# Nonstandard Martingales, Markov Chains and the Heat Equation

## TL;DR

This paper introduces a nonstandard martingale derived from a discrete Markov chain, providing a novel approach to solving the heat equation with both smooth and non-smooth initial conditions.

## Contribution

It develops a nonstandard martingale framework that connects discrete Markov chains to solutions of the heat equation, extending classical methods.

## Key findings

- Nonstandard martingale construction from Markov chains.
- Solution to heat equation with non-smooth initial conditions.
- Classical solution recovered for smooth initial conditions.

## Abstract

We construct a nonstandard martingale from a discrete Markov chain. This is shown to be useful for solving the heat equation with a non smooth initial condition. We show that the nonstandard solution to the heat equation with a smooth initial condition specialises to the classical solution.