Hörmander’s theorem for stochastic evolution equations driven by fractional Brownian motion.

Abstract

In this thesis, we prove the Hörmander’s theorem for a stochastic evolution equation driven by a trace-class fractional Brownian motion with Hurst exponent 1 2 < H < 1 and an analytical semigroup {S(t); t ≥ 0} on a given separable Hilbert space E. 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 Hörmander’s bracket condition on the vector fields of the stochastic PDE and the additional assumption that S(t)E is dense, we prove the law of finite-dimensional projections of the stochastic PDE at time t has a density w.r.t Lebesgue measure. The argument is based on rough path techniques in the sense of Gubinelli (Controlling rough paths. J. Funct. Anal (2004)) and a suitable analysis on the Gaussian space of the fractional Brownian motion.