# A support and density theorem for Markovian rough paths

## TL;DR

This paper proves a support theorem and a density existence theorem for Markovian rough paths derived from subelliptic Dirichlet forms, advancing the understanding of their probabilistic and analytical properties.

## Contribution

It establishes the first support theorem in Hölder rough path topology for Markovian rough paths and a H"ormander-type theorem for densities of solutions to rough differential equations.

## Key findings

- Support theorem in Hölder topology for all α in (0,1/2)
- Existence of densities for RDE solutions driven by these paths
- Extension of H"ormander's theorem to rough path setting

## Abstract

We establish two results concerning a class of geometric rough paths $\mathbf{X}$ which arise as Markov processes associated to uniformly subelliptic Dirichlet forms. The first is a support theorem for $\mathbf{X}$ in $\alpha$-H\"older rough path topology for all $\alpha \in (0,1/2)$, which answers in the positive a conjecture of Friz-Victoir (2010). The second is a H\"ormander-type theorem for the existence of a density of a rough differential equation driven by $\mathbf{X}$, the proof of which is based on analysis of (non-symmetric) Dirichlet forms on manifolds.