arXiv:quant-ph/9507002

Nonnegative Feynman-Kac Kernels in Schrödinger's Interpolation Problem

Ph. Blanchard, Piotr Garbaczewski, Robert Olkiewicz

Published Jul 5, 1995• 1 citation

Abstract

The existing formulations of the Schrödinger interpolating dynamics, which is constrained by the prescribed input-output statistics data, utilize strictly positive Feynman-Kac kernels. This implies that the related Markov diffusion processes admit vanishing probability densities only at the boundaries of the spatial volume confining the process. We extend the framework to encompass singular potentials and associated nonnegative Feynman-Kac-type kernels. It allows to deal with general nonnegative solutions of the Schrödinger boundary data problem. The resulting stochastic processes are capable of both developing and destroying nodes (zeros) of probability densities in the course of their evolution.

DOI: 10.48550/arxiv.quant-ph/9507002

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arXiv:quant-ph/9507002

Nonnegative Feynman-Kac Kernels in Schrödinger's Interpolation Problem

Ph. Blanchard, Piotr Garbaczewski, Robert Olkiewicz

Published Jul 5, 1995• 1 citation

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.quant-ph/9507002

Related papers

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{\cal{V}}(x)

, continuous, confining and bounded from below, may take negative values. If positive,

{\cal{V}}(x)

can be interpreted as the killing rate of the decaying diffusion process. In case of relaxing F-K kernels the killing effects are tamed (often overcompensated). The taming inavoidably appears in conjunction with the existence of the negativity subdomains of

{\cal{V}}(x)

R

. If locally

{\cal{V}}(x) < 0

, its sign inversion

- {\cal{V}}(x)

can be interpreted as the branching (cloning, alternatively bifurcation) rate in the course of the other wise free random motion. The arising killed diffusion processes with branching, we interpret as the possible path-wise background of tamed (relaxing) Feynman-Kac diffusions. We present acomputer-assisted path-wise arguments, towards a consistency of the killing/branching taming scenario, for a number of nonlinear model systems in one space dimension. Special attention is paid to Feynman-Kac potential shapes in the double well form, where an analytic access to eigenvalues and eigenfunctions is scarce. Throughout the paper the dynamics refers to the positive real time. Since the Newton-type equations of motion for admissible classical trajectories have a Euclidean form (due to the sign inverted force term), we give a brief resume of a couple of their explicit solutions, without recourse to the Euclidean time intuitions, and the instanton lore of related quantum model systems.

|\nabla |