Further properties on functional SDEs. - Jianhai Bao (2013)
In this work, we aim to study some fine properties for functional stochastic differential equation. The results consist of five main parts. In the second chapter, by constructing successful couplings, the derivative formula, gradient estimates and Harnack inequalities are established for the semigroup associated with a class of degenerate functional stochastic differential equations. In the third chapter, by using Malliavin calculus, explicit derivative formulae are established for a class of semi-linear functional stochastic partial differential equations with additive or multiplicative noise. As applications, gradient estimates and Harnack inequalities are derived for the semigroup of the associated segment process. In the fourth chapter, we apply the weak convergence approach to establish a large deviation principle for a class of neutral functional stochastic differential equations with jumps. In particular, we discuss the large deviation principle for neutral stochastic differential delay equations which allow the coefficients to be highly nonlinear with respect to the delay argument. In the fifth chapter, we discuss the convergence of Euler-Maruyama scheme for a class of neutral stochastic partial differential equations driven by alpha-stable processes, where the numerical scheme is based on spatial discretization and time discretization. In the last chapter, we discuss (i) the existence and uniqueness of the stationary distribution of explicit Euler-Maruyama scheme both in time and in space for a class of stochastic partial differential equations whenever the stepsize is sufficiently small, and (ii) show that the stationary distribution of the Euler-Maruyama scheme converges weakly to the counterpart of the stochastic partial differential equation.