Coupling, Geometry and Hypoellipticity

SpeakerDr. Sayan Banerjee
Organization University of North Carolina, Chapel Hill
LocationEBIII 2213
Start Date September 8, 2017 11:45 AM
End Date September 8, 2017 1:00 PM

Abstract: Coupling is a way of constructing Markov processes with given laws on the same space. The coupling is called Markovian if the coupled processes are co-adapted to the same filtration. We will first investigate Markovian couplings of elliptic diffusions and demonstrate how the rate of coupling (how fast you can make the coupled processes meet) is intimately connected to the geometry of the underlying space. Next, we will consider couplings of hypoelliptic diffusions (diffusions driven by vector fields whose Lie algebra span the whole tangent space). We will construct successful Markovian couplings for a large class of hypoelliptic diffusions. We will also investigate non-Markovian couplings for some hypoelliptic diffusions, namely the Kolmogorov diffusion and Brownian motion on the Heisenberg group, and demonstrate how these couplings yield sharp estimates for the total variation distance between the laws of the coupled diffusions when Markovian couplings fail. Furthermore, we will demonstrate how non-Markovian couplings can be used to furnish purely analytic gradient estimates of harmonic functions on the Heisenberg group by purely probabilistic means, providing yet another strong link between probability and geometric analysis.

Bio: Sayan Banerjee is an Assistant Professor in the Department of Statistics and Operations Research, University of North Carolina, Chapel Hill. He obtained his undergraduate and master’s education at the Indian Statistical Institute, Kolkata, from 2005-2010 before moving to the Mathematics Department, University of Washington, Seattle, for his Ph.D. in mathematics (2010-2013) under the supervision of Prof. Krzysztof Burdzy. Subsequently, he was a Research Fellow in the Statistics Department, University of Warwick, UK, from November 2013- June 2016, working with Prof. Wilfrid Kendall, before moving to UNC. Sayan works in probability theory. Recently his focus has been on interactions between probability (like couplings of diffusions) and the geometry of the underlying space. He also works in large deviations, queueing systems, interacting particle systems, random walks in random environments and random matrices.

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