dorsal/arxiv
View SchemaFeynman-Kac Kernels in Markovian Representations of the Schroedinger Interpolating Dynamics
| Authors | Piotr Garbaczewski, Robert Olkiewicz |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9505012 |
| URL | https://arxiv.org/abs/quant-ph/9505012 |
| DOI | 10.1063/1.531412 |
| Journal | J.Math.Phys. 37 (1996) 732-751 |
Abstract
Probabilistic solutions of the so called Schr\"{o}dinger boundary data problem provide for a unique Markovian interpolation between any two strictly positive probability densities designed to form the input-output statistics data for the process taking place in a finite-time interval. The key issue is to select the jointly continuous in all variables positive Feynman-Kac kernel, appropriate for the phenomenological (physical) situation. We extend the existing formulations of the problem to cases when the kernel is \it not \rm a fundamental solution of a parabolic equation, and prove the existence of a continuous Markov interpolation in this case. Next, we analyze the compatibility of this stochastic evolution with the original parabolic dynamics, while assumed to be governed by the temporally adjoint pair of (parabolic) partial differential equations, and prove that the pertinent random motion is a diffusion process. In particular, in conjunction with Born's statistical interpretation postulate in quantum theory, we consider stochastic processes which are compatible with the Schr\"{o}dinger picture quantum evolution.
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"abstract": "Probabilistic solutions of the so called Schr\\\"{o}dinger boundary data\nproblem provide for a unique Markovian interpolation between any two strictly\npositive probability densities designed to form the input-output statistics\ndata for the process taking place in a finite-time interval. The key issue is\nto select the jointly continuous in all variables positive Feynman-Kac kernel,\nappropriate for the phenomenological (physical) situation. We extend the\nexisting formulations of the problem to cases when the kernel is \\it not \\rm a\nfundamental solution of a parabolic equation, and prove the existence of a\ncontinuous Markov interpolation in this case. Next, we analyze the\ncompatibility of this stochastic evolution with the original parabolic\ndynamics, while assumed to be governed by the temporally adjoint pair of\n(parabolic) partial differential equations, and prove that the pertinent random\nmotion is a diffusion process. In particular, in conjunction with Born\u0027s\nstatistical interpretation postulate in quantum theory, we consider stochastic\nprocesses which are compatible with the Schr\\\"{o}dinger picture quantum\nevolution.",
"arxiv_id": "quant-ph/9505012",
"authors": [
"Piotr Garbaczewski",
"Robert Olkiewicz"
],
"categories": [
"quant-ph",
"adap-org",
"chao-dyn",
"chem-ph",
"hep-th",
"math.PR",
"nlin.AO",
"nlin.CD"
],
"doi": "10.1063/1.531412",
"journal_ref": "J.Math.Phys. 37 (1996) 732-751",
"title": "Feynman-Kac Kernels in Markovian Representations of the Schroedinger Interpolating Dynamics",
"url": "https://arxiv.org/abs/quant-ph/9505012"
},
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