Moments dynamics and stationary states for classical diffusion-type GKSL equations

TL;DR

This paper analyzes the moments dynamics and stationary states of classical diffusion-type GKSL equations, which model Wiener stochastic processes, providing explicit solutions and insights into their Gaussian stationary states.

Contribution

It introduces explicit methods for analyzing moments and stationary states of diffusion-type GKSL equations related to Wiener processes, highlighting their quantum analogs.

Findings

Explicit moments dynamics derived for the GKSL equation.

Identification of stationary Gaussian states in the model.

Connection between double commutator and second spatial derivative.

Abstract

The explicit dynamics of the moments for the GKSL equation and the approach in finding stationary Gaussian states are obtained. In our case the GKSL equation corresponds to Wiener stochastic processes. Such equations contain a double commutator which can be understood as a quantum analog of the second spatial derivative.