Noise Prevents Singularities in Linear Transport Equations

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 Hölder continuity of solutions. The proof is based on a careful analysis of the associated stochastic flow of characteristics.

References in corpus (1)

• Blow-up for the stochastic nonlinear Schrodinger equation with multiplicative noise

Cited by in corpus (2)

• The density of the solution to the stochastic transport equation with fractional noise
• A Note on the Malliavin Differentiability of One-Dimensional Reflected Stochastic Differential Equations with Discontinuous Drift