Density bounds for solutions to differential equations driven by Gaussian rough paths

TL;DR

This paper derives upper bounds and asymptotic estimates for the density of solutions to Gaussian-driven rough differential equations, using advanced stochastic calculus and rough path techniques.

Contribution

It introduces a general framework for density bounds of Gaussian rough differential equations under broad covariance conditions, combining Malliavin calculus and rough path methods.

Findings

Established upper bounds on the solution density for Gaussian rough differential equations.

Provided Varadhan estimates describing small noise asymptotics.

Applicable to a wide class of Gaussian processes with checkable covariance conditions.

Abstract

We consider finite dimensional rough differential equations driven by centered Gaussian processes. Combining Malliavin calculus, rough paths techniques and interpolation inequalities, we establish upper bounds on the density of the corresponding solution for any fixed time $t>0$. In addition, we provide Varadhan estimates for the asymptotic behavior of the density for small noise. The emphasis is on working with general Gaussian processes with covariance function satisfying suitable abstract, checkable conditions.