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 $U(x)\sim x^m$, $m=2n \geq 2$. 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 $L^*= D\Delta - \nabla [b(x)\, \cdot]$, into the affiliated Schr\"{o}dinger one $\hat{H}= - D\Delta + {\cal{V}}(x)$. Upon a proper adjustment of operator domains, the dynamics is set by semigroups $\exp(tL)$, $\exp(tL_*)$ and $\exp(-t\hat{H})$, with $t \geq 0$. 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 $L^*$ actually follow. The spectral "closeness" of the pertinent $\hat{H}$ and the Neumann Laplacian $-\Delta_{\cal{N}}$ 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 ${\cal{V}}(x) \sim x^m$, when $ \hat{H}$ becomes spectrally "close" to the Dirichlet Laplacian $-\Delta_{\cal{D}}$ for large $m$. For completness, a somewhat puzzling issue of the absence of negative eigenvalues for $\hat{H}$ with a bistable-looking potential ${\cal{V}}(x)= ax^{2m-2} - bx^{m-2}, a, b, >0, m>2$ has been addressed.