Expectation values in the random walk theory and the diffusion equation

TL;DR

This paper explains how expectation values in random walk theory relate to the heat kernel approach for the diffusion equation, supported by simulations and uncertainty analysis.

Contribution

It clarifies the connection between random walk expectation values and the diffusion equation's heat kernel method, including simulation validation.

Findings

Expectation values in random walks align with heat kernel solutions.

Simulations confirm theoretical relations with statistical uncertainty analysis.

Provides a concrete explanation linking stochastic processes and PDE solutions.

Abstract

The relation between the expectation values computed in the random walk theory, and the heat kernel method for the diffusion equation is explained concretely. The random walk is also realized by simulations and their statistical uncertainties are analyzed.