UW Probability Seminar
Time: Monday, October 7, 2013 at 2:30 pm.
Location: MEB 243
Speaker: Soumik Pal (University of Washington)
Title: Intertwining diffusions and wave equations
Abstract:
An intertwining of two Markov chains is a coupling of the two processes whereby one chain can be sampled from a fixed distribution by running the other. An analogous definition exists for diffusion processes. Although intertwining and the related concept of duality have been around for a long time, recently, there has been an upsurge in interest in new intertwined systems related to models of random matrices and random surfaces. The delicate construction of intertwined chains in this context hinge on skew-Cauchy identities satisfied by symmetric polynomials. For diffusions, their occurrences were a complete mystery and only a few miraculous examples (such as the Brownian-Bessel intertwining, the Warren process and the Whittaker model associated with the Hamiltonian of the quantum Toda lattice) were known in the literature.
We will present a complete characterization of intertwined diffusions through solutions of hyperbolic partial differential equations (e.g., the classical wave equations). This approach covers all the known examples, and leads to a systematic way of generating families of new ones. Moreover, this construction can be thought of as the first probabilistic representation of solutions of second order hyperbolic PDEs.
Joint work with Mykhaylo Shkolnikov.
Archive of previous talks
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