Existence of densities for stochastic evolution equations driven by fractional Brownian motion
Abstract
In this work, we prove a version of Hörmander’s theorem for a stochastic evolution equation driven by a trace-class fractional Brownian motion with Hurst exponent 12<H<112<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 Hörmander’s bracket condition and some algebraic constraints on the vector fields combined with the range of the semigroup, we prove that the law of finite-dimensional projections of such solutions has a density with respect to Lebesgue measure. The argument is based on rough path techniques and a suitable analysis on the Gaussian space of the fractional Brownian motion.
