Zero-noise selection and Large Deviations in $L^\infty_t L^p_x$ for the stochastic transport equation beyond DiPerna-Lions

TL;DR

This paper establishes strong existence, uniqueness, and large deviations principles for solutions of the stochastic transport equation in a non-separable function space, extending DiPerna-Lions theory beyond deterministic cases.

Contribution

It proves strong solutions and pathwise uniqueness for stochastic transport equations in regimes where deterministic solutions are non-unique, and characterizes the zero-noise limit via large deviations.

Findings

Solutions converge to the DiPerna-Lions renormalized solution as noise vanishes

Large Deviations Principle governs the convergence in non-separable space

Established strong existence and uniqueness in a broader parameter regime

Abstract

We consider $L^\infty_t L^p_x$ solutions of the stochastic transport equation with drift in $L^\infty_t W^{1,q}_x$. We show strong existence and pathwise uniqueness of solutions in a regime of parameters $p,q$ for which non-unique weak solutions of the deterministic transport equation exist. When the intensity of the noise goes to zero, we prove that the solutions of the stochastic transport equation converge to the unique renormalized solution of the transport equation in the sense of DiPerna-Lions. Furthermore, we show that the convergence is governed by a Large Deviations Principle in the space $L^\infty_t L^p_x$. Since the space $L^\infty_t L^p_x$ is not separable, the weak convergence approach to Large Deviations by Budhiraja, Dupuis, and Maroulas is not directly applicable.