Advances in Mathematical Physics
Volume 2010 (2010), Article ID 509326, 42 pages
doi:10.1155/2010/509326
Research Article

Introducing Randomness into First-Order and Second-Order Deterministic Differential Equations

John F. Moxnes1 and Kjell Hausken2

1Department for Protection, Norwegian Defence Research Establishment, P.O. Box 25, 2007 Kjeller, Norway
2Faculty of Social Sciences, University of Stavanger, 4036 Stavanger, Norway

Received 23 June 2009; Revised 1 January 2010; Accepted 1 March 2010

Academic Editor: Luigi Berselli

Copyright © 2010 John F. Moxnes and Kjell Hausken. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We incorporate randomness into deterministic theories and compare analytically and numerically some well-known stochastic theories: the Liouville process, the Ornstein-Uhlenbeck process, and a process that is Gaussian and exponentially time correlated (Ornstein-Uhlenbeck noise). Different methods of achieving the marginal densities for correlated and uncorrelated noise are discussed. Analytical results are presented for a deterministic linear friction force and a stochastic force that is uncorrelated or exponentially correlated.