Contributions to the Theory of Multiplicative Stochastic Processes

The theory of multiplicative stochastic processes is contrasted with the theory of additive stochastic processes. The case of multiplicative factors which are purely random, Gaussian, stochastic processes is treated in detail. In a spirit originally introduced by theoretical work in nuclear magnetic resonance and greatly extended by Kubo, dissipative behavior is demonstrated, on the average, for dynamical equations which do not show dissipa-tive behavior without averaging. It is suggested that multiplicative stochastic processes lead to a conceptual foundation for nonequilibrium thermodynamics and nonequilibrium statistical mechanics, of marked generality. The purpose of this paper is to present results in the theory of "multiplicative stochastic processes." The physical applications of this theory will be presented in a sequel to this paper. Effective use of stochastic processes in physics was first achieved in the theory of Brownian motion. 1 The basic ideas were generalized by Onsager and Machlup in their theory of fluctuations and irreversible processes. 2 Further generalizations, which resulted in a general stochastic theory for the linear dynamical behavior of classical thermodynamical syst'ems, close to but not yet in full equilibrium, were presented by Fox and Uhlenbeck. 3 ,4 The theory of Fox and Uhlenbeck includes the Langevin theory of Brownian motion and the Onsager and Machlup theory for irreversible processes as special cases. In addition , it includes the linearized fluctuating hydrodyna-mical equations of Landau and Lifshitz 5 and the linearized fluctuating Boltzmann equation as special cases. In each of these special cases, and in the general theory, the mathematical description used involves either linear partial integro-differential equations or linear matrix equations which are inhomogeneous. The inhomogeneity is the stochastic "driving force" of the process. Consequently, we shall refer to these processes as "additive stochastic processes." The processes to be presented in this paper will be seen to involve homogeneous equations in which the sto-chastic "driving force" enters in a multiplicative way. These processes will, consequently, be called "multiplicative stochastic processes." Multiplicative stochastic processes arise in a natural way in the field of nuclear magnetic resonance. The nature and history of this development may be found in a paper by Redfield. 6 Major generalizations of these ideas for other areas of physics have been presented by Kub07-9 Kubo has also pursued the mathematical foundations for a theory of multiplica-tive stochastic processes in his work. The speCial attention paid to purely random, Gaussian, stochastic processes in this paper will serve to …