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Game theoretical methods in PDE – tutorial By Prof. Marta Lewicka (University of Pittsburgh)

  • Class schedule:  Tuesday 26th & Thursday 28th, May 2015 from 1:30 – 3:00 pm
  • Location: Building 1, Room 4214

The objective of these lectures is to give a brief self-contained introduction to how random tug-of-war games are applied to gain new proofs and insights in the theory of nonlinear partial differential equations. We first revisit the classical linear case, where an interplay between harmonic functions and martingales is based on a common cancellation property, expressed via the mean value property. We then pass to the nonlinear case and explain the connection between tug-of-war games and the, so called, infinity harmonic functions as established by Peres, Schramm, Sheffield and Wilson in 2009, followed by the case of p-harmonic functions due to Peres and Sheffield and furthered by Manfredi, Parviainen and Rossi in 2011. We shall see how solutions to these and certain other well-known Pdes can be interpreted as values of a game, in the limit when the distance that the token is allowed to be 'tugged' by players in their preferred directions at each turn of the game, decreases to zero. This observation allows replacing some classical techniques in the study of nonlinear Pdes by relying instead on suitable choices of strategies for the competing players. Such approach has also inspired further studies in different directions, e.g. asymptotic mean value properties, a new proof of Harnack’s inequality for p-harmonic functions, connections with the optimal Lipschitz extension problem, control theory and economic modeling.

Marta Lewicka attended the University of Gdansk (Poland) and the Polytechnics of Czestochowa, receiving an MSc in Mathematics in 1996 and a BSc in Computer Science in 1998. She completed her PhD in 2000 in SISSA (Italy) under the supervision of A. Bressan. Subsequently, she was a postdoc in Max Planck Institute for Mathematical Sciences in Leipzig (Germany) and a Dickson Instructor at the University of Chicago. She has worked as an Assistant and Associate Professor at the University of Minnesota and Rutgers University, and as an Associate Professor at the University of Pittsburgh, where she is currently the Colloquium Chair. She has been the recipient of the McKnight Land Grant Award and the NSF Career Award.