Brownian motion in trapping enclosures: Steep potential wells, bistable wells and false bistability of induced Feynman-Kac (well) potentials

Abstract

We investigate signatures of convergence for a sequence of diffusion processes on a line, in conservative force fields stemming from superharmonic potentials $U(x)\sim x^m$, $m=2n\ge 2$. This is paralleled by a transformation of each $m$-th diffusion generator $L=D\Delta +b(x)\nabla$, and likewise the related Fokker-Planck operator $L^{\ast }=D\Delta -\nabla [b(x)\cdot ]$, into the affiliated Schrödinger one $\hat{H}=-D\Delta +\mathcal{V}(x)$. Upon a proper adjustment of operator domains, the dynamics is set by semigroups $\exp (tL)$, $\exp (t{L}_{\ast })$ and $\exp (-t\hat{H})$, with $t\ge 0$. The Feynman-Kac integral kernel of $\exp (-t\hat{H})$ is the major building block of the relaxation process transition probability density, from which $L$ and $L^{\ast }$ actually follow. The spectral "closeness" of the pertinent $\hat{H}$ and the Neumann Laplacian $-{\Delta }_{\mathcal{N}}$ in the interval is analyzed for $m$ even and large. As a byproduct of the discussion, we give a detailed description of an analogous affinity, in terms of the $m$-family of operators $\hat{H}$ with a priori chosen $\mathcal{V}(x)\sim x^m$, when $\hat{H}$ becomes spectrally "close" to the Dirichlet Laplacian $-{\Delta }_{\mathcal{D}}$ for large $m$. For completness, a somewhat puzzling issue of the absence of negative eigenvalues for $\hat{H}$ with a bistable-looking potential $\mathcal{V}(x)=a{x}^{2m-2}-b{x}^{m-2},a,b,>0,m>2$ has been addressed.

Rewritten, two new subsections, 32 pp, 16 Fig