This course will describe the basic tools of stochastic analysis and their applications in probability, while at the same time discussing their analogues in quantum physics. With the proper background the probabilistic description of these tools is relatively transparent and simple. The main topics to be discussed include Gaussian Hilbert Spaces, Wiener chaos, Wick products, hypercontractivity, and CameronMartin shifts. The ultimate goal of the course is to gain a working understanding of the Malliavin calculus, which is usually described as the stochastic calculus of variations (i.e. variational formulas over functions in Wiener space). A strong background in analysis will be assumed. A good background in probability would be helpful but a working knowledge of multidimensional Gaussians will probably be sufficient.
