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

Jorge Nascimento, Alberto Ohashi

Published 2019-02-21Version 1

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 $\{S(t); t \ge 0\}$ on a given separable Hilbert space. In contrast to the classical finite-dimensional case, the Jacobian operator in typical parabolic stochastic PDEs is not invertible which causes a severe difficulty in expressing the Malliavin matrix in terms of an adapted process. Under H\"{o}rmander's bracket condition on the vector fields and the additional assumption that $S(t)E$ is dense, we prove the law of finite-dimensinal projections of $X_t$ 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.