中科院数学与系统科学研究院
数学研究所
学术报告
动力系统研讨班
报告人： Professor Sami Mustapha (Sorbonne Université)
题 目：Potential theory and combinatorics
时 间：2023.10.24（星期二）15:00-16:00
地 点：数学院南楼N205
摘 要：“... But the most general and direct method for resolving questions of probability consists of making them depend on difference equations ...”
The aim of this talk is to illustrate this quote from Laplace (Philosophical essay on probabilities) by developing certain aspects of the theory of discrete potential theory attached to a random walk in $Z^d$; the difference equations playing in this framework the same role played by the PDEs in the theory of the classical potential theory. Although the presentation will be mainly limited to simple walks in quadrants, some extensions to walks in discrete Lipschitzian domains and to inhomogeneous walks will be discussed. The emphasis will be placed on the role that tools from discrete potential theory can play (harmonic functions and discrete caloric functions, maximum principle, Harnack inequalities, boundary Harnack inequalities at the boundary ) in establishing optimal estimates for the number of paths confined to a region as well as the number of excursions.
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