Stochastic flows for Hölder drifts and transport/continuity equations with noise

TL;DR

The paper develops a well-posedness theory for stochastic flows and linear SPDEs when the drift lies in $L^q_t C^{0,\alpha}_x$ with $q\ge 2$ and $\alpha\in(0,1)$. It uses a Zvonkin-type transformation to regularize the drift via a parabolic PDE, obtains existence and optimal regularity for the transformed problem, and deduces a differentiable stochastic flow of $C^{1,\beta}$-diffeomorphisms for $\beta<\alpha$. This flow yields representation formulas for BV$_{loc}$-solutions of the stochastic transport equation and weak solutions of the stochastic continuity equation with transport noise, and provides stability results under convergence of drift coefficients. The approach broadens stochastic flow theory to rough Hölder drifts and delivers robust tools for the analysis of transport/continuity phenomena under stochastic perturbations, with potential applications to fluid dynamics and related SPDEs.

Abstract

We prove existence of a stochastic flow of diffeomorphisms generated by SDEs with drift in $L^q_t C^{0, α}_x$ for any $q \in [2, \infty)$ and $α\in (0, 1)$. This result is achieved using a Zvonkin-type transformation for the SDE. As a key intermediate step, well-posedness and optimal regularity for a class of parabolic PDEs related to the transformation is established. Using the existence of a differentiable stochastic flow, we prove well-posedness of $BV_\text{loc}$-solutions of stochastic transport equations and weak solutions of stochastic continuity equations with so-called transport noise and velocity fields in $L^q_t C^{0, α}_x$. For both equations, solutions may fail to be unique in the deterministic setting.

Theorems & Definitions (48)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1: Kunita kunita_1990
• Definition 3.1
• Lemma 3.2
• proof
• Lemma 3.4
• proof
• Proposition 3.5: A priori regularity