The logarithmic Schr{\"o}dinger equation with spatial white noise on the full space

TL;DR

This paper addresses the challenge of solving the logarithmic Schrödinger equation with spatial white noise in low dimensions by employing an exponential transform, bypassing traditional regularity structures.

Contribution

It introduces a novel approach using exponential transform to solve a singular SPDE with logarithmic nonlinearity and multiplicative noise, where standard methods are inapplicable.

Findings

Successfully solves the equation in dimensions d ≤ 2.

Demonstrates the effectiveness of exponential transform in this context.

Provides a new method for handling similar singular SPDEs.

Abstract

We solve the Schr{\"o}dinger equation with logarithmic nonlinearity and multiplicative spatial white noise on R d with d $\le$ 2. Because of the nonlinearity, the regularity structures and the paracontrolled calculus can not be used. To solve the equation, we rely on an exponential transform that has proven useful in the context of other singular SPDEs.