HOẠT ĐỘNG TRONG TUẦN

Harmonic functions for singular quadrant walks
Người trình bày: GS. Kilian Raschel, CNRS France

Thời gian: 14h Thứ 5, ngày 07/11/2024

Địa điểm: Phòng 507 nhà A6

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Tóm tắt: We consider discrete random walks confined to the quarter plane, with jumps only in directions (i, j) with i + j ≥ 0 and small negative jumps. These walks are called singular, and were recently intensively studied from a combinatorial point of view. In this paper, we show how the compensation approach introduced in the 1990s by Adan, Wessels and Zijm may be applied to compute positive harmonic functions with Dirichlet boundary conditions. In particular, in case the random walks have a drift with positive coordinates, we derive an explicit formula for the escape probability, which is the probability to tend to infinity without reaching the boundary axes. These formulas typically involve famous recurrent sequences, such as the Fibonacci numbers. As a second step, we propose a probabilistic interpretation of the previously constructed harmonic functions and prove that they allow us to compute all positive harmonic functions of these singular walks. To that purpose, we derive the asymptotics of the Green functions in all directions of the quarter plane and use Martin boundary theory.

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