Principal eigenvalue characterization connected with stochasticparticle motion in a finite interval

[摘要] In this paper, we show that despite their distinction, both the Statonovich and Îto scalculi lead to the same reactive Fokker-Planck equation:∂p∂t−∂∂x[D∂p∂x−bp]=λmp,  (1)describing stochastic dynamics of a particle moving under the influence of anindefinite potentialm(x,t), a driftb(x,t), and a constant diffusionD. We treat the periodic-parabolic eigenvalue problem (1) for finite domains havingabsorbing barriers. We show that under conditions required by the maximumprinciple, the positive principal eigenvalueλ*(and the negative principalλ*eigenvalue) is connected to the probability eigendensity functionp(x,t)by a Raleigh-Ritz like formulation. In the process, we establish the manner of effect ofthe drift and any inducing potential on the size of the principal eigenvalue. Weshow that the degree of convexity of the potential plays a major role in thisregard.

[学科分类] 应用数学