Rough paths and 1d sde with a time dependent distributional drift. Application to polymers

TL;DR

This paper develops a rough paths approach to establish existence and uniqueness for 1D SDEs with distributional drifts, inspired by SPDEs like KPZ, and introduces a new stochastic calculus for such equations.

Contribution

It extends rough paths theory to handle SDEs with distributional drifts, providing new regularity results and a stochastic calculus framework.

Findings

Existence and uniqueness of solutions in the weak sense.

Regularity analysis of the distributional drift.

A new stochastic calculus for equations with distributional drifts.

Abstract

Motivated by the recent advances in the theory of stochastic partial differential equations involving nonlinear functions of distributions, like the Kardar-Parisi-Zhang (KPZ) equation, we reconsider the unique solvability of one-dimensional stochastic differential equations, the drift of which is a distribution, by means of rough paths theory. Existence and uniqueness are established in the weak sense when the drift reads as the derivative of a H{\"o}lder continuous function. Regularity of the drift part is investigated carefully and a related stochastic calculus is also proposed, which makes the structure of the solutions more explicit than within the earlier framework of Dirichlet processes.