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. 0. Introduction This is a review of various recent developments regarding the construction and properties of Bernstein processes, a class of diffusions originally introduced for the purpose of Euclidean Quantum Mechanics (EQM), a probabilistic analogue of Quantum Theory [1, 2]. The first section describes their construction, in a rather general setting, compatible with singular interactions. Most of Bernstein processes are not Markovian. The original characterization of the Markovian ones in terms of a maximal entropy principle goes back to E. Schrödinger [3], the originator of EQM, and has been mathematically substantiated by H. Föllmer [4]. An adaptation in the present setting is given in section 2. For the relations with quantum dynamics, however, the above characterization is not directly relevant. It is more natural to introduce a concept of action functional on a class of processes, along the line of Feynman’s path integral, and to look for the minimal point of this action. This is done in section 3. The next section considers the relations between a crucial factorization, which is the probabilistic counterpart of Born’s interpretation of the (complex) wave function ψt solving Schrödinger’s equation (ψ̄t(x)ψt(x) dx should be a probability), and a martingale problem associated with the probability measure of the Bernstein processes. Section 5 describes the regularity of the (positive) solutions of the pair of adjoint PDEs which are the basis of the construction. Section 6 is devoted to the dynamical characterization of the Bernstein processes, with some applications to the case where the state space E is finite dimensional, Euclidean or Riemannian, then to the case where E is the Wiener space C([0, 1];R). Finally, section 7 formulates in the simplest situation (E finite dimensional and Euclidean) the Noether Theorem associated with the action functional of section 3, together with some interesting open problems suggested by it. This Theorem relates the presence of symmetries of the action functional under some space-time transformations to the existence of some martingales of the Bernstein processes. The whole framework has been designed to be the closest possible analogue of quantum theory using (Kolmogorovian) probabilistic concepts. It has recently partially justified this claim in showing that, after the proper analytic continuation