We start with a review of Brownian motion,including basic properties and Levy`s construction, following which we willpresent the classical Wiener space as a space equipped with an infinitedimensional Gaussian measure (Lecture 1). After further examining the theory ofthe classical Wiener space, we will extend it to abstract settings, introducethe notion of abstract Wiener space (AWS), and give a general exposition ofsome structural properties of infinite dimensional Gaussian measures (Lecture2&3). One application of the theory of AWS is that it providesmathematically rigorous interpretations and treatments of the Gaussian freefield (GFF), an important object originated from statistical physics. Othermethods used in the study of GFFs will also be discussed (Lecture 4). The lastpart of the course will be dedicated to some recent developments in the fieldof GFFs, including the construction of Liouville quantum gravity measures, aproof of the KPZ relation, and the study of thick points of GFFs (Lecture5&6).
1. Probability Theory - An Analytic View(2nd edition), D. Stroock, Cambridge University Press, 2010. (SeeSection 4.3 for Levy`s construction of Brownian motion, and Section 8.1-8.5 forabstract Wiener space and Gaussian free fields.)
2. Abstract Wiener Space, by L.Gross, Proc. 5th Berkeley Symp. Math. Stat. and Prob., vol. 1(2), pp31-42, 1965.
3. Gaussian free fields formathematicians, S. Sheffield, Probability Theory Related Fields, vol. 139(3-4),pp 521-541, 2007.
4. Thick point of the Gaussian free field,X. Hu, J. Miller and Y. Peres, Ann. Prob., vol. 38(2), pp 896-926, 2010.
5. Liouville quantum gravity and KPZ, B. Duplantier and S. Sheffield,Invent. Math., vol. 185(2), pp 333-393, 2011.