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.
On the average of probability distributions of a discrete Markov chain
Probabilistic representations of solutions to the heat equation
On a new probabilistic representation for the solution of the heat equation
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