Symplectic Noise & The Classical Analog of the Lindblad Generator

TL;DR

This paper introduces symplectic noise and Poisson brackets for classical systems, showing how certain stochastic models preserve the full Poisson structure and relate to quantum Markovian models.

Contribution

It develops the concept of symplectic noise and derives the general form of phase space diffusions that preserve the combined system and noise Poisson structure.

Findings

Classical noise can be described using Poisson brackets and symplectic processes.

Preserving the joint Poisson structure links classical stochastic models to quantum Markovian models.

The framework unifies classical and quantum stochastic processes.

Abstract

We introduce the concepts of Poisson brackets for classical noise, and of canonically conjugate Wiener processes (symplectic noise). Phase space diffusions driven by these processes are considered and the general form of a stochastic process preserving the full (system and noise) Poisson structure is obtained. We show that, once the classical stochastic model is required to preserve the joint system and noise Poisson bracket, it has much in common with quantum markovian models.