Jean-Claude Zambrini

Variational processes and stochastic versions of mechanics

The dynamical structure of any reasonable stochastic version of classical mechanics is investigated, including the version created by Nelson [E. Nelson, Quantum Fluctuations (Princeton U.P., Princeton, NJ, 1985); Phys. Rev. 150, 1079 (1966)] for the description of quantum phenomena. Two different theories result from this common structure. One of them is the imaginary time version of Nelson’s theory, whose existence was unknown, and yields a radically new probabilistic interpretation of the heat equation. The existence and uniqueness of all the involved stochastic processes is shown under conditions suggested by the variational approach of Yasue [K. Yasue, J. Math. Phys. 22, 1010 (1981)].

Introduction to random time and quantum randomness

Introduction to Random Time: Prologue Stopping Martingale Stopped Random Past and Future Other Times From First to Last Gapless Time Markov Chain in Continuum Time The Trouble with Infinite Introduction to Quantum Randomness: Classical Prologue Standard Quantum Mechanics Probabilities in Standard Quantum Mechanics Feynmans Approach to Quantum Probabilities Schrodingers Euclidean Quantum Mechanics Beyond Feynmans Approach Time for a Dialogue.

Probabilistic Deformation of Contact Geometry, Diffusion Processes and Their Quadratures

Classical contact geometry is an odd-dimensional analogue of symplectic geometry. We show that a natural probabilistic deformation of contact geometry, compatible with the very irregular trajectories of diffusion processes, allows one to construct the stochastic version of a number of basic geometrical tools, like, for example, Liouville measure. Moreover, it provides an unified framework to understand the origin of explicit relations (cf. “quadrature”) between diffusion processes, useful in many fields. Various applications are given, including one in stochastic finance.

Bernstein Processes Associated with a Markov Process

A general description of Bernstein processes, a class of diffusion processes, relevant to the probabilistic counterpart of quantum theory known as Euclidean Quantum Mechanics, is given. It is compatible with finite or infinite dimensional state spaces and singular interactions. Although the relations with statistical physics concepts (Gibbs measure, entropy,…) is stressed here, recent developments requiring Feynman’s quantum mechanical tools (action functional, path integrals, Noether’s Theorem,…) are also mentioned and suggest new research directions, especially in the geometrical structure of our approach.

Reciprocal processes. A measure-theoretical point of view

This is a survey paper about reciprocal processes. The bridges of a Markov process are also Markov. But an arbitrary mixture of these bridges fails to be Markov in general. However, it still enjoys the interesting properties of a reciprocal process. The structures of Markov and reciprocal processes are recalled with emphasis on their time-symmetries. A review of the main properties of the reciprocal processes is presented. Our measure-theoretical approach allows for a unified treatment of the diffusion and jump processes. Abstract results are illustrated by several examples and counter-examples.

Probability and Quantum Symmetries. II. The Theorem of Nœther in quantum mechanics

For the largest class of physical systems having a classical analog, a new rigorous, but not probabilistic, Lagrangian version of nonrelativistic quantum mechanics is given, in terms of a notion of regularized action function. As a consequence of the study of the symmetries of this action, an associated Nœther theorem is obtained. All the quantum symmetries resulting from the canonical quantization procedure follow in this way, as well as a number of symmetries which are new even for the case of the simplest systems. The method is based on the study of a corresponding Lie algebra and an analytical continuation in the time parameter of the probabilistic construction given in paper I of this work. Generically, the associated quantum first integrals are time dependent and the probabilistic model provides a natural interpretation of the new symmetries. Various examples illustrate the physical relevance of our results.

Stochastic models in population biology: from dynamic noise to Bayesian description and model comparison for given data sets

We study some stochastic models from population biology, namely epidemiology, which can be solved analytically. From the solution of the stochastic processes we construct the likelihood function and compare the maximum likelihood parameter estimation procedure with the Bayesian approach of obtaining an explicit probability for the parameters given the available data. Finally, we compare one model against the other in the Bayesian framework, both models performing on the same simulated data set. In some cases of data obtained under one model with specific parameter values, the model comparison favours the model not underlying the simulated data. This apparently paradoxical situation arises in parameter regions which do not easily give sufficient information to the simulated data to reject simpler models.

A cinematic study of quantum kinematics

Abstract Within the framework of stochastic quantization, quantum mechanical trajectory of a particle is computed numerically. The fundamental kinematic equation is a stochastic differential equation which can be intergrated numerically by means of pseudo random numbers. Three typical examples in quantum mechanics, the superposition, the classical limit, and the two-slit interference, are considered in detail. The computer output of the numerical analysis of these examples is shown in a grapic display. Several samples of the quantum mechanical trajectories are illustrated there from which intuitive meaning of quantum phenomena appearing in the examples can be seen easily.

The research program of Stochastic Deformation (with a view toward Geometric Mechanics)

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

Stochastic analysis and mathematical physics

Preface * 1. Functorial Analysis in Geometric Probability Theory (H. Airault/P. Malliavin) * 2. Stochastic Volterra Equations with Singular Kernels (L. Coutin/L. Decreusefond) * 3. Stochastic Diffeology and Homotopy (R. Leandre) * 4. Some Results on Entropic Projections (C. Leonard) * 5. Mehler-type Semigroups on Hilbert Spaces and Their Generators (P. Lescot) * 6. Singular Limiting Behavior in Nonlinear Stochastic Wave Equations (M. Oberguggenberger/F. Russo) * 7. Complete Positivity and Open Quantum Systems (R. Rebolledo) * 8. Properties of Measure-preserving Shifts on the Wiener Space (A.S. Ustunel) * 9. Martingale and Markov Uniqueness of Infinite Dimensional Nelson Diffusions (L. Wu)