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 parabolicstochasticPDEsisnotinvertiblewhichcausesaseveredifficultyinexpressing 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 isdense, weprovethelawoffinite-dimensionalprojectionsofthestochasticPDE 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.