Brownian motion in trapping enclosures: steep potential wells, bistable wells and false bistability of induced Feynman–Kac (well) potentials. (24th July 2020) - Garbaczewski, Piotr
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 ) ∼ x m, m = 2 n ⩾ 2. This is paralleled by a transformation of each m th diffusion generator L = D Δ + b ( x )∇, and likewise the related Fokker–Planck operator L * = D Δ − ∇[ b ( x ) ⋅], into the affiliated Schrödinger one Ĥ = − D Δ + V ( x ) . Upon a proper adjustment of operator domains, the dynamics is set by semigroups exp( tL ), exp( tL *) and exp(− tĤ ), with t ⩾ 0. The Feynman–Kac integral kernel of exp(− tĤ ) 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 Ĥ and the Neumann Laplacian − Δ 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 Ĥ with a priori chosen V ( x ) ∼ x m, when Ĥ becomes spectrally 'close' to the Dirichlet Laplacian − Δ D for large m . For completness, a somewhat puzzling issue of the absence of negative eigenvalues for Ĥ with a bistable-looking potential V ( x ) = a x 2 m − 2 − b x m − 2, a, b, > 0, m > 2 has been addressed.