Lectures | Monday and Wednesday | 1:45-2:35 | Physics 205 | |
Sayan Mukherjee | sayan@stat.duke.edu | Feb 7-March 14 |
Stochastic Processes on Manifolds, Varieties, and Sheaves We will discuss how to place stochastic processes on geometric, algebraic, and topological objects. We will mainly consider Brownian motion and Gaussian processes as example processes. We first go through the manifold case where we introduce the Eells-Elworthy-Malliavin formulation of lifting Brownian motion on manifolds and then discuss a SPDE formulation of Gaussian processes on manifolds that ensure the covariance function is positive (semi) definite. We then consider the case of algebraic varieties and adapt manifold models to the setting where there are singularities induced by algebraic structures. There is very little on stochastic processes on sheaves. We will consider some recent work where we consider probability spaces as Grothendieck sites on which Brownian motions are defined via sheaves. I will try to make this short course as self-contained as possible. I am hoping knowledge of Markov chains and basic differential geometry is enough for a base requirement. >
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