Noise Prevents Singularities in Linear Transport Equations

TL;DR

This paper demonstrates that multiplicative noise in stochastic linear transport equations prevents singularities, maintaining regularity and Hölder continuity of solutions even with minimal drift regularity.

Contribution

It shows that noise can prevent blow-up in linear transport equations with low regularity drifts, preserving Sobolev regularity and continuity.

Findings

Solutions maintain Sobolev regularity under noise

Hölder continuity of solutions is preserved

Noise prevents singularity formation in the equation

Abstract

A stochastic linear transport equation with multiplicative noise is considered and the question of no-blow-up is investigated. The drift is assumed only integrable to a certain power. Opposite to the deterministic case where smooth initial conditions may develop discontinuities, we prove that a certain Sobolev degree of regularity is maintained, which implies H\"older continuity of solutions. The proof is based on a careful analysis of the associated stochastic flow of characteristics.