A Discrete Kac’s Formula and Optimal Quantitative Approximation in the Solution of Heat Equation

In this chapter we get 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 presented in terms of the first and second moduli of continuity of some first and second order partial derivatives of the exact solution. This is done probabilistically by using a suitable random walk. This treatment is based on [31].