Non-uniqueness in law of transport-diffusion equation forced by random noise

TL;DR

This paper demonstrates the non-uniqueness in law for solutions to a stochastic transport-diffusion equation with various types of random forcing, using a probabilistic convex integration approach.

Contribution

It introduces a novel probabilistic convex integration method to establish non-uniqueness in law for stochastic transport-diffusion equations with different noise types.

Findings

Existence of divergence-free vector fields with Sobolev regularity.

Solutions with Lebesgue space regularity for the transport-diffusion equation.

Proof of non-uniqueness in law for strong solutions globally in time.

Abstract

We consider a transport-diffusion equation forced by random noise of three types: additive, linear multiplicative in It$\hat{\mathrm{o}}$'s interpretation, and transport in Stratonovich's interpretation. Via convex integration modified to probabilistic setting, we prove existence of a divergence-free vector field with spatial regularity in Sobolev space and corresponding solution to a transport-diffusion equation with spatial regularity in Lebesgue space, and consequently non-uniqueness in law at the level of probabilistically strong solutions globally in time.