The High-Order Corrections of Discrete Harmonic Measures and Their Correction Constants

Abstract

By the dimension reduction idea, overshoot for random walks, coupling and martingale arguments, we obtain a simpler and easily computable expression for the first-order correction constant between discrete harmonic measures for random walks with rotationally invariant step distribution in \(\mathbb {R}^d\ (d\ge 2)\) and the corresponding continuous counterparts. This confirms and extends a conjecture in Jiang and Kennedy (J Theor Probab 30(4):1424–1444, 2017), and simplifies the related expression of Wang et al. (Bernoulli 25(3):2279–2300, 2019). Furthermore, we propose a universality conjecture on high-order corrections for error estimation between generalized discrete harmonic measures and their continuous counterparts, which generalizes the universality conjecture of the first-order correction in Kennedy (J Stat Phys 164(1):174–189, 2016); and we prove this conjecture heuristically for the rotationally invariant case, and also provide several examples of second-order error corrections to check the conjecture by a numerical simulation argument.