A discrete analog of Kac's formula and optimal approximation of the solution of the heat equation

Abstract

In this article we obtain a closed form solution of the Dirichlet problem of the discretized heat equation with potential. Sharp quantitative estimates of the difference between actual and approximate solutions are given in terms of the first and second moduli of continuity of some first and second order partial derivatives of the exact solution. This is achieved probabilistically by using the appropriate random walk.

Publication Title

Indian Journal of Pure and Applied Mathematics

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