The Research Program of Stochastic Deformation (with a View Toward Geometric Mechanics)

Zambrini, Jean-Claude

doi:10.1007/978-3-0348-0909-2_13

Cited by 29 publications

(32 citation statements)

References 65 publications

(78 reference statements)

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“…From arguments similar to those used to obtain (4.8), given ν ∈ M 1 (W ) such that ν 0 := W 0 ν ∈ M 1 (R d ) satisfies In Theorem 4.3 below, we use specific Lagrangians with a usual convention of E.Q.M. (see [36], [54]).…”

Section: An Entropic Approach On the Classical Wiener Spacementioning

confidence: 99%

“…In [21], a representation of the relative entropy with respect to the Wiener measure shows that the specific entropy minimization involved, can be seen equivalently as an action functional minimization. On the other hand, works in the line of [53], [54] have shown that these problems could be used to produce some mathematical tools which suggest a convincing analogy to classical mechanics; due to the specificities of the Schrödinger problem, this should be distinguished from works in the line of [8]. Ever since, there have been several contributions in this direction, among many see [37], [41], [42], [46], [48].…”

Section: Introductionmentioning

confidence: 99%

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On a Class of Average Preserving Semi-Martingale Laws Optimization Problems

Lassalle

2020

josa

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In this paper, within the specific framework of a recently developed information flows preserving calculus of variations, we investigate a class of average fixed optimization problems, over sets of laws of Itô semimartingales. We show that, within those conditions, the optimums are clearly ruled by a least action principle, which yields the corresponding Euler-Lagrange conditions; this enlightens their specific dynamics. In particular, this encompasses a specific class of semi-martingale optimal transportation problems, and specific entropy minimization problems in close connection to the Schrödinger problem.

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Section: An Entropic Approach On the Classical Wiener Spacementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On a Class of Average Preserving Semi-Martingale Laws Optimization Problems

Lassalle

2020

josa

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“…Nelson [95] introduced a process for the treatment of a non-relativistic quantum particle and a random field as an alternative approach to relativistic quantum fields [94], but he also presented a Euclidean approach to the study and construction of relativistic fields [93] which greatly influenced constructive quantum field theory. Inspired by the work of Feynman and Kolmogorov, J. C. Zambrini [107,51] developed a Euclidean approach to Schrödinger processes looked upon as Bernstein processes where a couple of forward and backward processes, suitably connected, are considered. C. Léonard [83], on one hand, and Cresson and Darses [49,50] managed in a way to connect Nelson's and Zambrini's work.…”

Section: Remarkmentioning

confidence: 99%

Infinite dimensional integrals and partial differential equations for stochastic and quantum phenomena

Albeverio¹,

Mazzucchi²

2019

Journal of Geometric Mechanics

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We present a survey of the relations between infinite dimensional integrals, both of the probabilistic type (e. g. Wiener path integrals) and of oscillatory type (e. g. Feynman path integrals). Besides their mutual relations (analogies and differences) we also discuss their relations with certain types of partial differential equations (parabolic resp. hyperbolic), describing time evolution with or without stochastic terms. The connection of these worlds of deterministic and stochastic evolutions with the world of quantum phenomena is also briefly illustrated. The survey spans a bridge from basic concepts and methods in these areas to recent developments concerning their relations.

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“…where F t is the filtration of the stochastic process, see [25,29,30] for more details on this derivative. If the noise is uniform, that is σ i = σ for all i, we can go further to find that…”

Section: Conservation Lawsmentioning

confidence: 99%

Structure preserving noise and dissipation in the Toda lattice

Arnaudon¹

2018

J. Phys. A: Math. Theor.

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In this paper, we use Flaschka's change of variables of the open Toda lattice and its interpretation in term of the group structure of the LU factorisation as a coadjoint motion on a certain dual of Lie algebra to implement a structure preserving noise and dissipation. Both preserve the structure of coadjoint orbit, that is the space of symmetric tri-diagonal matrices and arise as a new type of multiplicative noise and nonlinear dissipation of the Toda lattice. We investigate some of the properties of these deformations and in particular the continuum limit as a stochastic Burger equation with a nonlinear viscosity. This work is meant to be exploratory, and open more questions that we can answer with simple mathematical tools and without numerical simulations.

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The Research Program of Stochastic Deformation (with a View Toward Geometric Mechanics)

Abstract: We give an overview of a program of Stochastic Deformation of Classical Mechanics and the Calculus of Variations, strongly inspired by the quantization method.

Cited by 29 publications

References 65 publications

On a Class of Average Preserving Semi-Martingale Laws Optimization Problems

On a Class of Average Preserving Semi-Martingale Laws Optimization Problems

Infinite dimensional integrals and partial differential equations for stochastic and quantum phenomena

Structure preserving noise and dissipation in the Toda lattice

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