On the stochastic selection of integral curves of a rough vector field

TL;DR

This paper establishes the existence and uniqueness of an incompressible measure on integral curves for certain rough, divergence-free vector fields, and demonstrates stochastic behavior for a specific example within this class.

Contribution

It proves the existence and uniqueness of measures on integral curves for rough vector fields and shows stochasticity in a particular constructed example.

Findings

Unique incompressible measure exists for the class of vector fields considered.

The constructed example exhibits stochasticity in its integral curves.

The results extend understanding of flow behavior in rough vector fields.

Abstract

We prove that for bounded, divergence-free vector fields b in L^1_{loc}((0,1];BV(\T^d;\R^d)), there exists a unique incompressible measure on integral curves of b. We recall the vector field constructed by Depauw in [Depauw, C. R. Math. Acad. Sci. Paris, 2003], which lies in the above class, and prove that for this vector field, the unique incompressible measure on integral curves exhibits stochasticity.