题目：Potential theory and combinatorics
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.
Sami Mustapha现任法国索邦大学教授，数学学院院长。Sami Mustapha教授是世界著名调和分析专家，在群上的调和分析及随机游走等领域作出了杰出贡献。