# Regularization by noise in one-dimensional continuity equation

## TL;DR

This paper establishes existence, uniqueness, and stability of solutions for a stochastic one-dimensional continuity equation with irregular coefficients, extending classical theory to cases with random dependence.

## Contribution

It introduces a probabilistic approach to solve the continuity equation with low regularity and random vector fields, addressing an open problem in the field.

## Key findings

- Proved strong solution existence and uniqueness for irregular coefficients.
- Extended the theory to include random dependence in the vector field.
- Established stability of solutions under perturbations.

## Abstract

A linear stochastic continuity equation with non-regular coefficients is considered. We prove existence and uniqueness of strong solution, in the probabilistic sense, to the Cauchy problem when the vector field has low regularity, in which the classical DiPerna-Lions-Ambrosio theory of uniqueness of distributional solutions does not apply. We solve partially the open problem that is the case when the vector-field has random dependence. In addition, we prove a stability result for the solutions.