An Algebraic and Microlocal Approach to the Stochastic Non-linear Schr\"odinger Equation

TL;DR

This paper extends an algebraic and microlocal framework for stochastic PDEs to include the stochastic non-linear Schrödinger equation with Gaussian white noise, enabling intrinsic renormalization and computation of expectations.

Contribution

It introduces a novel extension of the algebraic approach to stochastic PDEs, specifically applying it to the stochastic non-linear Schrödinger equation with additive noise.

Findings

Framework successfully applied to the stochastic Schrödinger equation

Allows computation of expectation values and correlations with intrinsic renormalization

Broadens applicability of algebraic methods to complex stochastic PDEs

Abstract

In a recent work [DDRZ20], it has been developed a novel framework aimed at studying at a perturbative level a large class of non-linear, scalar, real, stochastic PDEs and inspired by the algebraic approach to quantum field theory. The main advantage is the possibility of computing the expectation value and the correlation functions of the underlying solutions accounting for renormalization intrinsically and without resorting to any specific regularization scheme. In this work we prove that it is possible to extend the range of applicability of this framework to cover also the stochastic non-linear Schroedinger equation in which randomness is codified by an additive, Gaussian, complex white noise.