Systems Containing Random Parameters with Small Correlation Times

Abstract

Systems described by differential equations involving delta-function-correlated (i.e. white noise) random parameters are discussed. To be physically meaningful, solutions of such equations should be interpreted as the limits of solutions of the corresponding equations with realistic, i.e. finite, correlation time random processes. The implications of this are explored here. We consider first linear systems. Using the Poisson case as the basic process, a “superposition” principle is derived, allowing one to treat any delta-function-correlated process. Closed sets of ordinary linear differential equations are found for the moments. The treatment is then generalized to the non-linear case. Finally, we derive conditions under which a delta-function-correlated process is a valid approximation to one with finite correlation time in a specific stochastic equation and we show how the appropriate approximation may be found.