Random periodic solutions of stochastic functional differential equations

In this thesis, we study the existence of random periodic solutions for both nonlinear dissipative stochastic functional differential equations (SFDEs) and semilinear nondissipative SFDEs in $\mathcal{C}([-r,0],\mathbb{R}^d)$. Under some sufficient conditions for the existence of global semiflows for SFDEs, by using pullback-convergence technique to SFDE, we obtain a general theorem about the existence of random periodic solutions. By applying coupled forward-backward infinite horizon integral equations method, we perform the argument of the relative compactness of Wiener-Sobolev spaces in $\mathcal{C}([0,\tau], \mathcal{C}([-r,0], \mathbf{L}^2 (\Omega)))$ and the generalized Schauder's fixed point theorem to show the existence of random periodic solutions.