Some application of Malliavin calculus to SPDE and convergence of densities

Abstract

Some applications of Malliavin calculus to stochastic partial differential equations (SPDEs) and to normal approximation theory are studied in this dissertation. In Chapter 3, a Feynman-Kac formula is established for a stochastic heat equation driven by Gaussian noise which is, with respect to time, a fractional Brownian motion with Hurst parameter H<1/2. To establish such a formula, we introduce and study a nonlinear stochastic integral of the Gaussian noise. The existence of the Feynman-Kac integral then follows from the exponential integrability of this nonlinear stochastic integral. Then, techniques from Malliavin calculus is used to show that the Feynman-Kac integral is the weak solution to the stochastic heat equation. In Chapter 4, the density formula in Malliavin calculus is used to study the joint H"{o}lder continuity of the solution to a nonlinear SPDE arising from the study of one dimensional super-processes. Dawson, Vaillancourt and Wang [Ann. Inst. Henri. Poincaré Probab. Stat., 36 (2000) 167-180] proved that the solution of this SPDE gives the density of the branching particles in a random environment. The time-space joint continuity of the density process was left as an open problem. Li, Wang, Xiong and Zhou [Probab. Theory Related Fields 153 (2012), no. 3-4, 441--469] proved that this solution is joint H"{o}lder continuous with exponent up to 1/10 in time and up to 1/2 in space. Using our new method of Malliavin calculus, we improve their result and obtain the optimal exponent 1/4 in time. In Chapter 5, we study the convergence of densities of a sequence of random variables to a normal density. The random variables considered are nonlinear functional of a Gaussian process, in particular, the multiple integrals. They are assumed to be non-degenerate so that their probability densities exist. The tool we use is the Malliavin calculus, in particular, the density formula, the integration by parts formula and the Stein's method. Applications to the convergence of densities of the least square estimator for the drift parameter in Ornstein-Ulenbeck is also considered. In Chapter 6, we apply an upper bound estimate from small deviation theory to prove the non-degeneracy of some functional of fractional Brownian motion.