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 [Formula: see text] 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.

Mild Stochastic Sewing Lemma, SPDE in Random Environment, and Fractional Averaging

“…See e.g. Garrido-Atienza-Lu-Schmalfuß [16], Pei-Xu-Bai [40], and Nascimento-Ohashi [10]. We emphasize that we do not impose this stronger condition.…”