A Course on Tug-of-War Games with Random Noise

This graduate textbook provides a detailed introduction to the probabilistic interpretation of nonlinear potential theory, relying on the recently introduced notion of tug-of-war games with noise. The book explores both basic and more advanced constructions, carefully explaining the parallel between linear and nonlinear cases. The presentation is self-contained with many exercises, making the book suitable as a textbook for a graduate course, as well as for self-study. Extensive background and auxiliary material allow the tailoring of courses to individual student levels.

Provides a rigorous, self-contained introduction to the emerging field of tug-of-war games Includes background and auxiliary material on probability and partial differential equations Includes exercises with solutions

Autorentext

Marta Lewicka is a Polish mathematician at the University of Pittsburgh, specializing in Mathematical Analysis. Marta obtained her PhD in 2000 from Scuola Internazionale Superiore di Studi Avanzati with a thesis on Hyperbolic Systems of Conservation Laws. She has subsequently worked on Calculus of Variations, Partial Differential Equations, Continuum Mechanics, and Tug-of-War Games in relation to Nonlinear Potential Theory.

Zusammenfassung

"The book is best suited for advanced graduate students and specialists in nonlinear potential theory. The author provides background appendices for probability, Brownian motion and PDEs." (Bill Satzer, MAA Reviews, April 4, 2021)

Inhalt
1 Introduction.- 2 The linear case: random walk and harmonic functions.- 3 Tug-of-War with noise. Case p [2,).- 4 Boundary aware tug-of-war with noise. Case p (2,).- 5 Game-regularity and convergence. Case p (2,).- 6 Mixed tug-of-war with noise. Case p (1,).- A Background in probability.- B Background in Brownian motion.- C Background in PDEs. D Solutions to selected exercises.- References.- Index.