We first consider the one-dimensional stochastic flow dx/dt = f(x) + g(x) xi(t), where xi(t) is a dichotomous Markov noise. A procedure involving the algebra of the relevant differential operators is used to identify the conditions under which the integro-differential equation satisfied by the total probability density P(x,t) of the driven variable can be reduced to a differential equation of finite order. This systematizes the enumeration of the "solvable" cases, of which the case of linear drift and additive noise is a notable one. We then revisit the known formula for the stationary density that exists under suitable conditions in dichotomous flow, and indicate how this expression may be derived and interpreted on direct physical grounds. Finally, we consider a diffusion process driven by an N-level extension of dichotomous noise, and explicitly derive the higher-order partial differential equation satisfied by P(x,t) in this case. This multi-level noise driven diffusion is a process that interpolates between the usual extremes of dichotomous diffusion and Brownian motion. We comment on the possible use of certain algebraic techniques to solve the master equation for this generalized diffusion.

Date Published: 7 May 2003
PDF: 7 pages
Proc. SPIE 5114, Noise in Complex Systems and Stochastic Dynamics, (7 May 2003); doi: 10.1117/12.497510

Published in SPIE Proceedings Vol. 5114:
Noise in Complex Systems and Stochastic Dynamics
Lutz Schimansky-Geier; Derek Abbott; Alexander Neiman; Christian Van den Broeck, Editor(s)