# Existence of densities for stochastic evolution equations driven by fractional Brownian motion

## TL;DR

This paper extends Hörmander's theorem to certain infinite-dimensional stochastic evolution equations driven by fractional Brownian motion, establishing the existence of densities for finite-dimensional projections under specific conditions.

## Contribution

It introduces a novel infinite-dimensional Hörmander-type result for equations driven by fractional Brownian motion with Hurst parameter H>1/2, using rough path analysis.

## Key findings

- Finite-dimensional projections have densities w.r.t. Lebesgue measure.
- Hörmander's bracket condition ensures regularity of solutions.
- The approach combines rough path techniques with Gaussian space analysis.

## Abstract

In this work, we prove a version of H\"{o}rmander's theorem for a stochastic evolution equation driven by a trace-class fractional Brownian motion with Hurst exponent $\frac{1}{2} < H < 1$ and an analytic semigroup on a given separable Hilbert space. In contrast to the classical finite-dimensional case, the Jacobian operator in typical solutions of parabolic stochastic PDEs is not invertible which causes a severe difficulty in expressing the Malliavin matrix in terms of an adapted process. Under a H\"{o}rmander's bracket condition and some algebraic constraints on the vector fields combined with the range of the semigroup, we prove the law of finite-dimensional projections of such solutions has a density w.r.t Lebesgue measure. The argument is based on rough path techniques and a suitable analysis on the Gaussian space of the fractional Brownian motion.