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International series of numerical mathematics

Two Results on Nelson Diffusions

Gianfausto Dell’Antonio

Published Jan 1, 1991• 0 citations
Publisher

Abstract

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.

DOI
10.1007/978-3-0348-6413-8_4

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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.
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.
In this lecture we give more details of an alternative approach to quantum scattering theory, initiated by V. Enss, This approach is based on a geometric analysis of the behavior for t→±∞t \rightarrow \pm \infty t→±∞ of the solutions of Schroedinger’s equation for initial data in the subspace of absolute continuity for the hamiltonian H . 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.
Bernstein diffusions belong to a new class of time symmetric (but not time homogeneous) stochastic processes associated with the quantum dynamics of nonrelativistic particles in potentials. It is shown in which sense Feynman’s formal path integral method can be interpreted in terms of those processes, specially for the subset of Gaussian Bernstein diffusions. The familiar Ornstein-Uhlenbeck process becomes, in this framework, a particular Gaussian process in a large class of Bernstein diffusions, all associated with the same dynamics. The method is also illustrated for some Hilbert space valued Gaussian Diffusions 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.