The standard model for the diffusion of heat uses the idea that heat
spreads randomly in all directions at some rate. The heat equation
is a deterministic (non-random), partial differential equation derived
from this intuition by averaging over the very large number of par-
ticles. This equation can and has been traditionally studied as a
deterministic equation. While much can be said from this perspec-
tive, one also loses much of the intutition that can be obtained by
considering the individual random particles.
The idea in these notes is to introduce the heat equation and the
closely related notion of harmonic functions from a probabilistic per-
spective. Our starting point is the random walk which in continuous
time and space becomes Brownian motion. We then derive equations
to understand the random walk. This follows the modern approach
where one tries to combine probabilistic and deterministic methods
to analyze diffusion.
Besides the random/deterministic dichotomy, another difference
in approach comes from choosing between discrete and continuous
models. The first chapter of this book starts with discrete random
walk and then uses it to define harmonic functions and the heat equa-
tions on the integer lattice. Here one sees that linear functions arise,
and the deterministic questions yield problems in linear algebra. In
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