Principal eigenvalue characterization connected with stochastic particle motion in a finite interval
Summary: We show that despite their distinction, both the Stratonovich and Itô calculi lead to the same reactive Fokker-Planck equation: \[ {\partial p\over\partial t}-{\partial\over \partial x}\left[ D{\partial p\over \partial x}-b p\right]= \lambda mp, \tag{1} \] 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 \(\lambda^*\) (and the negative principal eigenvalue \(\lambda_*)\) is connected to the probability eigendensity function \(p(x,t)\) by a Reileigh-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.
scientific article; zbMATH DE number 1933143