The Annals of Applied Probability

Initial-boundary value problem for the heat equation—A stochastic algorithm

Madalina Deaconu and Samuel Herrmann

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The initial-boundary value problem for the heat equation is solved by using an algorithm based on a random walk on heat balls. Even if it represents a sophisticated generalization of the Walk on Spheres (WOS) algorithm introduced to solve the Dirichlet problem for Laplace’s equation, its implementation is rather easy. The construction of this algorithm can be considered as a natural consequence of previous works the authors completed on the hitting time approximation for Bessel processes and Brownian motion [Ann. Appl. Probab. 23 (2013) 2259–2289, Math. Comput. Simulation 135 (2017) 28–38, Bernoulli 23 (2017) 3744–3771]. A similar procedure was introduced previously in the paper [Random Processes for Classical Equations of Mathematical Physics (1989) Kluwer Academic].

The definition of the random walk is based on a particular mean value formula for the heat equation. We present here a probabilistic view of this formula.

The aim of the paper is to prove convergence results for this algorithm and to illustrate them by numerical examples. These examples permit to emphasize the efficiency and accuracy of the algorithm.

Article information

Ann. Appl. Probab., Volume 28, Number 3 (2018), 1943-1976.

Received: October 2016
Revised: June 2017
First available in Project Euclid: 1 June 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K20: Initial-boundary value problems for second-order parabolic equations 65C05: Monte Carlo methods 60G42: Martingales with discrete parameter
Secondary: 60J22: Computational methods in Markov chains [See also 65C40]

Initial-boundary value problem heat equation random walk mean-value formula heat balls Riesz potential submartingale randomized algorithm


Deaconu, Madalina; Herrmann, Samuel. Initial-boundary value problem for the heat equation—A stochastic algorithm. Ann. Appl. Probab. 28 (2018), no. 3, 1943--1976. doi:10.1214/17-AAP1348.

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