Polyharmonic functions and random processes in cones

Marches aléatoires dans un cône et fonctions discrètes harmoniques

Random walks in cones have the double interest of being at the heart of many probabilistic problems and of being related to many mathematical fields, such as spectral theory, combinatorics, or discrete complex analysis. In this article, we present some key ideas associated with these processes: we will discuss their definition, the link with Brownian motion in cones, as well as some recent research topics such as the construction of discrete harmonic functions.

Martin boundary of random walks in convex cones

We determine the asymptotic behavior of the Green function for zero-drift random walks confined to multidimensional convex cones. As a consequence, we prove that there is a unique positive discrete harmonic function for these processes (up to a multiplicative constant); in other words, the Martin boundary reduces to a singleton.

Mould calculus, polyhedral cones, and characters of combinatorial Hopf algebras

We describe a method for constructing characters of combinatorial Hopf algebras by means of integrals over certain polyhedral cones. This is based on ideas from resurgence theory, in particular on the construction of well-behaved averages induced by diffusion processes on the real line. We give several interpretations and proofs of the main result in terms of noncommutative symmetric and quasisymmetric functions, as well as generalizations involving matrix quasi-symmetric functions. The interpretation of noncommutative symmetric functions as alien operators in resurgence theory is also discuss…

象限中离散热核展开里的对数项

Logarithmic terms in discrete heat kernel expansions in the quadrant

本文提出基于 Jacobi theta 函数与 Jacobi 变换的生成函数方法，计算四分之一平面格点游走计数的完整渐近展开，并首次在无限群模型中发现对数项。