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Journal of Applied Mathematics and Stochastic Analysis
Volume 15 (2002), Issue 4, Pages 349-356

Principal eigenvalue characterization connected with stochastic particle motion in a finite interval

Kuwait University, Department of Mathematics and Computer Science, P.O. Box 5969, Safat 130, Kuwait

Received 1 January 2002; Revised 1 October 2002

Copyright © 2002 Hindawi Publishing Corporation. 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.


In this paper, we show that despite their distinction, both the Statonovich and Îto s calculi lead to the same reactive Fokker-Planck equation: ptx[Dpxbp]=λmp,  (1) describing stochastic dynamics of a particle moving under the influence of an indefinite potential m(x,t), a drift b(x,t), and a constant diffusion D. We treat the periodic-parabolic eigenvalue problem (1) for finite domains having absorbing barriers. We show that under conditions required by the maximum principle, the positive principal eigenvalue λ* (and the negative principal λ* eigenvalue) is connected to the probability eigendensity function p(x,t) by a Raleigh-Ritz like formulation. In the process, we establish the manner of effect of the drift and any inducing potential on the size of the principal eigenvalue. We show that the degree of convexity of the potential plays a major role in this regard.