This chapter is reprinted from Physical Review, E, 15, June, 1994. Copyright © 1994 by The American Physical Society. Reprinted with permission.
MODULATED WIENER PROCESS
A. R. Bulsara
NCCOSC RDT&E Division, Code 573, San Diego, CA 92152-5000
S. B. Lowen
Dept. of Electrical Engineering, Columbia University, New York, NY 10027
C. D. Rees
NCCOSC RDT&E Division, Code 541, San Diego, CA 92152-5000
We consider a periodically modulated random walk ( Wiener process) to an absorbing barrier with a deterministic reset to the starting point following each barrier crossing. Cooperative effects arising from the interplay between the noise and periodic modulation are analysed as they manifest themselves in two statistical measures of the response: the passage time statistics of the process and the power spectral density of the output. Simple relationships exist between the extrema that occur in these two characterizations. The spectral properties of the response are seen to bear a striking resemblance to the stochastic resonance phenomenon that is known to occur in periodically driven noisy nonlinear systems.
The response of nonlinear dynamic systems to weak, deterministic, time-dependent stimuli in the presence of system noise has recently been of considerable interest to the statistical physics community. One of the most intriguing cooperative effects that arise out of the coupling between deterministic and random dynamics in a nonlinear system (usually taken to be bistable) is "Stochastic Resonance" (SR). This effect, originally reported by Benzi, Eckman and their co-workers  was proposed as a possible explanation for the Ice Ages . It consists of a noise-induced enhancement of the response of a nonlinear system to a weak, external, time-periodic modulation in the presence of background noise. The signal strength, measured in the output power spectral density at the stimulus frequency, can actually be enhanced over its input value through a coherent transfer of energy between the noise and stimulus-dominated hopping dynamics between the stable attractors of the system. The mechanism of SR is simple. Given a bistable dynamic system, for example, information is transmitted through the system in the form of switching events between the stable states, or attractors, of the potential function underlying the dynamics. The effect of an applied time-periodic signal is then to rock the potential, alternately raising and lowering the wells. However, should its amplitude be very low (compared to the height of the potential barrier), it will not be able to induce