Open Access
June 2023 Simplest random walk for approximating Robin boundary value problems and ergodic limits of reflected diffusions
Benedict Leimkuhler, Akash Sharma, Michael V. Tretyakov
Author Affiliations +
Ann. Appl. Probab. 33(3): 1904-1960 (June 2023). DOI: 10.1214/22-AAP1856


A simple-to-implement weak-sense numerical method to approximate reflected stochastic differential equations (RSDEs) is proposed and analysed. It is proved that the method has the first order of weak convergence. Together with the Monte Carlo technique, it can be used to numerically solve linear parabolic and elliptic PDEs with Robin boundary condition. One of the key results of this paper is the use of the proposed method for computing ergodic limits, that is, expectations with respect to the invariant law of RSDEs, both inside a domain in Rd and on its boundary. This allows to efficiently sample from distributions with compact support. Both time-averaging and ensemble-averaging estimators are considered and analysed. A number of extensions are considered including a second-order weak approximation, the case of arbitrary oblique direction of reflection, and a new adaptive weak scheme to solve a Poisson PDE with Neumann boundary condition. The presented theoretical results are supported by several numerical experiments.

Funding Statement

BL was supported by EPSRC grant no. EP/P006175/1 and by the Alan Turing Institute (EPSRC EP/N510129/1) as a Turing Fellow. AS was supported by the University of Nottingham Vice-Chancellor’s Scholarship for Research Excellence (International).


BL and MVT thank the Institute for Computational and Experimental Research in Mathematics (ICERM, Providence, RI) where this work was nucleated during a workshop hosted by ICERM. The authors thank anonymous referees for useful suggestions.


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Benedict Leimkuhler. Akash Sharma. Michael V. Tretyakov. "Simplest random walk for approximating Robin boundary value problems and ergodic limits of reflected diffusions." Ann. Appl. Probab. 33 (3) 1904 - 1960, June 2023.


Received: 1 August 2020; Revised: 1 February 2022; Published: June 2023
First available in Project Euclid: 2 May 2023

MathSciNet: MR4583661
zbMATH: 07692308
Digital Object Identifier: 10.1214/22-AAP1856

Primary: 60H35
Secondary: 37H10 , 60H10 , 65C30

Keywords: ergodic limits , Neumann boundary value problem , reflected Brownian dynamics , reflected stochastic differential equations , sampling from distributions with compact support , sampling on manifold , stochastic gradient system in bounded domains , weak approximation

Rights: This research was funded, in whole or in part, by [EPSRC, EP/P006175/1, EPSRC EP/N510129/1]. A CC BY 4.0 license is applied to this article arising from this submission, in accordance with the grant's open access conditions.

Vol.33 • No. 3 • June 2023
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