High Moment Asymptotics and its Applications in Intersection Local Times and Related Models

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This project aims to develop a powerful tool known as high moment asymptotics and to apply it to the study of the large deviations for intersection local times and for related models. The moment method has been explored in studying the intersection local times and ranges for about two decades. The classic applications include the existence and weak convergence of intersection local times. In this setting, the power of the moment is fixed. To study the large deviations for these models, however, one needs to understand the asymptotic behaviors of the moments as the power increases to infinity. In recent years, there have been new development in this direction. The method developed along this line is called high moment asymptotics. Even in its early stage, the method of high moment asymptotics has shown its power in the areas of intersection local times and of the local times of additive Levy processes.Understanding large deviations for intersection local times and related models is crucial for the applications to Euclidean quantum field theory, the growth of polymers, and the polaron problem. Carrying out this project will lead to the solution of some hard problems remaining in this area and to the development of new methodology.The problems addressed can be clearly stated without too much technical jargon and therefore can be easily explained to the non-experts. In this regard, the project will benefit mathematical education and help attracting more students to mathematics and probability in particular.