Well-posedness of the transport equation by stochastic perturbation

TL;DR

This paper demonstrates that adding a Brownian-type stochastic perturbation to a linear transport equation with a Holder continuous vector field makes the PDE well-posed, providing a novel example of noise-induced regularization.

Contribution

It introduces the first explicit example of a PDE becoming well-posed due to stochastic noise, using a differentiable stochastic flow and a special drift transformation.

Findings

Stochastic perturbation ensures well-posedness of the transport equation.

The approach involves a transformation of the drift of Ito-Tanaka type.

Provides a new perspective on noise-induced regularization of PDEs.

Abstract

We consider the linear transport equation with a globally Holder continuous and bounded vector field. While this deterministic PDE may not be well-posed, we prove that a multiplicative stochastic perturbation of Brownian type is enough to render the equation well-posed. This seems to be the first explicit example of partial differential equation that become well-posed under the influece of noise. The key tool is a differentiable stochastic flow constructed and analysed by means of a special transformation of the drift of Ito-Tanaka type.