Transportation Inequalities For The Law Of A Stochastic Differential Equation Driven By Multi-parameter

In this paper,we prove T₂-transportation inequalities on the continious path space with respect to the uniform norm for the law of a stochastic differential equation driven by fractional Brown Sheet and a stochastic heat equation driven by multiplicative noise on the unit sphere.In the first chapter,we introduce the background of the the progress that have been made so far in the research of transportation inequalities.In the second chapter,we prove T₂-transportation inequalities for the law of a stochastic differential equation driven by fractional Brown Sheet.In the first two sec-tions we introduce some preliminary concepts about fractional brown motion and frac-tional brown sheet and prove the existence and uniqueness of a solution to the equation;and we prove the main theory and obtain some useful corollaries in the last two sections.In the third chapter,we prove T₂-transportation inequalities for the law of a stochastic heat equation driven by multiplicative noise on the unit sphere.In the first section we introduce the definition and properties of Q-Wiener process,and we intro-duce some preliminary concepts about the spherical harmonics,function spaces in the second section.Also we prove that there exists a unique mild solution to the equation in the third section.To prove the main theorem,we introduce It(?)formula in infinite dimensions and martingale representation theorem in the fourth section and finally in the last section we prove the main theorem.In the fourth chapter,we summarize the progress we have made in this paper and the improvement that could be attained in our future work.