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Birkhäuser Boston eBooks

Some Results on Entropic Projections

Christian Léonard

Published Jan 1, 2001• 2 citations
Publisher

Abstract

We give a short survey of results related to the maximum entropy method. In this article we use the large deviations approach rather than the more direct convex analytical one. Indeed, the proposed applications are naturally stated in terms of large random particle systems, namely, the existence and construction problems for Schrödinger’s bridges and Nelson’s diffusion processes. These problems arise from probabilistic approaches to quantum mechanics. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-1-4612-0127-4_4

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We give a short survey of results related to the maximum entropy method. In this article we use the large deviations approach rather than the more direct convex analytical one. Indeed, the proposed applications are naturally stated in terms of large random particle systems, namely, the existence and construction problems for Schrödinger’s bridges and Nelson’s diffusion processes. These problems arise from probabilistic approaches to quantum mechanics. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
I will give here a short presentation of two results in the context of Nelson diffusions. The first is a Lyapunov-type theorem through which we estabilish conditions under which the nodes are not attained. The second proves continuity (in a suitable topology) of the map which associates to each Schroedinger evolution (defined by the potential and the initial data) the corresponding Nelson diffusion. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The interpretation provided by Schrödinger of quantum mechanics through classical stochastic processes is the root of Euclidean quantum mechanics founded on Bernstein processes. Open quantum systems have been extensively investigated during the last three decades and have motivated new probabilistic interpretations for quantum dynamics. This article, mainly addressed to classical probabilists, gives a short introduction to quantum dynamical semigroups and quantum Markov flows as a natural extension of Schrödinger ideas in noncommutative stochastic analysis. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
There are mainly used two basic approaches for probabilistic modeling of motion: stochastic in which the object literally makes succeeding random decisions using arbi-trarily chosen by us probabilities or ergodic in which we usually assume some chaotic classical evolution and probabilities appear while averaging over infinite trajectories. Both approaches assume we know the exact way the system evolves. In contrast, in this paper we will focus on thermodynamical motion models: assuming maximal uncertainty. Specifically, in the space of possible choices of transition probabil-ities, we take the optimizing entropy or free energy one. Equivalent condition appears to be calculating transition probabilities as proportions between single steps in canonical ensemble of trajectories going through a given point. It makes that these probabilities depend on the whole space- the walker cannot directly use them. This model is ther-modynamical: only we use it to predict the most probable behavior. Standard diffusion models like Brownian motion can be seen as obtained by locally maximizing uncertainty. For regular space it agrees with fully maximizing entropy choice of transition probabili-
In 1932, E.Wigner [1] introduced a phase-space method of computing physical observables in quantum (statistical) mechanics; some surveys have been published recently [2], [3]. See also a review article by Carruthers [4]. Here we give a brief synopsis of the method to make this paper self-contained, referring the reader to the references for details, including proofs. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.