Topics in the analysis and computation of stochastic differential equations

• Authors: Weinan E;Di Liu
• Venue: Topics in the analysis and computation of stochastic differential equations
• Year: 2003

This thesis consists of four parts on topics in the analysis and computation of stochastic partial and ordinary differential equations. In the first part, a general strategy for proving ergodicity for stochastically forced nonlinear dissipative PDEs is presented. It consists of two main steps. The first step is the reduction to a finite dimensional Gibbsian dynamics of the low modes. The second step is to prove the equivalence between measures induced by different past histories. As an application, ergodicity for Ginzburg-Landau, Kuramoto-Sivashinsky and Cahn-Hilliard equations with stochastic forcing is proved. In the second part, a spectral method is formulated as a numerical solution for the stochastic Ginzburg-Landau equation driven by space-time white noise. The rates of pathwise convergence and convergence in expectation in Sobolev spaces are given based on the convergence rates of the spectral approximation for the stochastic convolution. In the third part, a class of efficient numerical schemes for stochastic differential equations with multiple time scales is analyzed. Both advective and diffusive time scales are considered. Weak as well as strong convergence theorems are proved. The efficiency as well as optimal strategy for the method are discussed. In the fourth part, an analysis for bistable systems under random perturbations is given. Based on the time evolution of energy landscapes, a method of predicting the switching field in the hysteresis loop is provided. The system is reduced to a discrete finite Markov chain and the computation is greatly simplified. As an application, micromagnetics under the influence of thermal noise is investigated. The first part is published in Journal of Statistical Physics, Vol. 108, Nos. 5/6, 1125–1156, 2002 ([EL02]). The second part is published in Communications in Mathematical Sciences, Vol. 1, No. 2, 361–375, 2003 ([L03]). The third part is to be submitted ([ELV03]).