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Rigorous Malliavin Calculus versus Roman-Mafia’s stochastic action principles ∗

Gavriel Segre

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

Since brownian motion can’t be defined on a differentiable manifold without assigning on it a riemannian metric by which it depends, Guerra’s and coworkers ’ approach to the definition of the stochastic action in the framework of Stochastic Mechanics is completelly inconsistent ∗ This paper is dedicated to all the romans that, in different ways, ”suggested ” me not to

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Since brownian motion can’t be defined on a differentiable manifold without assigning on it a riemannian metric by which it depends, Guerra’s and coworkers ’ approach to the definition of the stochastic action in the framework of Stochastic Mechanics is completelly inconsistent ∗ This paper is dedicated to all the romans that, in different ways, ”suggested ” me not to
In the frame of Nelson stochastic quantization for dynamical systems on a manifold, we consider diffusion processes with Brownian covariance given by a Riemannian metric on the manifold. The dynamics is specified through a stochastic variational principle for a generalization of the classical action, with a given kinetic metric. The resulting programming equation, of the Hamilton-Jacobi type, depends on both metrics, the Brownian one and the kinetic one. We introduce a simple notion of compatibility between the two metrics, such that the programming equation and the continuity equation lead to the Schr\"odinger equation on the manifold.
We unify Brownian motion and quantum mechanics in a single mathematical framework. In particular, we show that non-relativistic quantum mechanics of a single spinless particle on a flat space can be described by a Wiener process that is rotated in the complex plane. We then extend this theory to relativistic stochastic theories on manifolds using the framework of second order geometry. As a byproduct, our results suggest that a consistent path integral based formulation of a quantum theory on a Lorentzian (Riemannian) manifold requires an Ito deformation of the Poincare (Galilean) symmetry, arising due to the coupling of the quadratic variation to the affine connection.