Stochastic analysis & discrete quantum systems

TL;DR

This paper investigates the deep connections between stochastic analysis and discrete quantum systems, deriving universal path integral measures, exploring quantum transformations, and linking integrable quantum models with stochastic differential equations.

Contribution

It introduces a unified framework for analyzing stochastic processes using quantum mechanical tools and establishes explicit links between quantum integrable systems and stochastic differential equations.

Findings

Universal path integral measure independent of diffusion coefficients

Drift acts as a super potential in supersymmetric quantum mechanics

Connections between quantum integrable models and stochastic PDEs

Abstract

We explore the connections between the theories of stochastic analysis and discrete quantum mechanical systems. Naturally these connections include the Feynman-Kac formula, and the Cameron-Martin-Girsanov theorem. More precisely, the notion of the quantum canonical transformation is employed for computing the time propagator, in the case of generic dynamical diffusion coefficients. Explicit computation of the path integral leads to a universal expression for the associated measure regardless of the form of the diffusion coefficient and the drift. This computation also reveals that the drift plays the role of a super potential in the usual super-symmetric quantum mechanics sense. Some simple illustrative examples such as the Ornstein-Uhlenbeck process and the multidimensional Black-Scholes model are also discussed. Basic examples of quantum integrable systems such as the quantum discrete non-linear hierarchy (DNLS) and the XXZ spin chain are presented providing specific connections between quantum (integrable) systems and stochastic differential equations (SDEs). The continuum limits of the SDEs for the first two members of the NLS hierarchy turn out to be the stochastic transport and the stochastic heat equations respectively. The quantum Darboux matrix for the discrete NLS is also computed as a defect matrix and the relevant SDEs are derived.