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- W2949131013 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 = DDelta + b(x)nabla $, and likewise the related Fokker-Planck operator $L^*= DDelta - nabla [b(x), cdot]$, into the affiliated Schrodinger one $hat{H}= - DDelta + {cal{V}}(x)$. Upon a proper adjustment of operator domains, the dynamics is set by semigroups $exp(tL)$, $exp(tL_*)$ and $exp(-that{H})$, with $t geq 0$. The Feynman-Kac integral kernel of $exp(-that{H})$ 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." @default.
- W2949131013 created "2019-06-27" @default.
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- W2949131013 date "2019-06-16" @default.
- W2949131013 modified "2023-09-27" @default.
- W2949131013 title "Brownian motion in trapping enclosures: Steep potential wells and false bistability of affiliated Schroedinger type systems" @default.
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