Title: SDE and BSDE on Hilbert spaces: applications to quasi-linear evolution equations and the asymptotic properties of the stochastic quasi-geostrophic equation
Abstract: In this thesis the following three related problems are considered. 1. We consider the following quasi-linear parabolic system of backward partial differential equations (∂t + L)u+ f(⋅, ⋅, u,∇uσ) = 0 on [0, T ]× R uT = φ, where L is a possibly degenerate second order differential operator with merely measurable coefficients. We solve this system in the framework of generalized Dirichlet forms and employ the stochastic calculus associated to the Markov process with generator L to obtain a probabilistic representation of the solution u by solving the corresponding backward stochastic differential equation. The solution satisfies the corresponding mild equation which is equivalent to being a generalized solution of the PDE. A further main result is the generalization of the martingale representation theorem using the stochastic calculus associated to the generalized Dirichlet form given by L. The nonlinear term f satisfies a monotonicity condition with respect to u and a Lipschitz condition with respect to ∇u. 2. We consider the following quasi-linear parabolic system of backward partial differential equations on a Banach space E (∂t + L)u+ f(⋅, ⋅, u, A1/2∇u) = 0 on [0, T ]× E, uT = φ, where L is a possibly degenerate second order differential operator with merely measurable coefficients. The results in 1 can be concluded in this case. 3. We study the 2D stochastic quasi-geostrophic equation in T for general parameter α ∈ (0, 1) and multiplicative noise. We prove it is uniquely ergodic provided the noise is non-degenerate for α > 2 3 . In this case, the convergence to the (unique) invariant measure is exponentially fast. In the general case, we prove the existence of Markov selections. vi Acknowledgements It is a pleasure to thank many people who helped make this thesis possible. First of all, I would like to express my sincere gratitude to my supervisors, Professor Dr. Michael Rockner and Professor Dr. Ma Zhiming . They continuously supported me in various ways with their enthusiasm, knowledge, inspiration and encouragement. Their constant encouragement and support gives me great motivation for moving forward the road of science. I would like to thank Professors Philippe Blanchard, Yuri Kondratiev, Michael Rockner, Ludwig Streit, Barbara Gentz, Moritz Kasmann and Gernot Akemann, who gave wonderful modularized courses and the visiting researchers of the IGK who gave short courses or talks in the IGK which opened to me the beautiful worlds of mathematics, physics and economics. I am indebted to my sister Xiangchan Zhu for many scientific discussions and daily help. During the whole procedure of writing this thesis, I have benefited from inspiring conversations with many people. Many thanks to Prof. Dr. L. Beznea, Prof. Dr. G. Da Prato, Prof. Dr. W. Stannat, Prof. Dr. Zhenqing Chen, Prof. Dr. Shige Peng, Dr. Qingyang Guan, Dr. Wei Liu, and Dr. Shunxiang Ouyang. Moreover, I also would like to thank my colleagues and friends in Bielefeld and Beijing, in particular Prof. Jia-An Yan, Prof. Shun-Long Luo, Prof. Fu-Zhou Gong, Prof. Zhao Dong, Prof. Xiangdong Li for their support and help. I am also very thankful to my colleagues in the IGK and the Chinese Academy of Science for their daily help in technical and scientific questions. I owe my special thanks to Rebecca Reischuk, Stephan Merks and Sven Wiesinger for their help during my studies in Bielefeld. Lastly, but most importantly, I wish to deeply thank my parents far away in China. They supported me throughout and taught me the philosophy of hard work and persistence. This thesis is dedicated to them. I appreciate very much the financial support from the DFG through the International Graduate College (IGK) at Bielefeld University. Bielefeld, Feb 17, 2012 Rongchan Zhu
Publication Year: 2012
Publication Date: 2012-01-01
Language: en
Type: article
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Cited By Count: 5
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