The transport equation and zero quadratic variation processes

TL;DR

This paper investigates the transport equation influenced by processes with zero quadratic variation, employing advanced stochastic calculus techniques to establish fundamental properties of solutions, including existence, uniqueness, and distributional regularity.

Contribution

It introduces a novel analysis of the transport equation driven by zero quadratic variation processes using stochastic calculus via regularization and Malliavin calculus.

Findings

Proves existence and uniqueness of solutions.

Establishes absolute continuity of the solution's law.

Analyzes the case with Hermite process noise.

Abstract

We analyze the transport equation driven by a zero quadratic variation process. Using the stochastic calculus via regularization and the Malliavin calculus techniques, we prove the existence, uniqueness and absolute continuity of the law of the solution. As an example, we discuss the case when the noise is a Hermite process.