Bernstein Processes Associated with a Markov Process

Abstract

Abstract. 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.

Minimization of entropy functionals

“…where P 0 and P 1 are the laws of the initial (t = 0) and final (t = 1) positions when P is the probability law of the whole path, see [9,14,28].…”

Bernstein Diffusions for a Class of Linear Parabolic Partial Differential Equations

“…More to the point, the Markov processes of [36] actually emerge as particular cases of reversible diffusions that belong to the larger class of the so-called reciprocal or Bernstein processes, whose theory was launched many years ago in [2] following Schrödinger's seminal contribution in [27]. The theory of Bernstein processes was subsequently further developed and systematically investigated in [19], and since then has played an important rôle in relating various fields such as the Malliavin calculus and Euclidean quantum mechanics, or Markov bridges with jumps and Lévy processes, to name only a few (see for instance [7], [8], [16], [25], [32] and the references therein for a more complete account).…”

Forward-Backward Stochastic Differential Equations Generated by Bernstein Diffusions

“…in the backward case, where u ϕ is the unique positive classical solution to (5) and v ψ the unique classical positive solution to (6). Finally Z τ ∈[0,T ] satisfies two Itô equations in the weak sense, namely, the forward equation…”