general symmetry conditions are presented and are easily understood in terms of generalized Lyapunov exponents.